# The pollution effect for the Ginzburg-Landau equation

Théophile Chaumont-Frelet, Patrick Henning

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Théophile Chaumont-Frelet, Patrick Henning. The pollution effect for the Ginzburg-Landau equation. Mathematics of Computation, In press. hal-05104390

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# The pollution effect for the Ginzburg-Landau equation*

Théophile Chaumont-Frelet<fnref n="1" />, Patrick Henning<fnref n="2" />

## Abstract

In this paper, we investigate the approximation properties of solutions to the Ginzburg-Landau equation (GLE) in finite element spaces. Special attention is given to how the errors are influenced by coupling the mesh size $h$ and the polynomial degree $p$ of the finite element space to the size of the so-called Ginzburg-Landau material parameter $\kappa$. As observed in previous works, the finite element approximations to the GLE are suffering from a numerical pollution effect, that is, the best-approximation error in the finite element space converges under mild coupling conditions between $h$ and $\kappa$, whereas the actual finite element solutions possess poor accuracy in a large pre-asymptotic regime which depends on $\kappa$. In this paper, we provide a new error analysis that allows us to quantify the pre-asymptotic regime and the corresponding pollution effect in terms of explicit resolution conditions. In particular, we are able to prove that higher polynomial degrees reduce the pollution effect, i.e., the accuracy of the best-approximation is achieved under relaxed conditions for the mesh size. We provide both error estimates in the $H^1$- and the $L^2$-norm and we illustrate our findings with numerical examples.

## 1 Introduction

The Ginzburg-Landau equation (GLE) is a common model to describe superconductors [23, 25], i.e., materials that allow to conduct electricity without electrical resistance. Mathematically, superconductors can be described by the order parameter $u: \Omega \to \mathbb{C}$ which is a complex function that acts on the body $\Omega \subset \mathbb{R}^d$ (for $d=2,3$) that is occupied by the superconducting material. From $u$, one can extract the density $|u|$ of superconducting charge carriers (so-called Cooper pairs). The density is a real-valued physical observable with $0 \le |u| \le 1$, where $|u| = 0$ represents the normal state (no superconducting charge carriers) and $|u| = 1$ represents the perfectly superconducting state. The order parameter $u$ is given as a solution to the GLE which reads in its basic form, cf. [18],

$$ (\frac{1}{\kappa}\nabla + \mathbf{A})^2 u + (|u|^2 - 1)u = 0 \quad \text{in } \Omega, \tag{1} \\ (\frac{1}{\kappa}\nabla u + \mathbf{A}u) \cdot \mathbf{n} = 0 \quad \text{on } \partial\Omega. $$

Here, $i$ is the complex unit, $\kappa$ is a real-valued positive material parameter and $\mathbf{A}: \Omega \to \mathbb{R}^d$ is a vector potential for the magnetic field $\mathbf{H} = \text{curl} \mathbf{A}$ within the superconductor. Under the Coulomb gauge, the vector potential is selected such that $\text{div} \mathbf{A} = 0$ and $\mathbf{A} \cdot \mathbf{n}|_{\partial\Omega} = 0$. Equation (1) is formally obtained as the first order optimality condition fulfilled by local and global minimizers of the Ginzburg-Landau energy.

The physically most interesting states of a superconductor are its mixed normal-superconducting states where the magnetic field penetrates the superconductor in isolated “points”. These “points” are vortices of a circulating superconducting current. The vortices can form complex structures and pattern known as Abrikosov vortex lattices [1]. The crucial parameter in this context is the material parameter $\kappa$. For fixed $\mathbf{A} \neq 0$ (and hence fixed magnetic field), the

<footnote>*P. Henning acknowledges the support by the Deutsche Forschungsgemeinschaft through the grant HE 2464/7-1.</footnote>

<footnote n="1">¹Inria, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, 59000 Lille, France, e-mail: theophile.chaumont@inria.fr.</footnote>

<footnote n="2">²Department of Mathematics, Ruhr University Bochum, DE-44801 Bochum, Germany, e-mail: patrick.henning@rub.de.</footnote>
number of vortices quickly increases with $\kappa$. At the same time, the diameter of the individual vortices is shrinking, cf. Figure 1 in the numerical experiments for an illustration. Apparently, larger $\kappa$-values require finer computational meshes to resolve the vortices. Hence, $\kappa$ is related to some numerical resolution condition.

When solving the GLE (1) in a finite element space $V_h$ of mesh size $h$ and polynomial degree $p$, we can ask: How do we have to select $h$ and $p$ relative to $\kappa$ to ensure meaningful numerical solutions? The question is easily answered for the best approximation, where we obtain with the stability estimate $\|u\|_{H^{s+1}(\Omega)} \lesssim \kappa^{s+1}$ for all sufficiently smooth solutions and a straightforward calculation that

$$ \inf_{v_h \in V_h} \left( \|u - v_h\|_{L^2(\Omega)} + \frac{1}{\kappa} \|\nabla u - \nabla v_h\|_{L^2(\Omega)} \right) \lesssim (h\kappa)^s, \quad \text{for } s \le p. $$

Hence, accurate numerical approximations require at least $h\kappa \lesssim 1$. However, in previous works [6, 15], numerical experiments as well as analytical results for $\mathbb{P}_1$-FEM indicated that the practical resolution condition is much more severe when the actual finite element solution is considered and that there is a numerical pollution effect similar to what is known for Helmholtz equations with large wave number and related problems [3, 5, 10, 24, 32, 36].

Error estimates on FEM approximations to the full Ginzburg-Landau equations (including an additional curl-curl problem for the vector potential) were first obtained in the seminal work by Du, Gunzburger and Peterson [18]. However, the parameter $\kappa$ was not yet traced in the estimates. The first attempt for $\kappa$-explicit error estimates for the GLE (1) was given in [15]. In particular, it was proved that the finite element error behaves asymptotically as well as the $H^1$-best-approximation error. Later the findings were generalized to discrete multiscale spaces in [6] and [14]. Nevertheless, despite being $\kappa$-explicit in almost all arguments, a full tracing of the $\kappa$-dependency could not be achieved in the aforementioned works. This is due to the proof technique which exploits an absorption argument for higher order terms which in turn requires the mesh size to be “sufficiently small” without being able to quantify how small it actually needs to be. The reason is that the argument is based on abstract compactness results that just ensure convergence but no rates. Hence, a full $\kappa$-tracing is still open. Furthermore, the existing results on $\kappa$-dependencies are restricted to $\mathbb{P}_1$-FEM and it is unclear if potential resolution conditions can be relaxed in higher order FEM spaces.

It is worth to mention that there has been also significant work on FEM approximations for the time-dependent Ginzburg-Landau equation with vector potential, cf. [12, 16, 17, 21, 22, 27, 28, 29, 30], however, there were no attempts to trace the dependency on $\kappa$ and to identify corresponding resolution conditions.

In this work, we will give a significant extension of the previous results for the GLE (1) by (i) providing a fully traced resolution condition under which the finite element error in $\mathbb{P}_p$-FEM spaces behaves like the best-approximation error and (ii) proving that the resolution condition (for achieving the accuracy of the best-approximation) indeed relaxes with increasing polynomial degrees $p$. Furthermore (iii), compared to previous works, we obtain improved $L^2$-error estimates and our analysis is no longer restricted to global minimizers.

To achieve this, we follow a new proof strategy that combines different techniques. On the one hand, we exploit that the GLE originates from an energy minimization problem, i.e. $u$ is a local minimizer of a (non-convex) energy functional $E(v)$. By the second order optimality condition, $E''(u)$ has a bounded inverse on some suitable subspace of $H^1(\Omega)$ and we can define the Galerkin-projection $R_h$ associated with $E''(u)$ on the finite element space. We then split the error between an exact minimizer $u$ and a FEM minimizer $u_h$ into $u - u_h = (u - R_h(u)) + (R_h(u) - u_h)$, investigating both error contributions individually.

For the first part, even though the operator $E''(u)$ was also used in previous works [15, 6]
the corresponding Galerkin-projection was avoided due to, at first glance, unfavourable $\kappa$-dependent stability bounds entering through $E''(u)$. We circumvent these issues by an abstract splitting technique proposed in [10] which allows us to decompose functions $E''(u)^{-1}g$ into a into a regular oscillatory contribution and a low-regularity contribution that behaves nicely for increasing $\kappa$-values. With this, we will be able to bound $u - R_h(u)$ with optimal order estimates under suitable resolution conditions.

The second component of the error also requires dedicated new arguments. To estimate $R_h(u) - u_h$, we establish an asymptotic superconvergence result with which we can bound $R_h(u) - u_h$ by $u - u_h$ up to higher order contributions. The higher order terms are controlled by a Banach fixed-point argument which allows us to identify a resolution condition under which $R_h(u) - u_h$ becomes so small that it can be absorbed into $u - u_h$ and ultimately yielding $\|u - u_h\|_{H_k^1(\Omega)} \lesssim \|u - R_h(u)\|_{H_k^1(\Omega)}$ in a suitable $\kappa$-weighted $H^1$-norm.

*Overview.* In Section 2 we introduce the basic notation and the precise mathematical setting of the Ginzburg-Landau equation. Furthermore, we revisit some important results on the existence, local uniqueness, and regularity of corresponding solutions. In Section 3 we define the finite element approximations and we present our main result on quasi-optimality under a suitable resolution condition for the mesh size that relaxes with higher order finite element spaces. The proof of the main result is split into two major parts covering Section 4 and Section 5. Section 4 deals with the aforementioned splitting of solutions involving the operator $E''(u)$ and Section 5 covers the core error estimates. Numerical experiments are presented in Section 6. In Appendix A, we also present the construction and approximation properties of an Oswald-type quasi-interpolation operator adapted to curved meshes that is used as a tool in our error analysis.

# 2 Analytical setting

## 2.1 Sobolev spaces

Throughout this manuscript, $\Omega \subset \mathbb{R}^d$ (for $d = 2, 3$) denotes a fixed domain with a Lipschitz boundary $\partial\Omega$. We assume for simplicity that the domain has a unit diameter, which can always be achieved by rescaling.

For $2 \le q \le +\infty$, $L^q(\Omega)$ is the usual Lebesgue space of (complex-valued) integrable functions, and we denote by $\|\cdot\|_{L^q(\Omega)}$ its usual norm. For $q = 2$, we equip $L^2(\Omega)$ with the real inner product

$$ (v, w)_{L^2(\Omega)} := \operatorname{Re} \left( \int_{\Omega} v \bar{w} \, dx \right) \quad \text{for } v, w \in L^2(\Omega), $$

where $\bar{w}$ is the complex conjugate of $w$. This makes $L^2(\Omega)$ formally an $\mathbb{R}$-Hilbert space (consisting of complex-valued functions). We also employ the same notation for vector fields, in which case a dot product is used in the inner product.

For any integer $k \ge 1$, we employ the notation $W^{k,q}(\Omega)$ for the usual Sobolev spaces of functions belonging to $L^q(\Omega)$ together with all their partial derivatives of order less than or equal to $k$. Classically, $|\cdot|_{W^{k,q}(\Omega)}$ is the semi-norm of $W^{k,q}(\Omega)$ collecting the $L^q(\Omega)$ norms of the partial derivatives of order $k$. We also employ the shorthand notation $H^1(\Omega) := W^{1,2}(\Omega)$ for the first-order Hilbert Sobolev space, which we equip with the real inner product given by $(v, w)_{L^2(\Omega)} + (\nabla v, \nabla w)_{L^2(\Omega)}$ for $v, w \in H^1(\Omega)$.

Together with the standard Sobolev norm

$$ \|v\|_{W^{k,q}(\Omega)} := \sum_{\ell=0}^{k} |v|_{W^{\ell,q}(\Omega)}, $$
we will use the weighted version

$$ \|v\|_{W_{\kappa}^{k,q}(\Omega)} := \sum_{\ell=0}^{k} \kappa^{-\ell} |v|_{W^{\ell,q}(\Omega)}. $$

Finally, $W^{k,q}(\Omega, \mathbb{R}^d)$ denotes the space of real-valued vector fields with every component belonging to $W^{k,q}(\Omega)$.

## 2.2 PDE coefficients

Throughout the manuscript we consider a vector field $A \in L^{\infty}(\Omega, \mathbb{R}^d)$ representing a magnetic potential. We are free to choose a Gauge condition, since only the curl of $A$ has physical relevance, and we therefore assume without loss of generality that $\text{div} A = 0$ in $\Omega$ and $A \cdot n = 0$ on $\partial\Omega$. This is known as the Coulomb gauge, cf. [18]. We will sometimes require more regularity on $A$, and specifically, we will demand that $A \in W^{\infty,p^{-1}}(\Omega, \mathbb{R}^d)$.

The material parameter is constant and represented by the real number $\kappa$. Since we are especially interested in the high $\kappa$ regime, we will assume without loss of generality that $\kappa \ge 1$.

**Remark 2.1** (Piecewise smooth coefficient). In principle, one could consider a piecewise smooth coefficient $A$ without major modifications. For the sake of simplicity, we avoid doing so here, but refer the reader to [10] for details. Similarly, (piecewise) smooth material coefficients $\kappa$ could be considered.

## 2.3 Hidden constant

Throughout this manuscript, we fix a polynomial degree $p \ge 1$, a constant $M > 0$ measuring the shape-regularity of the finite element mesh, and an arbitrarily small parameter $0 < \varepsilon \le 1$. We will write $a \lesssim b$ if there is a constant $C > 0$ solely depending on $\Omega, A, p, M$ and $\varepsilon$ such that $a \le Cb$. We also use the notation $a \gtrsim b$ if $a \ge Cb$. The hidden constant depends on $A$ only through $\|A\|_{W_{\kappa}^{p^{-1},\infty}(\Omega)}$, and only through $\|A\|_{L^{\infty}(\Omega)}$ in the results where we do not assume that $A$ is smooth. We note in particular that such a constant $C$ can be bounded independently of $\kappa$ and the mesh size $h$. Physically, the dependency on $A$ suggests that the amplitude of the magnetic potential is moderate, but $A$ is allowed to oscillate with frequency at most $\kappa$.

At a limited number of places, if $\alpha, \beta, \gamma, \dots$ are previously introduced symbols, we will use the notation $a \lesssim_{\alpha,\beta,\gamma,\dots} b$ to indicate that the constant $C$ may also depend on $\alpha, \beta, \gamma, \dots$.

## 2.4 Functional inequalities

In this section, we collect some useful inequalities linked with Sobolev norms.

We are first interested by the regularity of products of functions. The following estimates are easily obtained by combining the product rule for derivatives together with the Hölder inequality. For all $m \ge 0$ and $s, t \in [2, +\infty]$, we have

$$ \|uv\|_{W_{\kappa}^{m,q}(\Omega)} \lesssim_{m,s,t} \|u\|_{W_{\kappa}^{m,s}(\Omega)} \|v\|_{W_{\kappa}^{m,t}(\Omega)} \quad (2a) $$

for all $u \in W^{m,s}(\Omega)$, $v \in W^{m,t}(\Omega)$ and $1/q = 1/s + 1/t$. Applying this inequality twice, for all $q \ge 1$, we can further show that

$$ \|uvw\|_{W_{\kappa}^{m,q}(\Omega)} \lesssim_{m,q} \|u\|_{W_{\kappa}^{m,3q}(\Omega)} \|v\|_{W_{\kappa}^{m,3q}(\Omega)} \|w\|_{W_{\kappa}^{m,3q}(\Omega)} \quad (2b) $$
for all $u, v, w \in W^{m,3q}(\Omega)$ and

$$ \|uv\|_{W_{\kappa}^{m,q}(\Omega)} \lesssim_{m,q} \|u\|_{W_{\kappa}^{m,\infty}(\Omega)} \|v\|_{W_{\kappa}^{m,\infty}(\Omega)} \|w\|_{W_{\kappa}^{m,q}(\Omega)}. \quad (2c) $$

whenever $u, v \in W^{m,\infty}(\Omega)$ and $w \in W^{m,q}(\Omega)$.

We then turn to Sobolev inequalities for a single function. By combining the fractional Sobolev embedding with a Banach space interpolation inequality (see e.g. [34, Theorem 8.2], [39, Lemma 36.1] and [31, Lemma B.1]), we have

$$ \|\phi\|_{L^{\infty}(\Omega)} \lesssim_q \|\phi\|_{L^q(\Omega)}^{1-s} \|\phi\|_{W^{1,q}(\Omega)}^s \lesssim_q \kappa^s \|\phi\|_{W_{\kappa}^{1,q}(\Omega)} \quad (3) $$

for all $\phi \in W^{1,q}(\Omega)$ whenever $0 < s < 1$ and $sq > d$. Similarly, we have the Gagliardo-Nirenberg inequalities

$$ \|\phi\|_{L^4(\Omega)} \lesssim \|\phi\|_{L^2(\Omega)}^{1-d/4} \|\nabla\phi\|_{L^2(\Omega)}^{d/4} \lesssim \kappa^{d/4} \|\phi\|_{H_{\kappa}^1(\Omega)} \quad (4) $$

for all $\phi \in H^1(\Omega)$, and

$$ \|\nabla\phi\|_{L^{2q}(\Omega)} \lesssim \|\phi\|_{L^{\infty}(\Omega)}^{1/2} \|\phi\|_{W^{2,q}(\Omega)}^{1/2} \lesssim \kappa \|\phi\|_{L^{\infty}(\Omega)}^{1/2} \|\phi\|_{W_{\kappa}^{2,q}(\Omega)}^{1/2}. \quad (5) $$

for all $q \ge 1$ and $\phi \in W^{2,q}(\Omega)$, see e.g. [7].

## 2.5 Minimization problem

The Ginzburg-Landau energy functional $E: H^1(\Omega) \to \mathbb{R}$ is defined by

$$ E(v) = \frac{1}{2} \int_{\Omega} \left| \frac{i}{\kappa} \nabla v + \mathbf{A} v \right|^2 + \frac{1}{2} (|v|^2 - 1)^2 \, dx \quad (6) $$

and we seek the corresponding lowest energy states of the system, i.e., minimizers $u \in H^1(\Omega)$ with

$$ E(u) = \min_{v \in H^1(\Omega)} E(v). \quad (7) $$

The existence of such global minimizers and their fundamental properties were first established in [18]. In this work, we shall not only consider global minimizers, but also isolated local minimizers of $E$ (in a sense made precise below) and we will show that in a small neighborhood of any such isolated local minimizer of $E$ there is a corresponding discrete local minimizer in the finite element space, provided that the mesh is sufficiently fine. As a first step, we formulate the conditions fulfilled by minimizers and properly introduce what *isolation* means in this context.

## 2.6 First and second order conditions for local and global minimizers

If $u$ is a local minimizer of the energy functional $E$, it is classical that the first order condition for minimizers implies $E'(u) = 0$ and the second order condition implies a non-negative spectrum for $E''(u)$. Here, $E'$ and $E''$ denote the first and second Fréchet derivatives of $E$ which can be computed as

$$ \langle E'(v), w \rangle = \left( \frac{i}{\kappa} \nabla v + \mathbf{A} v, \frac{i}{\kappa} \nabla w + \mathbf{A} w \right)_{L^2(\Omega)} + \left( (|v|^2 - 1)v, w \right)_{L^2(\Omega)} \quad (8) $$

and

$$ \langle E''(v)z, w \rangle = \left( \frac{i}{\kappa} \nabla z + \mathbf{A} z, \frac{i}{\kappa} \nabla w + \mathbf{A} w \right)_{L^2(\Omega)} + \left( (|v|^2 - 1)z + v^2 \bar{z} + |v|^2 z, w \right)_{L^2(\Omega)} \quad (9) $$
for $v, w, z \in H^1(\Omega)$.

Exploiting the formula for $E'$, the first order condition $E'(u) = 0$ is nothing but the Ginzburg-Landau equation in variational form, i.e., $u \in H^1(\Omega)$ solves

$$ (\frac{i}{\kappa}\nabla u + \mathbf{A}u, \frac{i}{\kappa}\nabla v + \mathbf{A}v)_{L^2(\Omega)} + (|u|^2 - 1)u, v)_{L^2(\Omega)} = 0 \quad \text{for all } v \in H^1(\Omega). $$

With the Gauge condition $\operatorname{div} \mathbf{A} = 0$ and $\mathbf{A} \cdot \mathbf{n}|_{\partial\Omega} = 0$, the Ginzburg-Landau equation can be written in strong form as

$$ -\frac{1}{\kappa^2}\Delta u + \frac{2i}{\kappa}\mathbf{A} \cdot \nabla u + |\mathbf{A}|^2 u + (|u|^2 - 1)u = 0 \quad (10) $$

accompanied by the homogenous Neumann boundary condition

$$ \nabla u \cdot \mathbf{n} = 0 \quad \text{on } \partial\Omega. $$

Turning now to the necessary second order condition for minimizers, we recall that $E''(u)$ must have a nonnegative spectrum. One might wonder if the spectrum is strictly positive, so that $E''(u)$ has a bounded inverse. However, this is unfortunately not the case and zero is in fact an eigenvalue of $E''(u)$. This is caused by the invariance of the energy under complex phase shifts or, in other words, the missing local uniqueness of minimizers. This is easily seen by considering an arbitrary minimizer $u$ and the corresponding class of gauge-transformed states $\exp(\omega i)u$ for any $\omega \in [-\pi, \pi)$. A simple calculation shows $E(\exp(\omega i)u) = E(u)$ for all $\omega$ and hence, $\exp(\omega i)u$ is another minimizer. Practically, this is not an issue because all these minimizers share the same density $|u| = |\exp(\omega i)u|$ and can therefore be considered equivalent. Nevertheless, this requires special attention in the error analysis.

One of the direct implications is the aforementioned singularity of $E''(u)$. In fact, if we move from $u$ along the circle line parametrized by $\gamma: \omega \mapsto \exp(\omega i)u$, then the energy is constant along that line and the corresponding directional derivatives must vanish. In $u$ (i.e. $\omega = 0$), the corresponding tangential direction is $\gamma'(0) = iu$ and we therefore expect $E''(u)iu = 0$. Indeed, using (9), a direct calculation shows

$$ \begin{aligned} \langle E''(u)iu, v \rangle &= \left(\frac{i}{\kappa}\nabla(iu) + \mathbf{A}(iu), \frac{i}{\kappa}\nabla v + \mathbf{A}v\right)_{L^2(\Omega)} + (|u|^2 - 1)(iu) + u^2 \overline{iu} + |u|^2iu, v)_{L^2(\Omega)} \\ &\stackrel{(8)}{=} \langle E'(iu), v \rangle + 2(\underbrace{\operatorname{Re}(\overline{u}iu)}_{=0}u, v)_{L^2(\Omega)} = 0 \end{aligned} $$

for any $v \in H^1(\Omega)$. A realistic sufficient second order condition for local minimizers must therefore exclude the orthogonal complement of the kernel, i.e., the $L^2$-orthogonal complement of $iu$ in $H^1(\Omega)$ given by

$$ (iu)^{\perp} := \{v \in H^1(\Omega) \mid (iu, v)_{L^2(\Omega)} = 0\}. $$

Note that $u \in (iu)^{\perp}$ since $(\cdot, \cdot)_{L^2(\Omega)}$ only includes real parts. In summary, we can hence work with the following sufficient second order condition as suggested in previous works (cf. [15, 6]):

(A1) The state $u \in H^1(\Omega)$ is a local minimizer of $E$ that fulfils the necessary first order condition $E'(u) = 0$ and the sufficient second order condition

$$ \langle E''(u)v, v \rangle > 0 \quad \text{for all } v \in (iu)^{\perp} \setminus \{0\}. $$

Condition (A1) implies that zero is a simple eigenvalue of $E''(u)$ and that the remaining spectrum is positive. Hence, such a local minimizer is isolated from any other local minimizer of $E$. Exploiting this, it can be proved that $E''(u)$ induces a coercive operator:
**Lemma 2.2.** If (A1) holds true, then the constant

$$ \rho(\kappa) := \max_{v \in (iu)^\perp \setminus \{0\}} \frac{\|v\|_{H_\kappa^1(\Omega)}^2}{\langle E''(u)v, v \rangle} \gtrsim 1 \quad (11) $$

is finite, and only depends on $u$ and $\kappa$.

A proof that $\rho(\kappa)$ is finite can be found in [6, Proposition 2.2]. Note that the explicit dependency of $\rho(\kappa)$ on $\kappa$ is unknown, but numerical experiments suggest a polynomial growth $\kappa^\alpha$ for some $\alpha > 0$ (with typically $\alpha = \mathcal{O}(1)$, cf. [15, 6]). The fact that $\rho(\kappa) \gtrsim 1$ is a direct consequence of the uniform continuity of $E''(u)$, see bound (23) in Lemma 4.1 below.

As we will see, the operator $E''(u)$ plays a key role in the error analysis which we will revisit in detail in Section 4.

**Remark 2.3.** We stress that for any $v \in H^1(\Omega)$, the condition $v \in (iu)^\perp$ simply amounts to choosing a particular convention for the phase of $v$. Indeed, if $w \in H^1(\Omega)$ is arbitrary, then for $\alpha \in \mathbb{C}$

$$ (\alpha w, iu)_{L^2(\Omega)} = \operatorname{Re} \int_{\Omega} \alpha v \bar{i} u = \operatorname{Re} (-i\alpha) \int_{\Omega} v \bar{u} $$

and we can always choose $\alpha$ with $|\alpha| = 1$ so that $\alpha w \in (iu)^\perp$. In particular, requiring that $u_h \in (iu)^\perp$ for a finite element solution $u_h$ is simply a convenient choice of phase convention to easily provide error estimates. In practice, this condition does not need to be fulfilled, and the solution obtained is accurate, up to a possible phase shift.

## 2.7 $\kappa$-dependent stability estimates

Next, we shall establish $\kappa$-dependent stability estimates for any solution to the GLE $E'(u) = 0$. The following $\kappa$-dependent $H^1$-bounds are established for global minimizers in [18, 14, 15]. Here, we show that such bounds are actually valid for any critical point. The proof only needs slight modifications and is reported here for completeness.

**Theorem 2.4.** Assume that $u \in H^1(\Omega)$ is a critical point, i.e. $E'(u) = 0$. Then, we have

$$ |u(x)| \le 1 \quad \text{a.e. in } \Omega \quad \text{and} \quad \frac{1}{\kappa} \|\nabla u\|_{L^2(\Omega)} \lesssim \|u\|_{L^2(\Omega)} \lesssim 1. \quad (12) $$

*Proof.* The bound $0 \le |u| \le 1$ for any critical point with $E'(u) = 0$ can be proved analogously to the corresponding result in [18, Proposition 3.11]. For the $H^1$-bounds we slightly deviate from [14] where the proof exploits $E(u) \le E(0)$, which is only guaranteed for minimizers. Instead, we directly use the Ginzburg-Landau equation. From $E'(u) = 0$ we obtain with (8) that

$$ \| \frac{i}{\kappa} \nabla u + \mathbf{A} u \|_{L^2(\Omega)}^2 = ((1 - |u|^2)u, u)_{L^2(\Omega)} \le \|u\|_{L^2(\Omega)}^2. $$

Hence, we have the remaining two estimates as

$$ \frac{1}{\kappa} \| \nabla u \|_{L^2(\Omega)} \le \| u \|_{L^2(\Omega)} + \| \mathbf{A} u \|_{L^2(\Omega)} \le (1 + \| \mathbf{A} \|_{L^\infty(\Omega)}) \| u \|_{L^2(\Omega)} \le (1 + \| \mathbf{A} \|_{L^\infty(\Omega)}) \sqrt{|\Omega|}. \quad \square $$

By demanding more smoothness on $\mathbf{A}$ and $\Omega$, we can establish additional bounds on critical points. Similar estimates were obtained previously in [14], but they are extended here to higher order Sobolev spaces.
**Theorem 2.5.** Assume that $\mathbf{A} \in W^{p-1,\infty}(\Omega)$ and that $\partial\Omega$ is of class $C^{p+1}$. Then, for all $u \in H^1(\Omega)$ such that $E'(u) = 0$, we have $u \in W^{p+1,4}(\Omega)$ with

$$ \|u\|_{W_{\kappa}^{p+1,4}(\Omega)} \lesssim 1 \quad \text{and} \quad \|u\|_{W_{\kappa}^{p-1,\infty}(\Omega)} \lesssim \kappa^{\varepsilon/(2p)}. \qquad (13) $$

In addition, the estimate

$$ \|u\|_{H_{\kappa}^{p+1}(\Omega)} \lesssim \kappa^{\varepsilon/2} \|u\|_{L^2(\Omega)} $$

holds true.

*Proof.* We start by noting that $u$ solves the PDE

$$ -\Delta u = -2i\kappa \mathbf{A} \cdot \nabla u - \kappa^2 |\mathbf{A}|^2 u - |u|^2 u + u =: F $$

in $\Omega$, together with the Neumann boundary condition $\nabla u \cdot \mathbf{n} = 0$ on $\partial\Omega$. The remainder of the proof will therefore largely hinge on the regularity shift

$$ \|u\|_{W^{\ell+1,q}(\Omega)} \lesssim_q \|F\|_{W^{\ell-1,q}(\Omega)} + \|u\|_{L^q(\Omega)} \qquad (14) $$

which holds for all $q \in [1, +\infty)$ and $1 \le \ell \le p$ provided that the right-hand side is finite. Such regularity shift indeed does hold true due to our assumption on $\partial\Omega$, see [2, Theorem 15.2] and [8, Theorem on p. 143]. An easy consequence is that we also have

$$ \|u\|_{W_{\kappa}^{\ell+1,q}(\Omega)} \lesssim_{\ell,q} \kappa^{-2} \|F\|_{W_{\kappa}^{\ell-1,q}(\Omega)} + \|u\|_{W_{\kappa}^{1,q}(\Omega)} \qquad (15) $$

uniformly in $\kappa$. We first show that for any $q \in [1, +\infty)$, we have

$$ \|u\|_{W_{\kappa}^{\ell,q}(\Omega)} \lesssim_{\ell,q} 1 \qquad (16) $$

for all $\ell \le p + 1$ by induction on $\ell$.

Let us observe that (16) holds true for $\ell = 0$ due to (12).

For the case $\ell = 1$, we proceed by induction on $q$. We first see that

$$ \|F\|_{L^2(\Omega)} \lesssim \kappa \|\mathbf{A}\|_{L^\infty(\Omega)} \|\nabla u\|_{L^2(\Omega)} + \kappa^2 \|\mathbf{A}\|_{L^\infty(\Omega)}^2 \|u\|_{L^2(\Omega)} + (1 + \|u\|_{L^\infty(\Omega)}^2) \|u\|_{L^2(\Omega)} \lesssim \kappa^2 $$

due to (12). This shows that the result holds true for $q = 2$, and in fact for any $1 \le q \le 2$. Furthermore, thanks to the regularity shift (14), we have $u \in W^{2,2}(\Omega)$ with $\|u\|_{W^{2,2}(\Omega)} \lesssim \kappa^2$. Let us now assume that the result has been established for some $q \ge 2$. Proceeding as above, we have

$$ \|F\|_{L^{2q}(\Omega)} \lesssim \|\nabla u\|_{L^{2q}(\Omega)} + \kappa^2 $$

using the $L^\infty(\Omega)$ bound on $u$. We now apply the Gagliardo-Nirenberg inequality in (5), to show that

$$ \|F\|_{L^{2q}(\Omega)} \lesssim_q \kappa \|u\|_{L^\infty(\Omega)}^{1/2} \|u\|_{W^{2,q}(\Omega)}^{1/2} + \kappa^2 \lesssim_q \kappa^2 $$

using again the $L^\infty(\Omega)$ bound on $u$ and the induction hypothesis in combination with the regularity shift (14). This now shows the result for all exponents less than $2q$, which concludes the proof by induction on $q$ for the case $\ell = 1$.

We now assume that (16) holds true up to some $\ell \le p$ and for any $q \in [1, +\infty)$. We show that it still holds true for $\ell + 1$ and again any $q \in [1, +\infty)$. In view of (15), we simply need to show that

$$ \|F\|_{W_{\kappa}^{\ell-1,q}(\Omega)} \lesssim_{\ell,q} \kappa^2. $$
Using the product rules in (2) and the induction assumption, we have

$$
\begin{align*}
\|F\|_{W_{\kappa}^{\ell-1,q}(\Omega)} & \\
& \lesssim_{\ell,q} \kappa \|\mathbf{A} \cdot \nabla u\|_{W_{\kappa}^{\ell-1,q}(\Omega)} + \kappa^2 \|\mathbf{A}\|^2 \|u\|_{W_{\kappa}^{\ell-1,q}(\Omega)} + \|u\|^2 \|u\|_{W_{\kappa}^{\ell-1,q}(\Omega)} + \|u\|_{W_{\kappa}^{\ell-1,q}(\Omega)} \\
& \lesssim_{\ell,q} \kappa \|\mathbf{A}\|_{W_{\kappa}^{\ell-1,\infty}(\Omega)} \|\nabla u\|_{W_{\kappa}^{\ell-1,q}(\Omega)} + \kappa^2 \|\mathbf{A}\|^2_{W_{\kappa}^{\ell-1,\infty}(\Omega)} \|u\|_{W_{\kappa}^{\ell-1,q}(\Omega)} + \|u\|^3_{W_{\kappa}^{\ell-1,3q}(\Omega)} + \|u\|_{W_{\kappa}^{\ell-1,q}(\Omega)} \\
& \lesssim_{\ell,q} \kappa^2 \|\mathbf{A}\|_{W_{\kappa}^{\ell-1,\infty}(\Omega)} \|u\|_{W_{\kappa}^{\ell,q}(\Omega)} + \kappa^2 \|\mathbf{A}\|^2_{W_{\kappa}^{\ell-1,\infty}(\Omega)} \|u\|_{W_{\kappa}^{\ell-1,q}(\Omega)} + \|u\|^3_{W_{\kappa}^{\ell-1,3q}(\Omega)} + \|u\|_{W_{\kappa}^{\ell-1,q}(\Omega)} \\
& \lesssim_{\ell,q} \kappa^2 + 1 \lesssim_{\ell,q} \kappa^2,
\end{align*}
$$

from which the desired bound on F follows. The regularity shift in (15) finishes the argument.

With (16) established, the first estimate in (13) readily follows by taking $q = 4$. For the second estimate in (13), we simply employ the Sobolev embedding in (3), and write that

$$
\|\phi\|_{L^{\infty}(\Omega)} \lesssim \kappa^{\varepsilon/2p} \|\phi\|_{W_{\kappa}^{1,q\varepsilon}(\Omega)}
$$

for all $\phi \in W^{1,q\epsilon}(\Omega)$ with $q_\epsilon := 2p(d+1)/\epsilon$, $s_\epsilon = \epsilon/(2p)$ so that $s_\epsilon q_\epsilon > d$. A direct consequence is that

$$
\|u\|_{W_{\kappa}^{p-1,\infty}(\Omega)} \lesssim \kappa^{\epsilon/2p} \|u\|_{W_{\kappa}^{p,q\epsilon}(\Omega)} \lesssim \kappa^{\epsilon/2p}
$$

due to (16).

Similarly to what we did above, we will show that

$$
\|u\|_{H_{\kappa}^{\ell+1}(\Omega)} \lesssim_{\ell} \kappa^{\epsilon/(2p)} \|u\|_{L^2(\Omega)}
$$

for $0 \le \ell \le p$ by induction on $\ell$. The result then follows with $\ell = p$. For $\ell = 0$, the result was established in Theorem 2.4. Assuming the result for some $1 \le \ell \le p - 1$, we are going to show that it is still valid for $\ell + 1$. Following the proof of the first two estimates above, it suffices to show that

$$
\|F\|_{H_{\kappa}^{\ell-1}(\Omega)} \lesssim \kappa^2 \kappa^{\ell\epsilon/p} \|u\|_{L^2(\Omega)}.
$$

We easily do so using the product rules in (2) and the induction hypothesis as follows:

$$
\begin{align*}
\|F\|_{H_{\kappa}^{\ell-1}(\Omega)} & \lesssim \kappa \|\mathbf{A} \cdot \nabla u\|_{H_{\kappa}^{\ell-1}(\Omega)} + \kappa^2 \|\mathbf{A}\|^2 \|u\|_{H_{\kappa}^{\ell-1}(\Omega)} + \kappa^2 \|u\|_{H_{\kappa}^{\ell-1}(\Omega)} + \kappa^2 \|u\|^2 \|u\|_{H_{\kappa}^{\ell-1}(\Omega)} \\
& \lesssim \kappa \|\mathbf{A}\|_{W_{\kappa}^{\ell-1,\infty}(\Omega)} \|\nabla u\|_{H_{\kappa}^{\ell-1}(\Omega)} + \kappa^2 \|\mathbf{A}\|^2_{W_{\kappa}^{\ell-1,\infty}(\Omega)} \|u\|_{H_{\kappa}^{\ell-1}(\Omega)} \\
& \phantom{\lesssim \kappa \|\mathbf{A}\|_{W_{\kappa}^{\ell-1,\infty}(\Omega)} \|\nabla u\|_{H_{\kappa}^{\ell-1}(\Omega)} +} + \kappa^2 \|u\|_{H_{\kappa}^{\ell-1}(\Omega)} + \kappa^2 \|u\|^2_{W_{\kappa}^{\ell-1,\infty}(\Omega)} \|u\|_{H_{\kappa}^{\ell-1}(\Omega)} \\
& \stackrel{(13)}{\lesssim} \kappa^2 \|\mathbf{A}\|_{W_{\kappa}^{\ell-1,\infty}(\Omega)} \|u\|_{H_{\kappa}^{\ell}(\Omega)} + \kappa^2 (\|\mathbf{A}\|^2_{W_{\kappa}^{\ell-1,\infty}(\Omega)} + 1) \|u\|_{H_{\kappa}^{\ell-1}(\Omega)} + \kappa^2 \kappa^{\epsilon/p} \|u\|_{H_{\kappa}^{\ell-1}(\Omega)} \\
& \stackrel{(13)}{\lesssim} \kappa^2 \|u\|_{H_{\kappa}^{\ell}(\Omega)} + \kappa^2 \kappa^{\epsilon/p} \|u\|_{H_{\kappa}^{\ell-1}(\Omega)} \lesssim \kappa^2 (\kappa^{(\ell-1)\epsilon/(2p)} + \kappa^{\epsilon/p} \kappa^{(\ell-2)\epsilon/(2p)}) \|u\|_{L^2(\Omega)} \\
& \stackrel{(13)}{\lesssim} \kappa^2 \kappa^{\ell\epsilon/(2p)} \|u\|_{L^2(\Omega)}.
\end{align*}
$$

□

3 Discrete minimizers in finite element spaces and $H_\kappa^1$-optimality

In this section, we describe the finite element discretization of the Ginzburg-Landau minimiza-
tion problem and present our main result regarding optimality of the corresponding discrete
minimizers in the $H_\kappa^1$-norm.
## 3.1 Finite element setting

In the following we consider a shape-regular, conforming partition $\mathcal{T}_h$ of $\Omega$ into (curved) simplices $T \in \mathcal{T}_h$ of (maximum) mesh size $h := \max\{\text{diam}(T) | T \in \mathcal{T}_h\}$. Each mesh element $K \in \mathcal{T}_h$ is obtained from a fixed reference simplex $\hat{K}$ through a bilipschitz mapping $\mathcal{F}_K : \hat{K} \to K$.

The corresponding $\mathbb{P}_p$ Lagrange finite element space is given by

$$V_h := \{v_h \in H^1(\Omega) \mid v_h|_K \circ \mathcal{F}_K \in \mathbb{P}_p(\hat{K}) \text{ for all } K \in \mathcal{T}_h\}, \quad (17)$$

where $\mathbb{P}_p(\hat{K})$ is the set of polynomials of total degree less than or equal to $p$. In the following, we assume that there exists a constant $M > 0$ and a projection operator $I_h : H^1(\Omega) \to V_h$ such that

$$|I_h(v)|_{H^1(\Omega)} \leq M|v|_{H^1(\Omega)} \quad (18a)$$

for all $v \in H^1(\Omega)$, and

$$h^{-1}\|v - I_h(v)\|_{L^q(\Omega)} + |v - I_h(v)|_{W^{1,q}(\Omega)} \leq M h^{\ell} \|v\|_{W^{\ell+1,q}(\Omega)} \quad (18b)$$

for $0 \le \ell \le p$ and $2 \le q \le 4$.

The existence of such an interpolation operator is standard for meshes consisting of straight elements (see e.g. [19]), but has not been addressed for curved elements to the best of our knowledge. In Appendix A below, we show that such condition can indeed be fulfilled with a constant $M$ independent of $h$ under natural regularity assumptions on the element mappings $\mathcal{F}_K$. In turn, such mappings may be constructed under the natural smoothness assumption on the boundary that $\partial\Omega$ is of class $C^{p+1}$, see [4, 11, 26].

For simplicity, we will further assume that $h\kappa \le 1$. This assumption is only made to simplify expressions like $h\kappa + 1 \lesssim 1$. In addition, it is in fact not restrictive, since we will impose stronger conditions on the mesh size later on.

## 3.2 Discrete minimizer

In what follows, we either consider a discrete global minimizer $u_h \in V_h$ with the property

$$E(u_h) = \min_{v_h \in V_h} E(v_h) \quad (19)$$

or a discrete local minimizer with $\langle E'(u_h), v_h \rangle = 0$ for all $v_h \in V_h$. Global discrete minimizers always exist in finite-dimensional spaces $V_h$ (cf. [6, Lemma 3.2]) which directly follows from the non-negativity of $E(v)$ and the inequality $\|v\|_{H^1(\Omega)}^2 \lesssim 1 + E(v)$ for all $v \in H^1(\Omega)$.

However, just like continuous minimizers in $H^1(\Omega)$, $u_h$ cannot be unique and any complex phase shift of $u_h$, i.e. $\exp(i\omega)u_h$ for $\omega \in [-\pi, \pi)$, is a (local) minimizer on the same energy level. This has to be considered when comparing some $u_h$ with an exact local minimizer $u$.

Assuming that $A \in W^{p-1,\infty}(\Omega)$ and that $\partial\Omega$ is $C^{p+1}$ and using the $\kappa$-dependent bounds in Theorems 2.4 and 2.5 together with (18), we can conclude that the $H^1$-best-approximation error for any critical point $u$ of $E$ (and hence also for any local or global minimizer) behaves as

$$\min_{v_h \in V_h} \|u - v_h\|_{H^1(\Omega)} \lesssim h^{\ell+1} \|u\|_{H^{\ell+1}(\Omega)} + \kappa^{-1} h^{\ell} \|u\|_{H^{\ell+1}(\Omega)} \lesssim (1 + h\kappa) (h\kappa)^{\ell}$$

provided $u \in H^{\ell+1}(\Omega)$ and $\ell \le p$. This shows that the mesh resolution condition $h\kappa \le 1$ we imposed above is natural to obtain meaningful numerical approximations. However, as we will see later, there is a numerical pollution effect and the condition $h\kappa \lesssim 1$ is no longer sufficient for $u_h$ to be a reasonable approximation of $u$ in the $L^2$- or $H^1$-norm. On the contrary, the discrete energy level of global minimizers is not affected by the pollution effect. This is seen by the following result.
**Proposition 3.1** (Error in the minimal energy). Assume that $\mathbf{A} \in W^{p-1,\infty}(\Omega)$ and $\partial\Omega$ is $C^{p+1}$. Then it holds

$$0 \le \min_{u_h \in V_h} E(u_h) - \min_{u \in H^1(\Omega)} E(u) \lesssim (h\kappa)^{2p}.$$

*Proof.* Let $u$ and $u_h$ respectively denote any global minimizers in $H^1(\Omega)$ and $V_h$. Since $u_h \in V_h$ is a discrete global minimizer, we have $E(u) \le E(u_h) \le E(I_h(u))$, and hence

$$0 \le E(u_h) - E(u) \le E(I_h(u)) - E(u).$$

Analogously to the proof of [15, Lemma 5.9] it holds

$$E(I_h(u)) - E(u) \lesssim \|u - I_h(u)\|_{H_{\kappa}^1(\Omega)}^2 + \left( \|u - I_h(u)\|_{L^2(\Omega)}^2 + \|u - I_h(u)\|_{L^3(\Omega)}^3 + \|u - I_h(u)\|_{L^4(\Omega)}^4 \right)$$

The interpolation error estimates for $I_h$ in (18) together with the stability bounds for $u$ from Theorem 2.5 finish the proof. $\square$

### 3.3 Main results and $H_{\kappa}^1$-optimality

In this section, we present the main result of this paper, which is formulated in Theorem 3.2 below. Essentially, the theorem states that if the polynomial degree $p$ and the mesh size $h$ of the finite element space $V_h$ are selected relative to $\kappa$ with the resolution condition $\kappa^{d/2}\rho(\kappa)(h\kappa)^p \ll 1$ for a local minimizer $u \in H^1(\Omega)$ satisfying (A1), then there is a discrete (local) minimizer $u_h \in V_h$, such that the error $\|u - u_h\|_{H_{\kappa}^1(\Omega)}$ converges with optimal order $O((h\kappa)^p)$ to zero. Similarly, a corresponding $L^2$-error estimate holds.

**Theorem 3.2.** Assume that $\mathbf{A} \in W^{p-1,\infty}(\Omega)$ and $\partial\Omega$ is $C^{p+1}$ and consider a (local) minimizer $u \in H^1(\Omega)$ of $E$ satisfying (A1). Then, there exists a constant $K_* \gtrsim 1$ such that if

$$\kappa^{d/2} \rho(\kappa) (h\kappa)^p \le K_*, \quad (20)$$

then there exists a discrete (local) minimizer $u_h \in V_h$ of $E$ with

$$\begin{array}{l@{\qquad}l}
\langle E'(u_h), v_h \rangle = 0 & \text{for all } v_h \in V_h \quad \text{and} \\
\langle E''(u_h)v_h, v_h \rangle \gtrsim \rho(\kappa)^{-1} \|v_h\|_{H_{\kappa}^1(\Omega)}^2 & \text{for all } v_h \in V_h \cap (iu)^{\perp}
\end{array}$$

such that

$$\|u - u_h\|_{H_{\kappa}^1(\Omega)} \lesssim (h\kappa)^p \quad (21)$$

and

$$\|u - u_h\|_{L^2(\Omega)} \lesssim \kappa^{\varepsilon} (h\kappa)^{p+1} + \kappa^{d/2} \rho(\kappa) (h\kappa)^{2p}. \quad (22)$$

The proof of the theorem takes place in several steps. The essential preparations are introduced in Section 4 and the main proofs follow in Section 5.

Before we start with the proofs, let us briefly discuss the main findings of Theorem 3.2. First of all, the theorem predicts a numerical pollution effect (later observed in our numerical experiments), where the mesh size $h$ needs to fulfill the stronger resolution condition $\kappa^{d/2}\rho(\kappa)(h\kappa)^p \lesssim 1$ before the numerical approximations $u_h$ start to converge with the expected rate of $O((h\kappa)^p)$ in the $H_{\kappa}^1$-norm to the correct minimizer $u$. Since $\rho(\kappa)$ is a priori unknown,
the precise necessary resolution for $h$ is also a priori unknown. However, the theorem also shows that whatever the growth of $\rho(\kappa)$ w.r.t. $\kappa$, the resolution condition improves with the polynomial degree $p$ of the finite element space $V_h$. For example, for $d=2$ and $p=1$, the resolution condition becomes $h\rho(\kappa)\kappa^2 \lesssim 1$, whereas it reduces significantly to $h\rho(\kappa)^{1/2}\kappa^{3/2} \lesssim 1$ for $p=2$. Hence, higher order finite element spaces can effectively reduce the pollution effect for the Ginzburg-Landau equation. It remains open if the factor $\kappa^{d/2}$ in (20) is sharp or if it can be removed with different proof techniques.

Finally, Theorem 3.2 also establishes an $L^2$-error estimate of optimal order in $h$. However, the estimate appears still suboptimal for $p=1$ with respect to its $\kappa$-dependency. In fact, for $p=1$, (22) essentially reduces to

$$ \|u - u_h\|_{L^2(\Omega)} \lesssim (1 + \kappa^{d/2}\rho(\kappa))(h\kappa)^2, $$

where $\rho(\kappa)$ appears as a multiplicative factor. For larger $p$, the term $\kappa^{d/2}\rho(\kappa)(h\kappa)^{2p}$ is of higher order and the scaling with $\rho(\kappa)$ vanishes asymptotically in the same way as for the $H_\kappa^1$-estimate. To be precise, for $\kappa^{d/2}\rho(\kappa)(h\kappa)^{p-1} \lesssim 1$, estimate (22) becomes asymptotically optimal with

$$ \|u - u_h\|_{L^2(\Omega)} \lesssim (h\kappa)^{p+1}. $$

The remaining part of the paper is devoted to the proof of Theorem 3.2. Furthermore, we will present corresponding numerical experiments to illustrate the pollution effect and how it is reduced in higher order FE spaces.

# 4 The operator $E''(u)$ and the shift property

As an essential preparation for the error analysis, this section is concerned with various auxiliary results regarding the properties of the operator $E''(u)$ and its inverse. For that, recall from Lemma 2.2 that $E''(u)$ is coercive on $(iu)^\perp$, however, with a coercivity constant $\rho(\kappa)^{-1}$ that scales unfavourably with $\kappa$. For that reason, we will mainly exploit a Gårding inequality in our analysis, where the arising constants do no longer involve $\kappa$. In the following, we collect all the necessary auxiliary results. We start with the uniform continuity of $E''(u)$ in the $H_\kappa^1$-norm and the aforementioned Gårding inequality.

**Lemma 4.1.** Consider a critical point $u \in H^1(\Omega)$ with $E'(u) = 0$. Then $E''(u)$ is continuous on $H^1(\Omega)$ w.r.t. $\| \cdot \|_{H_\kappa^1(\Omega)}$ with a continuity constant independent of $\kappa$, i.e.,

$$ \langle E''(u)v, w \rangle \lesssim \|v\|_{H_\kappa^1(\Omega)} \|w\|_{H_\kappa^1(\Omega)} \quad \text{for all } v, w \in H^1(\Omega) \quad (23) $$

and, for all $v \in H^1(\Omega)$, we have the Gårding inequality

$$ \langle E''(w)v, v \rangle \geq \frac{1}{2} \|v\|_{H_\kappa^1(\Omega)}^2 - \left( \frac{3}{2} + \|\mathbf{A}\|_{L^\infty(\Omega)}^2 \right) \|v\|_{L^2(\Omega)}^2. $$

*Proof.* The proof of the continuity is straightforward using $|u| \le 1$ a.e. and the Gårding inequality follows with $|\frac{1}{\kappa}\nabla v + \mathbf{A}v|^2 \ge \frac{1}{2\kappa^2}|\nabla v|^2 - |\mathbf{A}v|^2$ as

$$ \begin{align*} \langle E''(w)v, v \rangle & \stackrel{(9)}{=} \int_{\Omega} |\frac{1}{\kappa}\nabla v + \mathbf{A}v|^2 + (2|w|^2-1)|v|^2 + \text{Re}(w^2\bar{v}^2) \\ & \geq \int_{\Omega} \frac{1}{2\kappa^2}|\nabla v|^2 - |\mathbf{A}v|^2 + (|w|^2-1)|v|^2 \\ & \geq \int_{\Omega} \frac{1}{2\kappa^2}|\nabla v|^2 + \frac{1}{2}|v|^2 + (|w|^2 - |\mathbf{A}|^2 - \frac{3}{2})|v|^2. \end{align*} $$

□
Assuming that $u$ is a local minimizer satisfying (A1), the Lax-Milgram lemma ensures that for each $\psi \in ((iu)^{\perp})^{*}$, there exists a unique $E''(u)|_{(iu)^{\perp}}^{-1} \psi \in (iu)^{\perp}$ such that

$$ \langle E''(u)(E''(u)|_{(iu)^{\perp}}^{-1} \psi), v \rangle = \langle \psi, v \rangle $$

for all $v \in (iu)^{\perp}$. Let us recall a well-posedness and regularity result formulated in [15, Corollary 2.9] which essentially follows from the coercivity in Lemma 2.2.

**Proposition 4.2.** Consider a local minimizer $u \in H^1(\Omega)$ of $E$ satisfying (A1). Then, for each $f \in L^2(\Omega)$, it holds

$$ \|E''(u)|_{(iu)^{\perp}}^{-1} f\|_{H_{\kappa}^{1}(\Omega)} \lesssim \rho(\kappa) \|f\|_{L^{2}(\Omega)}. \quad (24) $$

Furthermore, if $\partial\Omega$ is $C^2$ (or if $\Omega$ is convex), then we have $z \in H^2(\Omega)$ and it holds

$$ \|E''(u)|_{(iu)^{\perp}}^{-1} f\|_{H_{\kappa}^{2}(\Omega)} \lesssim \rho(\kappa) \|f\|_{L^{2}(\Omega)}. \quad (25) $$

Following [10], the main goal of this section is to refine the above result by deriving an expansion of $z$ in terms of $\kappa$ of the form

$$ z = \sum_{j=0}^{\ell-1} \kappa^j z_j + r_{\ell}, $$

where the regularity of $z_j$ increases with $j$, that is $z_j \in H^{j+2}(\Omega)$ with $|z_j|_{H^{j+2}(\Omega)} \lesssim \kappa^{j+2} \|f\|_{L^2(\Omega)}$ independent of $\rho(\kappa)$, and where $r_{\ell} \in H^{\ell+2}(\Omega)$ is a rest term fulfilling $|r_{\ell}|_{H^{\ell+2}(\Omega)} \lesssim \rho(\kappa)\kappa^{\ell+2} \|f\|_{L^2(\Omega)}$. In a first step towards such an expansion, we split $E''(u)$ according to its powers in $\kappa$, where we obtain the following result.

**Lemma 4.3.** For $u \in H^1(\Omega)$, we have

$$ \langle E''(u)v, w \rangle = \frac{1}{\kappa^2} (\nabla v, \nabla w)_{L^2(\Omega)} + \frac{1}{\kappa} (\mathcal{L}_1(\nabla v), w)_{L^2(\Omega)} + (\mathcal{L}_{0,u}(v), w)_{L^2(\Omega)}, $$

for all $v, w \in H^1(\Omega)$, where

$$ \mathcal{L}_1(\nabla v) := 2i \nabla v \cdot \mathbf{A} \quad \text{and} \quad \mathcal{L}_{0,u}(v) := |\mathbf{A}|^2 v + (|u|^2 - 1)v + 2 \operatorname{Re}(\bar{u}v)u. \quad (26) $$

*Proof.* With (9) and $u^2\bar{v} + |u|^2v = 2 \operatorname{Re}(\bar{u}v)u$ it holds

$$ \langle E''(u)v, w \rangle = \left( \frac{i}{\kappa} \nabla v + A v, \frac{i}{\kappa} \nabla w + A w \right)_{L^2(\Omega)} + \left( (|u|^2 - 1)v + 2 \operatorname{Re}(\bar{u}v)u, w \right)_{L^2(\Omega)}. $$

It remains to check the first term, for which we obtain

$$ \begin{aligned} \left(\frac{i}{\kappa}\nabla v + A v, \frac{i}{\kappa}\nabla w + A w\right)_{L^2(\Omega)} &= \operatorname{Re} \int_{\Omega} \left(\frac{i}{\kappa}\nabla v + Av\right) \cdot \overline{\left(\frac{i}{\kappa}\nabla w + Aw\right)} \\ &= \frac{1}{\kappa^2} \operatorname{Re} \int_{\Omega} \nabla v \cdot \overline{\nabla w} - \operatorname{Re} \int_{\Omega} \frac{i}{\kappa}Av \cdot \overline{\nabla w} + \operatorname{Re} \int_{\Omega} \frac{i}{\kappa}\nabla v \cdot A\bar{w} + \operatorname{Re} \int_{\Omega} |A|^2 v\bar{w} \\ &= \frac{1}{\kappa^2} \operatorname{Re} \int_{\Omega} \nabla v \cdot \overline{\nabla w} + 2 \operatorname{Re} \int_{\Omega} \frac{i}{\kappa}\nabla v \cdot A\bar{w} + \operatorname{Re} \int_{\Omega} |A|^2 v\bar{w} \\ &= \operatorname{Re} \int_{\Omega} \left(-\frac{1}{\kappa^2}\Delta v + \frac{2i}{\kappa}\nabla v \cdot A + |A|^2 v\right) \bar{w}, \end{aligned} $$

where we used $A \cdot n|_{\partial\Omega} = 0$ and $\operatorname{div} A = 0$ in the penultimate step. $\square$

<figure><img src="image_2.png" /></figure>
**Proposition 4.4.** Consider a critical point $u \in H^1(\Omega)$ such that $E'(u) = 0$. Then, for $1 \le \ell \le p$, $\tilde{f} \in H^{\ell-1}(\Omega)$ and $\tilde{g} \in H^\ell(\Omega)$ we let $\tilde{u} \in (iu)^\perp$ denote the solution to

$$ (\nabla \tilde{u}, \nabla v)_{L^2(\Omega)} + \kappa^2(\tilde{u}, v)_{L^2(\Omega)} = -\kappa^2(\mathcal{L}_{0,u}(\tilde{f}) - \tilde{f}, v)_{L^2(\Omega)} - \kappa (\mathcal{L}_1(\nabla \tilde{g}), v)_{L^2(\Omega)} \quad (27) $$

for all $v \in (iu)^\perp$ and where $\mathcal{L}_{0,u}$ and $\mathcal{L}_1$ are given by (26). Then, $\tilde{u} \in H^{\ell+1}(\Omega)$ with

$$ \|\tilde{u}\|_{H_{\kappa}^{\ell+1}(\Omega)} \lesssim \kappa^{\varepsilon/p} \left( \|\tilde{f}\|_{H_{\kappa}^{\ell-1}(\Omega)} + \|\tilde{g}\|_{H_{\kappa}^{\ell}(\Omega)} \right). \quad (28) $$

*Proof.* We first establish the following stability estimate

$$ \|\tilde{u}\|_{H_{\kappa}^{1}(\Omega)} \lesssim \|\tilde{f}\|_{L^{2}(\Omega)} + \|\tilde{g}\|_{H_{\kappa}^{1}(\Omega)} \quad (29) $$

by taking $v = \tilde{u}$ in (27). Here, we used $\|\mathcal{L}_{0,u}(\tilde{f})\|_{L^2(\Omega)} \lesssim \|\tilde{f}\|_{L^2(\Omega)}$ due to the estimate $|u| \le 1$.

Our goal is then to apply regularity shifts for the Neumann problem. More precisely, we will use that if $\psi \in H^1(\Omega)$ satisfies $\kappa^2\psi - \Delta\psi \in H^{\ell-1}(\Omega)$ and $\nabla\psi \cdot \mathbf{n} = 0$ on $\partial\Omega$, then

$$ \|\psi\|_{H_{\kappa}^{\ell+1}(\Omega)} \lesssim \kappa^{-2} \|\kappa^2\psi - \Delta\psi\|_{H_{\kappa}^{\ell-1}(\Omega)} \quad (30) $$

for all $1 \le \ell \le p$. The estimate is completely standard without the $\kappa^2\psi$ term in the right-hand side (see e.g. [33, Thrm. IV.2.4]), and the present version is easily obtained by inductive application of the standard shift result.

Let us now verify that $\tilde{u}$ solves such a Neumann problem. For shortness, we introduce

$$ \tilde{F}_{\kappa} := -\kappa^2 \mathcal{L}_{0,u}(\tilde{f}) + \kappa^2 \tilde{f} - \kappa \mathcal{L}_1(\nabla \tilde{g}) \in L^2(\Omega), $$

so that (27) becomes $\kappa^2(\tilde{u}, v) + (\nabla \tilde{u}, \nabla v)_{L^2(\Omega)} = (\tilde{F}_{\kappa}, v)_{L^2(\Omega)}$ for all $v \in (iu)^\perp$. In order to transform this equation to a regular Neumann problem on $H^1(\Omega)$, it is natural to decompose an arbitrary test function $v \in H^1(\Omega)$ uniquely in $v = v^\perp + \alpha_v(iu)$, where $v^\perp \in (iu)^\perp$ is the $L^2$-projection of $v$ onto $(iu)^\perp$ and $\alpha_v := (v, iu)_{L^2(\Omega)} \|u\|_{L^2(\Omega)}^{-2}$ (cf. [15]). This decomposition can be now used in $(\nabla \tilde{u}, \nabla v)_{L^2(\Omega)}$ to obtain

$$ \begin{aligned} (\nabla \tilde{u}, \nabla v)_{L^2(\Omega)} &= (\nabla \tilde{u}, \nabla v^\perp)_{L^2(\Omega)} + \alpha_v (\nabla \tilde{u}, \nabla(iu))_{L^2(\Omega)} \\ &= (-\kappa^2 \tilde{u} + F_\kappa, v^\perp)_{L^2(\Omega)} + \alpha_v (\nabla \tilde{u}, \nabla(iu))_{L^2(\Omega)} \\ &= (F_\kappa - \kappa^2 \tilde{u}, v)_{L^2(\Omega)} - \alpha_v (F_\kappa - \kappa^2 \tilde{u}, iu)_{L^2(\Omega)} + \alpha_v (\nabla \tilde{u}, \nabla(iu))_{L^2(\Omega)} \\ &= \left( F_\kappa - \kappa^2 \tilde{u} - \frac{(F_\kappa, iu)_{L^2(\Omega)}}{\|u\|_{L^2(\Omega)}^2} iu + \frac{(\nabla \tilde{u}, \nabla(iu))_{L^2(\Omega)}}{\|u\|_{L^2(\Omega)}^2} iu, v \right)_{L^2(\Omega)}. \end{aligned} $$

Hence, defining

$$ F_{\kappa} := \tilde{F}_{\kappa} + \alpha iu, $$

where

$$ \alpha := \frac{(\nabla \tilde{u}, \nabla(iu))_{L^2(\Omega)} - (\tilde{F}_{\kappa}, iu)_{L^2(\Omega)}}{\|u\|_{L^2(\Omega)}^2}, $$

we verified that $\tilde{u} \in (iu)^\perp \subset H^1(\Omega)$ solves

$$ \kappa^2(\tilde{u}, v)_{L^2(\Omega)} + (\nabla \tilde{u}, \nabla v)_{L^2(\Omega)} = (F_{\kappa}, v)_{L^2(\Omega)} \quad \text{for all } v \in H^1(\Omega), \quad (31) $$

and it remains to estimate the Sobolev norms of $F_\kappa$.
On the one hand, using (12) and (29), we have

$$|\alpha| \le \frac{\|\nabla \tilde{u}\|_{L^2(\Omega)} \|\nabla u\|_{L^2(\Omega)} + \|\tilde{F}_\kappa\|_{L^2(\Omega)} \|u\|_{L^2(\Omega)}}{\|u\|_{L^2(\Omega)}^2} \le \frac{\kappa^2}{\|u\|_{L^2(\Omega)}} (\|f\|_{L^2(\Omega)} + \|g\|_{H_\kappa^1(\Omega)}).$$

As result, using again (12), it follows that

$$\|\alpha i u\|_{H_\kappa^1(\Omega)} \lesssim \kappa^2 \frac{\|u\|_{H_\kappa^1(\Omega)}}{\|u\|_{L^2(\Omega)}} (\|f\|_{L^2(\Omega)} + \|g\|_{H_\kappa^1(\Omega)}) \lesssim \kappa^2 (\|f\|_{L^2(\Omega)} + \|g\|_{H_\kappa^1(\Omega)}).$$

On the other hand we obtain using the product rules from (2) that

$$\begin{align*}
\|F_\kappa\|_{H^{\ell-1}(\Omega)} &= \kappa^2 \|\mathcal{L}_{0,u}(\tilde{f}) - \tilde{f} + \kappa^{-1} \mathcal{L}_1(\nabla \tilde{g})\|_{H^{\ell-1}(\Omega)} \\
&\lesssim \kappa^2 \left( (1 + \|u\|_{W_\kappa^{\infty,\ell-1}(\Omega)}^2) \|\tilde{f}\|_{H_\kappa^{\ell-1}(\Omega)} + \|\tilde{g}\|_{H_\kappa^\ell(\Omega)} \right) \tag{13} \\
&\lesssim \kappa^2 \kappa^{\varepsilon/p} \left( \|\tilde{f}\|_{H_\kappa^{\ell-1}(\Omega)} + \|\tilde{g}\|_{H_\kappa^\ell(\Omega)} \right),
\end{align*}$$

which concludes the proof due to (30). □

We are now ready to prove a splitting of $z$ into low-regularity contributions $z_j$ that do not depend on $\rho(\kappa)$ and a high-regularity remainder $r_\ell$ that scales with $\rho(\kappa)$. Here, we closely follow the procedure introduced in [10].

**Theorem 4.5.** Assume that $A \in W^{p-1,\infty}(\Omega)$ and $\partial\Omega$ is $C^{p+1}$, and consider a local minimizer $u \in H^1(\Omega)$ of $E$ satisfying (A1). Then, for all $f \in L^2(\Omega)$, we have

$$E''(u)|_{(iu)^\perp}^{-1} f = \sum_{j=0}^{p-2} z_j + r_{p-1},$$

where, for $j = 0, \dots, p-2$ we define recursively $z_j \in (iu)^\perp$ by

$$(\nabla z_0, \nabla v)_{L^2(\Omega)} + \kappa^2 (z_0, v)_{L^2(\Omega)} = \kappa^2 (f, v)_{L^2(\Omega)} \quad \text{for all } v \in (iu)^\perp$$

and

$$(\nabla z_1, \nabla v)_{L^2(\Omega)} + \kappa^2 (z_1, v)_{L^2(\Omega)} = -\kappa (\mathcal{L}_1(\nabla z_0), v)_{L^2(\Omega)} \quad \text{for all } v \in (iu)^\perp$$

and

$$(\nabla z_j, \nabla v)_{L^2(\Omega)} + \kappa^2 (z_j, v)_{L^2(\Omega)} = -\kappa^2 (\mathcal{L}_{0,u}(z_{j-2}) - z_{j-2}, v)_{L^2(\Omega)} - \kappa (\mathcal{L}_1(\nabla z_{j-1}), v)_{L^2(\Omega)}$$

for all $v \in (iu)^\perp$. Then we have $z_j \in H^{j+2}(\Omega)$ and

$$\|z_j\|_{H_{\kappa}^{j+2}(\Omega)} \lesssim \kappa^{\varepsilon/p} \|f\|_{L^2(\Omega)}$$

and with

$$r_{p-1} = z - \sum_{j=0}^{p-2} z_j, \quad \text{or equivalently } r_{p-1} = r_{p-2} - z_{p-2}$$

we have $r_{p-1} \in H^{p+1}(\Omega)$ and

$$\|r_{p-1}\|_{H_{\kappa}^{p+1}(\Omega)} \lesssim \kappa^{\varepsilon} \rho(\kappa) \|f\|_{L^2(\Omega)}.$$
Proof. First we note that $z_j \in H^{j+2}(\Omega)$ and the corresponding regularity estimate follows by a recursive application of Proposition 4.4. Applying Lemma 4.3 to $z := E''(u)|_{(iu)^\perp} f$, i.e. the solution $z \in (iu)^\perp$ to $\langle E''(u)z, v \rangle = (f, v)_{L^2(\Omega)}$ for all $v \in (iu)^\perp$, yields

$$ (\nabla z, \nabla v)_{L^2(\Omega)} + \kappa^2(z, v)_{L^2(\Omega)} = -\kappa(\mathcal{L}_1(\nabla z), v)_{L^2(\Omega)} - \kappa^2(\mathcal{L}_{0,u}(z), v)_{L^2(\Omega)} + \kappa^2(f + z, v)_{L^2(\Omega)}. $$

To prove the desired result for arbitrary $\ell \in \{1, \dots, p-1\}$, we plug $r_\ell = z - \sum_{j=0}^{\ell-1} z_j$ into the above characterization of $z$. For $\ell = 1$, i.e. $r_1 = z - z_0$, we obtain

$$ \begin{aligned} (\nabla r_1, \nabla v)_{L^2(\Omega)} + \kappa^2(r_1, v)_{L^2(\Omega)} &= (\nabla(z-z_0), \nabla v)_{L^2(\Omega)} + \kappa^2(z-z_0, v)_{L^2(\Omega)} \\ &= -\kappa^2(\mathcal{L}_{0,u}(z) - z, v)_{L^2(\Omega)} - \kappa(\mathcal{L}_1(\nabla z), v)_{L^2(\Omega)}. \end{aligned} $$

For $r_2 = r_1 - z_1$ we obtain

$$ \begin{aligned} (\nabla r_2, \nabla v)_{L^2(\Omega)} + \kappa^2(r_2, v)_{L^2(\Omega)} &= (\nabla r_1 - \nabla z_1, \nabla v)_{L^2(\Omega)} + \kappa^2(r_1 - z_1, v)_{L^2(\Omega)} \\ &= -\kappa(\mathcal{L}_1(\nabla z), v)_{L^2(\Omega)} - \kappa^2(\mathcal{L}_{0,u}(z), v)_{L^2(\Omega)} + \kappa^2(z, v)_{L^2(\Omega)} + \kappa(\mathcal{L}_1(\nabla z_0), v)_{L^2(\Omega)} \\ &= -\kappa(\mathcal{L}_1(\nabla r_1), v)_{L^2(\Omega)} - \kappa^2(\mathcal{L}_{0,u}(z) - z, v)_{L^2(\Omega)}. \end{aligned} $$

For $r_\ell = r_{\ell-1} - z_{\ell-1}$ we obtain the equation inductively: Let $r_0 := z$ and let, for $\ell \ge 3$ hold

$$ (\nabla r_{\ell-1}, \nabla v)_{L^2(\Omega)} + \kappa^2(r_{\ell-1}, v)_{L^2(\Omega)} = -\kappa(\mathcal{L}_1(\nabla r_{\ell-2}), v)_{L^2(\Omega)} - \kappa^2(\mathcal{L}_{0,u}(r_{\ell-3}) - r_{\ell-3}, v)_{L^2(\Omega)}. $$

Then we obtain

$$ \begin{aligned} (\nabla r_\ell, \nabla v)_{L^2(\Omega)} + \kappa^2(r_\ell, v)_{L^2(\Omega)} &= (\nabla r_{\ell-1} - \nabla z_{\ell-1}, \nabla v)_{L^2(\Omega)} + \kappa^2(r_{\ell-1} - z_{\ell-1}, v)_{L^2(\Omega)} \\ &= -\kappa(\mathcal{L}_1(\nabla r_{\ell-2}), v)_{L^2(\Omega)} - \kappa^2(\mathcal{L}_{0,u}(r_{\ell-3}) - r_{\ell-3}, v)_{L^2(\Omega)} \\ &\quad + \kappa^2(\mathcal{L}_{0,u}(z_{\ell-3}) - z_{\ell-3}, v)_{L^2(\Omega)} + \kappa(\mathcal{L}_1(\nabla z_{\ell-2}), v)_{L^2(\Omega)} \\ &= -\kappa(\mathcal{L}_1(\nabla r_{\ell-2} - \nabla z_{\ell-2}), v)_{L^2(\Omega)} - \kappa^2(\mathcal{L}_{0,u}(r_{\ell-3} - z_{\ell-3}) - (r_{\ell-3} - z_{\ell-3}), v)_{L^2(\Omega)} \\ &= -\kappa(\mathcal{L}_1(\nabla r_{\ell-1}), v)_{L^2(\Omega)} - \kappa^2(\mathcal{L}_{0,u}(r_{\ell-2} - r_{\ell-2}, v)_{L^2(\Omega)}. \end{aligned} $$

From the equation for $r_1$ we have

$$ \|r_1\|_{H_{\kappa}^2(\Omega)} \lesssim \|z\|_{H_{\kappa}^1(\Omega)} \stackrel{(24)}{\lesssim} \rho(\kappa) \|f\|_{L^2(\Omega)} $$

and by applying Proposition 4.4 to the equation for $r_1$, we also obtain

$$ \|r_1\|_{H_{\kappa}^3(\Omega)} \stackrel{(28)}{\lesssim} \kappa^{\varepsilon/p} \|z\|_{H_{\kappa}^2(\Omega)} \stackrel{(25)}{\lesssim} \kappa^{\varepsilon/p} \rho(\kappa) \|f\|_{L^2(\Omega)}. $$

For $r_2$ we obtain analogously with Proposition 4.4

$$ \|r_2\|_{H_{\kappa}^4(\Omega)} \stackrel{(28)}{\lesssim} \kappa^{\varepsilon/p} (\|z\|_{H_{\kappa}^2(\Omega)} + \|r_1\|_{H_{\kappa}^3(\Omega)}) \stackrel{(28)}{\lesssim} \kappa^{\varepsilon/p} (1 + \kappa^{\varepsilon/p}) \rho(\kappa) \|f\|_{L^2(\Omega)}. $$

Repeating the argument recursively at most $p-1$ times and using $(\kappa^{\varepsilon/p})^{p-1} \le \kappa^\varepsilon$ proves

$$ \|r_\ell\|_{H_{\kappa}^{\ell+2}(\Omega)} \lesssim \kappa^\varepsilon \rho(\kappa) \|f\|_{L^2(\Omega)}. $$

The result is obtained for $\ell = p-1$. $\square$
# 5 Proofs of the $L^2$ and $H_\kappa^1$-error estimates

With the considerations in Section 4 regarding the shift-property of the operator $E''(u)$, we are now prepared to turn towards the error estimates for the minimizers. This takes place in three steps. In the first step (Section 5.2), we will introduce the Ritz projection $R_h$ based on $E''(u)$ and quantify the corresponding projection error, as well as the error between $R_h(u)$ and a discrete minimizer $u_h$. In the second step (Section 5.3), we will derive uniform $\kappa$-bounds for the distance between an exact local minimizer $u$ and the closest discrete local minimizer $u_h$. These uniform bounds are suboptimal, but will be sufficient to get control over higher order error contributions that enter through the nonlinearity of the Ginzburg-Landau equation. Finally, in step three (Section 5.4), we combine all our findings to derive the desired error estimates between $u$ and $u_h$ in the $L^2$- and $H_\kappa^1$-norm.

## 5.1 Approximation factor

We are now ready to quantify the worst-case best-approximation error $\eta_h$ associated with solutions to problems involving the invertible operator $E''(u)|_{(iu)^\perp}$. Specifically, we define the worst-case best-approximation error by

$$ \eta_h := \sup_{\substack{f \in L^2(\Omega) \\ \|f\|_{L^2(\Omega)} = 1}} \min_{\substack{v_h \in V_h \cap (iu)^\perp \\ \|E''(u)|_{(iu)^\perp}^{-1} f - v_h\|_{H_\kappa^1(\Omega)} = 1}} \|E''(u)|_{(iu)^\perp}^{-1} f - v_h\|_{H_\kappa^1(\Omega)}. \quad (32) $$

**Theorem 5.1.** Let $u \in H^1(\Omega)$ denote a local minimizer of $E$ such that (A1) is fulfilled. Then,

$$ \lim_{h \to 0} \eta_h = 0. $$

In addition, if $A \in W^{p-1,\infty}(\Omega)$ and $\partial\Omega$ is $C^{p+1}$, then it holds that

$$ \eta_h \lesssim \kappa^\varepsilon (h\kappa + \rho(\kappa)(h\kappa)^p). \quad (33) $$

*Proof.* The fact that $\eta_h \to 0$ as $h \to 0$ without further regularity assumption is a simple consequence of the fact that the injection $E''(u)|_{(iu)^\perp}^{-1}(L^2(\Omega)) \hookrightarrow H^1(\Omega)$ is compact. Indeed $E''(u)|_{(iu)^\perp}^{-1}$ is continuous from $(H^1(\Omega))' \to H^1(\Omega)$ and the injection $L^2(\Omega) \hookrightarrow (H^1(\Omega))'$ is compact.

We then focus on (33). For $f \in L^2(\Omega)$, we consider $z := E''(u)|_{(iu)^\perp}^{-1} f \in (iu)^\perp$. To exploit Theorem 4.5 we let $z_j \in (iu)^\perp$ denote the solutions to

$$ \begin{align*} (\nabla z_0, \nabla v)_{L^2(\Omega)} + \kappa^2 (z_0, v)_{L^2(\Omega)} &= \kappa^2 (f, v)_{L^2(\Omega)} \\ (\nabla z_1, \nabla v)_{L^2(\Omega)} + \kappa^2 (z_1, v)_{L^2(\Omega)} &= -\kappa (\mathcal{L}_1(\nabla z_0), v)_{L^2(\Omega)} \\ (\nabla z_j, \nabla v)_{L^2(\Omega)} + \kappa^2 (z_j, v)_{L^2(\Omega)} &= -\kappa^2 (\mathcal{L}_{0,u}(z_{j-2}) - z_{j-2}, v)_{L^2(\Omega)} - \kappa (\mathcal{L}_1(\nabla z_{j-1}), v)_{L^2(\Omega)} \end{align*} $$

for all $v \in (iu)^\perp$. Setting

$$ r_{p-1} := z - \sum_{j=0}^{p-2} z_j, $$

Theorem 4.5 yields $z_j \in H^{j+2}(\Omega)$ and $r_{p-1} \in H^{p+1}(\Omega)$ with

$$ \|z_j\|_{H_{\kappa}^{j+2}(\Omega)} \lesssim \kappa^{\varepsilon/p} \|f\|_{L^2(\Omega)} \lesssim \kappa^{\varepsilon} \|f\|_{L^2(\Omega)} \quad \text{and} \quad \|r_{p-1}\|_{H_{\kappa}^{p+1}(\Omega)} \lesssim \kappa^{\varepsilon} \rho(\kappa) \|f\|_{L^2(\Omega)}. $$
We now construct an interpolation operator $I_h^{\perp} : (iu)^{\perp} \to V_h \cap (iu)^{\perp}$ by setting

$$I_h^{\perp}(v) := I_h(v) - \frac{(I_h(v), iu)_{L^2(\Omega)}}{(I_h(iu), iu)_{L^2(\Omega)}} I_h(iu)$$

for all $v \in (iu)^{\perp}$. Since $h\kappa \le 1$ by assumption, for every $v \in (iu)^{\perp}$, we have

$$\|v - I_h^{\perp}(v)\|_{H_{\kappa}^{1}(\Omega)} \lesssim \|v - I_h(v)\|_{H_{\kappa}^{1}(\Omega)}.$$

The argument is e.g. elaborated in [6, Proof Lem. 4.6] and exploits $\frac{1}{\kappa}\|\nabla u\|_{L^2(\Omega)} \lesssim \|u\|_{L^2(\Omega)}$. With this, we obtain

$$
\begin{align*}
\inf_{v_h \in V_h \cap (iu)^{\perp}} \|z - v_h\|_{H_{\kappa}^{1}(\Omega)} &\le \|z - I_h^{\perp}(z)\|_{H_{\kappa}^{1}(\Omega)} \lesssim \|z - I_h(z)\|_{H_{\kappa}^{1}(\Omega)} \\
&= \|r_{p-1} + \sum_{j=0}^{p-2} z_j - I_h(r_{p-1}) - \sum_{j=0}^{p-2} I_h(z_j)\|_{H_{\kappa}^{1}(\Omega)} \\
&\le \|r_{p-1} - I_h(r_{p-1})\|_{H_{\kappa}^{1}(\Omega)} + \sum_{j=0}^{p-2} \|z_j - I_h(z_j)\|_{H_{\kappa}^{1}(\Omega)} \\
&\lesssim h^p \kappa^p \|r_{p-1}\|_{H_{\kappa}^{p+1}(\Omega)} + \sum_{j=0}^{p-2} h^{j+1} \kappa^{j+1} \|z_j\|_{H_{\kappa}^{j+2}(\Omega)} \\
&\lesssim (h\kappa)^p \kappa^{\varepsilon} \rho(\kappa) \|f\|_{L^2(\Omega)} + \sum_{j=0}^{p-2} h^{j+1} \kappa^{j+1} \kappa^{\varepsilon} \|f\|_{L^2(\Omega)} \\
&\lesssim \kappa^{\varepsilon} (h\kappa + (h\kappa)^p \rho(\kappa)) \|f\|_{L^2(\Omega)}.
\end{align*}
$$

This finishes the proof. $\square$

## 5.2 Estimates for the $E''(u)$-Ritz projection

The Ritz projection associated with the elliptic operator $E''(u)|_{(iu)^{\perp}}$ will play a central role in what follows. For any $v \in (iu)^{\perp}$, we define $R_h(v) \in V_h \cap (iu)^{\perp}$ by requiring that

$$\langle E''(u)R_h(v), v_h \rangle = \langle E''(u)v, v_h \rangle \quad \text{for all } v_h \in V_h \cap (iu)^{\perp}. \qquad (34)$$

Note that $R_h$ is well-defined by the coercivity of $E''(u)$ on $(iu)^{\perp}$. The remainder of this section is dedicated to corresponding error estimates that are important for our analysis.

We start by showing an error estimate for the Ritz projection $R_h$. Since this projection corresponds to a coercive bilinear form with a “small” coercivity constant, naive arguments lead to error estimates with unfavorable $\kappa$ dependence. Since on the other hand, this bilinear form satisfies a Gårding inequality with favorable constants, we can employ the standard “Schatz argument” [38] to obtain sharp estimate for sufficiently fine meshes.

**Lemma 5.2.** Consider a local minimizer $u \in H^1(\Omega)$ of $E$ satisfying (A1). Recall $\eta_h$ from (32). There exists $\eta_* \ge 1$ (i.e. independent of $\kappa$ and $h$) such that if $\eta_h \le \eta_*$ then, for any $v \in (iu)^{\perp}$, it holds

$$\|v - R_h(v)\|_{H_{\kappa}^{1}(\Omega)} \lesssim \inf_{v_h \in V_h} \|v - v_h\|_{H_{\kappa}^{1}(\Omega)}.$$

and

$$\|v - R_h(v)\|_{L^2(\Omega)} \lesssim \eta_h \|v - R_h(v)\|_{H_{\kappa}^{1}(\Omega)}.$$
Proof. First, the Gårding inequality in Lemma 4.1 yields

$$ \|v - R_h(v)\|_{H_{\kappa}^1(\Omega)}^2 \lesssim \langle E''(u)(v - R_h(v)), v - R_h(v) \rangle + \|v - R_h(v)\|_{L^2(\Omega)}^2. $$

If $\xi \in (iu)^{\perp}$ solves

$$ \langle E''(u)\xi, \phi \rangle = (v - R_h(v), \phi)_{L^2(\Omega)} \quad \text{for all } \phi \in (iu)^{\perp}, $$

then we obtain for arbitrary $\xi_h, v_h \in V_h \cap (iu)^{\perp}$ that

$$ \begin{aligned} \|R_h(v) - v\|_{H_{\kappa}^1(\Omega)}^2 &\lesssim \langle E''(u)(R_h(v) - v + \xi), R_h(v) - v \rangle \\ &= \langle E''(u)(v_h - v), R_h(v) - v \rangle + \langle E''(u)(\xi - \xi_h), R_h(v) - v \rangle \\ &\lesssim (\|v_h - v\|_{H_{\kappa}^1(\Omega)} + \|\xi - \xi_h\|_{H_{\kappa}^1(\Omega)}) \|R_h(v) - v\|_{H_{\kappa}^1(\Omega)}. \end{aligned} $$

The definition of $\eta_h$ in (32) yields

$$ \|R_h(v) - v\|_{H_{\kappa}^1(\Omega)} \lesssim \inf_{v_h \in V_h \cap (iu)^{\perp}} \|v_h - v\|_{H_{\kappa}^1(\Omega)} + \eta_h \|R_h(v) - v\|_{L^2(\Omega)}. $$

Since $\|R_h(v) - v\|_{L^2(\Omega)} \le \|R_h(v) - v\|_{H_{\kappa}^1(\Omega)}$ and since (analogously as in the proof of Theorem 5.1)

$$ \inf_{v_h \in V_h \cap (iu)^{\perp}} \|v_h - v\|_{H_{\kappa}^1(\Omega)} \lesssim \inf_{v_h \in V_h} \|v_h - v\|_{H_{\kappa}^1(\Omega)}, $$

we obtain

$$ \left(1 - \frac{1}{2} \frac{\eta_h}{\eta_*}\right) \|R_h(v) - v\|_{H_{\kappa}^1(\Omega)} \lesssim \inf_{v_h \in V_h} \|v_h - v\|_{H_{\kappa}^1(\Omega)} $$

for some generic $\eta_* \ge 1$. For the $L^2$-estimates we write

$$ \begin{aligned} \|R_h(v) - v\|_{L^2(\Omega)}^2 &= \langle E''(u)(\xi), R_h(v) - v \rangle \\ &= \inf_{\xi_h \in V_h \cap (iu)^{\perp}} \langle E''(u)(\xi - \xi_h), R_h(v) - v \rangle \lesssim \eta_h \|R_h(v) - v\|_{L^2(\Omega)} \|R_h(v) - v\|_{H_{\kappa}^1(\Omega)}. \end{aligned} $$

□

Next, we establish an estimate for the error between a discrete minimizer $u_h$ and the Ritz projection $R_h(u)$ of an exact minimizer $u$.

**Lemma 5.3.** Assume that $u \in H^1(\Omega)$ is a (local) minimizer of E satisfying (A1) and let $u_h \in V_h \cap (iu)^{\perp}$ be an arbitrary discrete critical point such that $\langle E'(u_h), v_h \rangle = 0$ for all $v_h \in V_h$. It holds

$$ \begin{aligned} \|R_h(u) - u_h\|_{H_{\kappa}^1(\Omega)} \\ \lesssim \eta_h \|R_h(u) - u_h\|_{L^2(\Omega)} + \rho(\kappa) \left( \|u - u_h\|_{L^4(\Omega)}^2 + \kappa^{d/4} \|u - u_h\|_{L^4(\Omega)}^3 \right). \end{aligned} $$

Proof. We start as in the proof of Lemma 5.2 to estimate $R_h(u) - u_h$. Lemma 4.1 yields

$$ \|R_h(u) - u_h\|_{H_{\kappa}^1(\Omega)}^2 \lesssim \langle E''(u)(R_h(u) - u_h), R_h(u) - u_h \rangle + \|R_h(u) - u_h\|_{L^2(\Omega)}^2 $$

and accordingly we let $\xi \in (iu)^{\perp}$ denote the unique solution to

$$ \langle E''(u)\xi, v \rangle = (R_h(u) - u_h, v)_{L^2(\Omega)} \quad \text{for all } v \in (iu)^{\perp}. $$
Recalling that $\langle E''(u) \cdot, \cdot \rangle$ is symmetric, we conclude for arbitrary $\xi_h \in V_h \cap (iu)^\perp$ that

$$
\begin{align*}
\|R_h(u) - u_h\|_{H_{\kappa}^1(\Omega)}^2 &\lesssim \langle E''(u)(R_h(u) - u_h + \xi), R_h(u) - u_h \rangle \\
&= \langle E''(u)\xi, R_h(u) - u_h \rangle + \langle E''(u)(R_h(u) - u_h), R_h(u) - u_h \rangle \\
&= \underbrace{\langle E''(u)(\xi - \xi_h), R_h(u) - u_h \rangle}_{=: I} + \underbrace{\langle E''(u)(R_h(u) - u_h + \xi_h), u - u_h \rangle}_{=: II}.
\end{align*}
$$

We now estimate the two terms separately. For that, let us select $\xi_h \in V_h \cap (iu)^\perp$ through the $H_\kappa^1$-orthogonal projection, i.e.,

$$
\|\xi - \xi_h\|_{H_{\kappa}^{1}(\Omega)} = \inf_{v_h \in V_h \cap (iu)^{\perp}} \|\xi - v_h\|_{H_{\kappa}^{1}(\Omega)}.
$$

Term I. The definition of $\eta_h$ in (32) yields directly

$$
\begin{align*}
|\langle E''(u)(\xi - \xi_h), R_h(u) - u_h \rangle| &\lesssim \|\xi - \xi_h\|_{H_\kappa^1(\Omega)} \|R_h(u) - u_h\|_{H_\kappa^1(\Omega)} \\
&\le \eta_h \|R_h(u) - u_h\|_{H_\kappa^1(\Omega)} \|R_h(u) - u_h\|_{L^2(\Omega)}.
\end{align*}
$$

Term II. We start with rewriting

$$
\langle E''(u)v_h, (u - u_h) \rangle = \langle E''(u)(u - u_h), v_h \rangle
$$

for arbitrary $v_h \in V_h \cap (iu)^\perp$. Using the continuous and the discrete Ginzburg-Landau equations
we have

$$
\langle E'(u), v_h \rangle = \langle E'(u_h), v_h \rangle = 0 \quad \text{for all } v_h \in V_h.
$$

We therefore obtain from (8) and (9) (cf. [15, Lem. 5.5])

$$
\begin{align*}
\langle E''(u)(u - u_h), v_h \rangle &= \langle E''(u)u - E''(u_h)u_h, v_h \rangle + \langle (E''(u_h) - E''(u))u_h, v_h \rangle \\
&= 2(|u|^2u, v_h)_{L^2(\Omega)} + (|u_h|^2u_h, v_h)_{L^2(\Omega)} - (2|u|^2u_h + u^2\overline{u_h}, v_h)_{L^2(\Omega)} \\
&= (2u|u - u_h|^2 + (u - u_h)^2\bar{u} - |u - u_h|^2(u - u_h), v_h)_{L^2(\Omega)}.
\end{align*}
$$

We conclude with $v_h = R_h(u) - u_h + \xi_h \in V_h \cap (iu)^\perp$ that

$$
\begin{align*}
|\langle E''(u)(u - u_h), R_h(u) - u_h + \xi_h \rangle| \\
&\lesssim \|u - u_h\|_{L^4(\Omega)}^2 \|R_h(u) - u_h + \xi_h\|_{L^2(\Omega)} + \|u - u_h\|_{L^4(\Omega)}^3 \|R_h(u) - u_h + \xi_h\|_{L^4(\Omega)}.
\end{align*}
$$

In order to estimate $\|R_h(u) - u_h + \xi_h\|_{L^4(\Omega)}$ we exploit the Gagliardo–Nirenberg inequality for $d=2,3$, which yields for any $v \in H^1(\Omega)$ that

$$
\|v\|_{L^4(\Omega)} \lesssim \|\nabla v\|_{L^2(\Omega)}^{d/4} \|v\|_{L^2(\Omega)}^{1-d/4} \quad \text{and hence} \quad \|v\|_{L^4(\Omega)}^2 \lesssim \kappa^{d/2} \|v\|_{H_{\kappa}^1(\Omega)}^2.
$$

We conclude

$$
\begin{align*}
|\langle E''(u)(u - u_h), R_h(u) - u_h + \xi_h \rangle| & \\
&\lesssim (\|u - u_h\|_{L^4(\Omega)}^2 + \kappa^{d/4} \|u - u_h\|_{L^4(\Omega)}^3) \|R_h(u) - u_h + \xi_h\|_{H_{\kappa}^1(\Omega)} \\
&\lesssim (\|u - u_h\|_{L^4(\Omega)}^2 + \kappa^{d/4} \|u - u_h\|_{L^4(\Omega)}^3) (\|R_h(u) - u_h\|_{H_{\kappa}^1(\Omega)} + \|\xi\|_{H_{\kappa}^1(\Omega)}) \\
&\lesssim (\|u - u_h\|_{L^4(\Omega)}^2 + \kappa^{d/4} \|u - u_h\|_{L^4(\Omega)}^3) (1 + \rho(\kappa)) \|R_h(u) - u_h\|_{H_{\kappa}^1(\Omega)}.
\end{align*}
$$

The estimate $\rho(\kappa) \gtrsim 1$ as per (11), finishes the proof.

<figure><img src="image_3.png" /></figure>
In order to replace one of the $L^4$-error contributions in Lemma 5.3 by a contribution in the $H_\kappa^1$-norm, we can use again the Gagliardo-Nirenberg interpolation inequality. Note that it is important to only replace one $L^4$-contribution by an $H_\kappa^1$-term and to keep the other ones. Together with some $L^4$-bounds that follow later in Theorem 5.8, this will allow us to remove some artificial $\kappa$-dependencies which would otherwise remain. With this in mind, we formulate the following super-convergence result.

**Conclusion 5.4** (Asymptotic superconvergence). Assume that $u \in H^1(\Omega)$ is a (local) minimizer of $E$ and $u_h \in V_h \cap (iu)^\perp$ is an arbitrary discrete critical point such that $\langle E'(u_h), v_h \rangle = 0$ for all $v_h \in V_h$. Then there exists $\eta_\star \gtrsim 1$ such that if $\eta_h \le \eta_\star$ we have

$$ \|R_h(u) - u_h\|_{H_\kappa^1(\Omega)} \lesssim \rho(\kappa) \left( \kappa^{d/4} \|u - u_h\|_{L^4(\Omega)} + \kappa^{d/2} \|u - u_h\|_{L^4(\Omega)}^2 \right) \|u - u_h\|_{H_\kappa^1(\Omega)}. $$

*Proof.* By Lemma 5.3 we have

$$ \begin{aligned} & \|R_h(u) - u_h\|_{H_\kappa^1(\Omega)} \\ & \le \frac{1}{2\eta_\star} \left( \eta_h \|R_h(u) - u_h\|_{L^2(\Omega)} + \rho(\kappa) \left( \|u - u_h\|_{L^4(\Omega)}^2 + \kappa^{d/4} \|u - u_h\|_{L^4(\Omega)}^3 \right) \right) \end{aligned} $$

for some $\eta_\star \gtrsim 1$. Due to our assumption on $\eta_h$, we can absorb the $\|R_h(u) - u_h\|_{L^2(\Omega)}$-contribution and the estimate reduces to

$$ \|R_h(u) - u_h\|_{H_\kappa^1(\Omega)} \lesssim \rho(\kappa) \left( \|u - u_h\|_{L^4(\Omega)}^2 + \kappa^{d/4} \|u - u_h\|_{L^4(\Omega)}^3 \right). \quad (35) $$

Using again the Gagliardo-Nirenberg inequality $\|v\|_{L^4(\Omega)} \lesssim \kappa^{d/4} \|v\|_{H_\kappa^1(\Omega)}$ on one of the $L^4$-terms, the $H_\kappa^1$ estimate (35) becomes

$$ \|R_h(u) - u_h\|_{H_\kappa^1(\Omega)} \lesssim \rho(\kappa) \left( \kappa^{d/4} \|u - u_h\|_{L^4(\Omega)} + \kappa^{d/2} \|u - u_h\|_{L^4(\Omega)}^2 \right) \|u - u_h\|_{H_\kappa^1(\Omega)}. $$

□

Lemma 5.2 and Conclusion 5.4 finish the $H^1$-estimate, if we can show that the higher order contribution $\rho(\kappa)$ $(\kappa^{d/4} \|u - u_h\|_{L^4(\Omega)} + \kappa^{d/2} \|u - u_h\|_{L^4(\Omega)}^2) \|u - u_h\|_{H_\kappa^1(\Omega)}$ is negligible provided that $u$ and $u_h$ are related in a suitable way.

## 5.3 Preliminary error bounds

In this section we want to derive a preliminary (pessimistic) estimate for the error $\|u - u_h\|_{H_\kappa^1(\Omega)}$. More precisely, we want to identify a resolution condition such that it holds

$$ \rho(\kappa) \left( \kappa^{d/4} \|u - u_h\|_{L^4(\Omega)} + \kappa^{d/2} \|u - u_h\|_{L^4(\Omega)}^2 \right) \|u - u_h\|_{H_\kappa^1(\Omega)} \ll \|u - u_h\|_{H_\kappa^1(\Omega)}. $$

If such an estimate is available, we can later absorb the term in the final estimates. In order to obtain the desired bound, we take inspiration from [37] where an abstract formalism for the solution of nonlinear differential equations is developed. However, we cannot directly use the findings and arguments of [37] in our work, as this would result in far too pessimistic estimates (with respect to $\kappa$) to be applicable in our setting.

In the first step towards the desired estimates, we introduce an alternative characterization of a discrete local minimizer. For this we will employ a linearization of $E'$ around $u$, $E'_h : H^1(\Omega) \to H^1(\Omega)^*$, defined by

$$ \langle E'_h(v), w \rangle := \langle E'(v), R_h(w) \rangle + \langle E''(u)v, w - R_h(w) \rangle $$
for all $v, w \in H^1(\Omega)$, where $R_h$ is the Ritz projection introduced in (34). Note that the prime in $E'_h$ is purely notational: $E'_h$ is not the Fréchet derivative of a functional $E_h$. Later on, the notation $E_h''$ and $E_h'''$ will respectively refer to the first and second derivatives of $E'_h$.

**Lemma 5.5.** Consider a (local) minimizer $u \in H^1(\Omega)$ of $E$ such that (A1) holds. Then,

$$u_h \in (iu)^{\perp} \text{ solves } E'_h(u_h) = 0$$

if and only if

$$u_h \in V_h \cap (iu)^{\perp} \quad \text{and} \quad \langle E'(u_h), v_h \rangle = 0 \quad \text{for all } v_h \in V_h \cap (iu)^{\perp}.$$

*Proof.* The direction “$\Leftarrow$” is obvious. Conversely, let $u_h \in (iu)^{\perp}$ solve $E'_h(u_h) = 0$, then for all $w_h \in V_h \cap (iu)^{\perp}$ we have $\langle E'(u_h), w_h \rangle = 0$. This in turn implies $\langle E''(u)u_h, w - R_h(w) \rangle = 0$ for all $w \in (iu)^{\perp}$ and hence, since $R_h$ is the $\langle E''(u), \cdot \rangle$-orthogonal projection and (A1) holds, we have $u_h \in V_h \cap (iu)^{\perp}$. $\square$

Our goal is to identify a neighborhood of $u$ such that, in that neighborhood, there is a unique solution $u_h \in (iu)^{\perp}$ with $E'_h(u_h) = 0$. The size of the neighborhood will allow us to give a pessimistic estimate for $\|u - u_h\|_{H^1(\Omega)}$ provided that another resolution condition is fulfilled that remains to be identified.

Before we can prove the existence of a suitable neighborhood, we require two more auxiliary results. The first one collects various properties of $E_h''$ and its inverse.

**Lemma 5.6.** Consider a (local) minimizer $u \in H^1(\Omega)$ that fulfills (A1). It holds that

$$\langle E_h''(v)w, z \rangle = \langle E''(v)w, R_h(z) \rangle + \langle E''(u)w, z - R_h(z) \rangle \quad (36)$$

for all $v, w, z \in (iu)^{\perp}$. In particular, we have

$$\langle E_h''(u)w, z \rangle = \langle E''(u)w, z \rangle$$

and hence coercivity of $\langle E_h''(u), \cdot \rangle$ on $(iu)^{\perp}$ with

$$\langle E_h''(u)w, w \rangle \geq \rho(\kappa)^{-1} \|w\|_{H_{\kappa}^{1}(\Omega)}^{2} \quad \text{for all } w \in (iu)^{\perp}.$$

Furthermore, we have

$$E''(u)|_{(iu)^{\perp}}^{-1} E'_h(u) = u - R_h(u). \quad (37)$$

*Proof.* All properties follow by straightforward calculations. Only identity (37) requires verification. As the coercivity ensures that $E_h''(u)|_{(iu)^{\perp}}^{-1}$ exists, we let $\phi_u := E''(u)|_{(iu)^{\perp}}^{-1} E'_h(u) \in (iu)^{\perp}$ to shorten notation. By definition, we have

$$\langle E''(u)\phi_u, v \rangle = \langle E'_h(u), v \rangle \quad \text{for all } v \in (iu)^{\perp}.$$

Using the expression of $E'_h(u)$, this yields

$$\begin{aligned} \langle E''(u)\phi_u, v \rangle &= \langle E'(u), R_h(v) \rangle + \langle E''(u)u, v - R_h(v) \rangle = \langle E''(u)u, v - R_h(v) \rangle \\ &= \langle E''(u)(u - R_h(u)), v - R_h(v) \rangle = \langle E''(u)(u - R_h(u)), v \rangle. \end{aligned}$$

Since this holds for arbitrary $v \in (iu)^{\perp}$ and since both $\phi_u \in (iu)^{\perp}$ and $u - R_h(u) \in (iu)^{\perp}$, we conclude $\phi_u = u - R_h(u)$. $\square$
Next, we quantify the continuity of $E_h''$.

**Lemma 5.7.** Let $u \in H^1(\Omega)$ denote a (local) minimizer of $E$ such that (A1) holds. If $\eta_h \le \eta_\star$ as per Lemma 5.2, then for arbitrary $v, w, z \in (iu)^\perp$ it holds

$$ \langle (E_h''(u) - E_h''(z))w, v \rangle \lesssim \left( \|u-z\|_{L^4(\Omega)} + \kappa^{d/4} \|u-z\|_{L^4(\Omega)}^2 \right) \|w\|_{L^4(\Omega)} \|v\|_{H_\kappa^1(\Omega)} \quad (38) $$

and

$$ \langle (E_h''(u) - E_h''(z))w, v \rangle \lesssim \left( \kappa^{d/4} \|u-z\|_{L^4(\Omega)} + \kappa^{d/2} \|u-z\|_{L^4(\Omega)}^2 \right) \|w\|_{H_\kappa^1(\Omega)} \|v\|_{H_\kappa^1(\Omega)}. \quad (39) $$

*Proof.* Let $v, w \in (iu)^\perp$, according to (36) from Lemma 5.6 it holds

$$ \langle E_h''(z)w, v \rangle = \langle E''(z)w, R_h(v) \rangle + \langle E''(u)w, v - R_h(v) \rangle. $$

Hence, for all $r \in (iu)^\perp$, we have

$$ \begin{aligned} \langle E_h'''(z)r, (w, v) \rangle &= \langle E'''(z)r, (w, R_h(v)) \rangle \quad (40) \\ &= 2 \int_{\Omega} \mathrm{Re}(z\bar{r}) \mathrm{Re}(w\overline{R_h(v)}) + \mathrm{Re}(w\bar{z}) \mathrm{Re}(r\overline{R_h(v)}) + \mathrm{Re}(r\bar{w}) \mathrm{Re}(z\overline{R_h(v)}) \, dx. \end{aligned} $$

Note that $\langle E_h'''(z)r, (w, v) \rangle$ is symmetric w.r.t. to the second and third argument, i.e.

$$ \langle E_h'''(z)r, (w, v) \rangle = \langle E_h'''(z)w, (r, v) \rangle. $$

We shall exploit this in the following calculations. Also note that the fourth derivative $E_h''''(z)$ is constant (i.e. independent of $z$) with

$$ \langle E_h''''(z)q, (r, w, v) \rangle = \langle E'''(q)r, (w, R_h(v)) \rangle. $$

By Taylor-expanding $E_h''(z)$ in $u$ we therefore obtain the exact representation

$$ \begin{aligned} \langle E_h''(z)w, v \rangle &= \langle E_h''(u)w, v \rangle + \langle E_h'''(u)(z-u), (w, v) \rangle + \frac{1}{2} \langle E_h''''(u)(z-u), (z-u, w, v) \rangle \\ &= \langle E_h''(u)w, v \rangle + \langle E'''(u)(z-u), (w, R_h(v)) \rangle + \frac{1}{2} \langle E'''(z-u)(z-u), (w, R_h(v)) \rangle. \end{aligned} $$

With (40) and $\|u\|_{L^\infty(\Omega)} \le 1$ we conclude

$$ \begin{aligned} \langle (E_h''(u) - E_h''(z))w, v \rangle &= \langle E'''(u)(u-z), (w, R_h(v)) \rangle - \frac{1}{2} \langle E'''(z-u)(z-u), (w, R_h(v)) \rangle \\ &\le 6 \|u-z\|_{L^4(\Omega)} \|w\|_{L^4(\Omega)} \|R_h(v)\|_{L^2(\Omega)} + 3 \|u-z\|_{L^4(\Omega)}^2 \|w\|_{L^4(\Omega)} \|R_h(v)\|_{L^4(\Omega)}. \end{aligned} $$

With the Gagliardo–Nirenberg estimate $\|R_h(v)\|_{L^4(\Omega)} \lesssim \kappa^{d/4} \|R_h(v)\|_{H_\kappa^1(\Omega)}$ we further obtain

$$ \begin{aligned} \langle (E_h''(u) - E_h''(v))w, v \rangle &\lesssim \|u-z\|_{L^4(\Omega)} \|w\|_{L^4(\Omega)} \|R_h(v)\|_{L^2(\Omega)} + \kappa^{d/4} \|u-z\|_{L^4(\Omega)}^2 \|w\|_{L^4(\Omega)} \|R_h(v)\|_{H_\kappa^1(\Omega)} \\ &\le \left( \|u-z\|_{L^4(\Omega)} + \kappa^{d/4} \|u-z\|_{L^4(\Omega)}^2 \right) \|w\|_{L^4(\Omega)} \|R_h(v)\|_{H_\kappa^1(\Omega)}. \end{aligned} $$

The $H_\kappa^1$-stability of $R_h$ for $\eta_h \le \eta_\star$ according to Lemma 5.2 then finishes the proof of (38). The second estimate in (39) then simply follows from (38) by another application of the Gagliardo–Nirenberg estimate on $w$. $\square$
We are now ready to prove the existence of a unique zero of $E'_h$ in a neighborhood of any local minimizer $u$ that satisfies (A1).

**Theorem 5.8.** Consider a local minimizer $u \in H^1(\Omega)$ of $E$ satisfying (A1). Then, there exist constants $\tau_*, c_* \gtrsim 1$ such that for all $0 < \tau \le \tau_*$, if

$$ \kappa^{d/2} \rho(\kappa) \|u - R_h(u)\|_{H_{\kappa}^1(\Omega)} \le c_* \tau \quad (41) $$

then there exists a unique $u_h \in V_h \cap (iu)^{\perp}$ with

$$ \rho(\kappa) \|u_h - u\|_{H_{\kappa}^1(\Omega)} \le \tau \quad \text{and} \quad \kappa^{d/4} \rho(\kappa) \|u_h - u\|_{L^4(\Omega)} \le \tau $$

such that

$$ \langle E'(u_h), v_h \rangle = 0 \quad \text{for all } v_h \in V_h. $$

Finally, it also holds

$$ \langle E''(u_h)v_h, v_h \rangle \gtrsim \rho(\kappa)^{-1} \|v_h\|_{H_{\kappa}^1(\Omega)}^2 \quad \text{for all } V_h \cap (iu)^{\perp}. \quad (42) $$

i.e., $u_h$ is a (local) minimizer of $E$ in $V_h$.

*Proof.* Let $0 < \tau \le \tau_*$ (with an upper bound on $\tau_* \gtrsim 1$ to be specified later) and consider the following neighborhood of $u$:

$$ K_{\tau,\kappa}(u) := \{z \in (iu)^{\perp} \mid \rho(\kappa)\|u-z\|_{H_{\kappa}^1(\Omega)} \le \tau \text{ and } \kappa^{d/4}\rho(\kappa)\|u-z\|_{L^4(\Omega)} \le \tau \}. $$

The set $K_{\tau,\kappa}(u)$ is closed in $(iu)^{\perp}$ because of the continuous embedding of $H^1(\Omega)$ into $L^4(\Omega)$ (here, $d \le 3$). We consider the Newton-type (fixed-point) function $F: K_{\tau,\kappa}(u) \to (iu)^{\perp}$ given by

$$ F(z) := z - E''(u)_{|(iu)^{\perp}}^{-1} E'_h(z), $$

which is motivated by the local quadratic convergence of Newton's method. For a sufficiently small $\tau_*$ we want to prove that $F$ has a unique fixed-point $u_h \in K_{\tau,\kappa}(u)$ by using the Banach fixed-point theorem. For that, we need to identify a resolution condition (depending on $h$ and $\kappa$) such that $F$ is a contraction, i.e. $\|F(w) - F(z)\|_{H_{\kappa}^1(\Omega)} \le L\|w-z\|_{H_{\kappa}^1(\Omega)}$ on $K_{\tau,\kappa}(u)$ for some $L < 1$ and $F(K_{\tau,\kappa}(u)) \subset K_{\tau,\kappa}(u)$.

We start by deriving a condition for $\tau_*$ such that the contraction property holds. Let $w, z \in K_{\tau,\kappa}(u)$ be arbitrary. In this setting, we write

$$ \gamma(s) := F(sw + (1-s)z) \quad \text{and hence} \quad F(w) - F(z) = \int_0^1 \gamma'(s) \, ds. $$

Using the fact that $F(z) = E_h''(u)_{|(iu)^{\perp}}^{-1} (E_h''(u)z - E_h'(z))$ we obtain

$$ \gamma'(s) = E_h''(u)_{|(iu)^{\perp}}^{-1} (E_h''(u)(w-z) - E_h''(sw + (1-s)z)(w-z)). $$

Since the coercivity of $E_h''(u)$ with constant $\rho(\kappa)^{-1}$ implies for all $\mathcal{F} \in ((iu)^{\perp})^*$ that

$$ \|E_h''(u)^{-1} \mathcal{F}\|_{H_{\kappa}^1(\Omega)} \le \rho(\kappa) \sup_{v \in (iu)^{\perp}} \frac{\langle \mathcal{F}, v \rangle}{\|v\|_{H_{\kappa}^1(\Omega)}}, $$
we conclude with Lemma 5.7

$$
\begin{align*}
\|F(w) - F(z)\|_{H_{\kappa}^{1}(\Omega)} &\leq \rho(\kappa) \int_{0}^{1} \sup_{v \in (iu)^{\perp}} \frac{\langle (E_{h}''(u) - E_{h}''(sw + (1-s)z))(w-z), v \rangle}{\|v\|_{H_{\kappa}^{1}(\Omega)}} \, ds \\
&\lesssim \rho(\kappa) \sup_{s \in [0,1]} \left( \kappa^{d/4} \|u - sw - (1-s)z\|_{L^{4}(\Omega)} + \kappa^{d/2} \|u - sw - (1-s)z\|_{L^{4}(\Omega)}^{2} \right) \|w-z\|_{H_{\kappa}^{1}(\Omega)} \\
&\lesssim \rho(\kappa) (\rho(\kappa)^{-1}\tau + \rho(\kappa)^{-2}\tau^{2}) \|w-z\|_{H_{\kappa}^{1}(\Omega)} \lesssim (1+\tau^{*})\tau \|w-z\|_{H_{\kappa}^{1}(\Omega)}
\end{align*}
$$

since $\rho(\kappa) \gtrsim 1$. We conclude that for a sufficiently small $\tau_*$ (independent of $\kappa$ and $h$) it holds

$$
\|F(w) - F(z)\|_{H_{\kappa}^{1}(\Omega)} \leq \frac{1}{2} \|w - z\|_{H_{\kappa}^{1}(\Omega)},
$$

i.e., $F$ is a contraction.

It remains to show $\rho(\kappa)\|u-F(z)\|_{H_{\kappa}^{1}(\Omega)} \le \tau$ and $\kappa^{d/4}\rho(\kappa)\|u-F(z)\|_{L^{4}(\Omega)} \le \tau$ if $z \in K_{\tau,\kappa}(u)$.
For this, consider $u-F(z)$ for arbitrary $z \in (iu)^{\perp}$. As before, we have

$$
F(u) - F(z) = \int_{0}^{1} E_{h}''(u)^{-1} (E_{h}''(u)(u-z) - E_{h}''(su + (1-s)z)(u-z)) \, ds.
$$

As (37) implies $u - F(u) = E''(u)|_{(iu)^{\perp}}^{-1} E'_h(u) = u - R_h(u)$ we have altogether

$$
\begin{align*}
u - F(z) &= u - F(u) + F(u) - F(z) \\
&= u - R_h(u) + E''(u)|_{(iu)^{\perp}}^{-1} \int_0^1 (E_h''(u) - E_h''(su + (1-s)z))(u-z) \, ds.
\end{align*}
$$

Using the short-hand notation

$$
\varepsilon := \|u - R_h(u)\|_{H_{\kappa}^{1}(\Omega)}.
$$

we obtain again with Lemma 5.7 and $u - (su + (1-s)z) = (1-s)(u-z)$ that

$$
\begin{align*}
\rho(\kappa) \|u - F(z)\|_{H_{\kappa}^{1}(\Omega)}
&\le \rho(\kappa) \|u - R_h(u)\|_{H_{\kappa}^{1}(\Omega)} + \rho(\kappa)^2 \sup_{v \in (iu)^{\perp}} \frac{|\int_{0}^{1} (\langle E_h''(u) - E_h''(su + (1-s)z) \rangle (u-z), v) \, ds|}{\|v\|_{H_{\kappa}^{1}(\Omega)}} \\
&\lesssim \rho(\kappa) \|u - R_h(u)\|_{H_{\kappa}^{1}(\Omega)} + \rho(\kappa)^2 \left( \kappa^{d/4} \|u-z\|_{L^4(\Omega)} + \kappa^{d/2} \|u-z\|_{L^4(\Omega)}^2 \right) \|u-z\|_{H_{\kappa}^{1}(\Omega)} \\
&\lesssim \rho(\kappa) \varepsilon + (\tau + \rho(\kappa)^{-1}\tau^2)\tau \lesssim (c_* + \tau_* + \tau_*^2)\tau
\end{align*}
$$

where we used the condition involving $c_*$ in (41), i.e., that it holds $\varepsilon \le c_* \kappa^{-d/2} \rho(\kappa)^{-1} \tau \lesssim c_* \tau$.
We thus obtain for all sufficiently small $c_*$ and $\tau_*$ that

$$
\rho(\kappa) \|u - F(z)\|_{H_{\kappa}^{1}(\Omega)} \leq \tau.
$$

It remains to show $\kappa^{d/4}\rho(\kappa)\|u-F(z)\|_{L^4(\Omega)} \le \tau$. For this, we use again the Gagliardo-
Nirenberg inequality from (4) together with the previous $H_\kappa^1$-estimate to obtain

$$
\begin{align*}
\kappa^{d/4} \rho(\kappa) \|u - F(z)\|_{L^4(\Omega)}
&\le \kappa^{d/4} \rho(\kappa) \|u - R_h(u)\|_{L^4(\Omega)} \\
&\quad + \kappa^{d/4} \rho(\kappa) \|E''(u)|_{(iu)^\perp}^{-1} \int_0^1 (E_h''(u) - E_h''(su + (1-s)z))(u-z) \, ds \|_{L^4(\Omega)} \\
&\lesssim \kappa^{d/2} \rho(\kappa) \varepsilon + \kappa^{d/2} \rho(\kappa)^2 \int_0^1 \sup_{v \in (iu)^\perp} \frac{\langle (E_h''(u) - E_h''(su + (1-s)z))(u-z), v \rangle}{\|v\|_{H_\kappa^1(\Omega)}} \, ds
\end{align*}
$$
$$ (38) \qquad \lesssim \kappa^{d/2} \rho(\kappa) \varepsilon + \rho(\kappa)^2 \kappa^{d/2} \left( \|u-z\|_{L^4(\Omega)}^2 + \kappa^{d/4} \|u-z\|_{L^4(\Omega)}^3 \right) $$

$$ (41) \qquad \lesssim c_{\star} \tau + \tau^2 + \rho(\kappa)^{-1} \tau^3 \lesssim (c_{\star} + \tau_{\star} + \tau_{\star}^2) \tau. $$

We then have for all sufficiently small $c_{\star}$ and $\tau_{\star}$ that

$$ \kappa^{d/4} \rho(\kappa) \|u - F(z)\|_{L^4(\Omega)} \le \tau. $$

Consequently, $F$ is a contraction on $K_{\tau,\kappa}(u)$ with $F(K_{\tau,\kappa}(u)) \subset K_{\tau,\kappa}(u)$ and we have existence of a unique function $u_h \in (iu)^{\perp}$ such that $E_h'(u_h) = 0$. By Lemma 5.5 we conclude that $u_h \in V_h \cap (iu)^{\perp}$ fulfills

$$ \langle E'(u_h), v_h \rangle = 0 \quad \text{for all } v_h \in V_h \cap (iu)^{\perp}. $$

Before we can verify that the test functions can be arbitrary in $V_h$ (and do not have to belong to $(iu)^{\perp}$), we first verify the coercivity of $\langle E''(u_h) \cdot, \cdot \rangle$, i.e., property (42). To get the desired estimate, we use one more time Lemma 5.7 to obtain

$$ \begin{aligned} |\langle (E_h''(u) - E_h''(u_h))v_h, v_h \rangle| & \lesssim (\kappa^{d/4} \|u - u_h\|_{L^4(\Omega)} + \kappa^{d/2} \|u - u_h\|_{L^4(\Omega)}^2) \|v_h\|_{H_k^1(\Omega)}^2 \\ & \lesssim \tau \rho(\kappa)^{-1} \|v_h\|_{H_k^1(\Omega)}^2. \end{aligned} $$

Since $\langle E_h''(w)v_h, v_h \rangle = \langle E''(w)v_h, v_h \rangle$ for all $v_h \in V_h \cap (iu)^{\perp}$ and arbitrary $w \in (iu)^{\perp}$ (cf. Lemma 5.6), we obtain for sufficiently small $\tau$ that

$$ \begin{aligned} \langle E''(u_h)v_h, v_h \rangle &= \langle E''(u)v_h, v_h \rangle + \langle (E''(u_h) - E''(u))v_h, v_h \rangle \\ &\ge \rho(\kappa)^{-1} \|v_h\|_{H_k^1(\Omega)}^2 - |\langle (E''(u_h) - E''(u))v_h, v_h \rangle| \\ &\ge \rho(\kappa)^{-1}(1 - C\tau) \|v_h\|_{H_k^1(\Omega)}^2 \gtrsim \rho(\kappa)^{-1} \|v_h\|_{H_k^1(\Omega)}^2. \end{aligned} $$

It remains to verify that it holds $E'(u_h) = 0$ on $V_h$. We already know that $E'(u_h) = 0$ on $V_h \cap (iu)^{\perp}$ and (by the coercivity) that $E''(u_h)$ has a positive spectrum on $V_h \cap (iu)^{\perp}$. Consequently, $u_h$ is the unique local minimizer of $E$ in $V_h \cap (iu)^{\perp}$ with $\rho(\kappa)\|u - u_h\|_{H_k^1(\Omega)} \le \tau$ and $\kappa^{d/4}\rho(\kappa)\|u - u_h\|_{L^4(\Omega)} \le \tau$. However, the constraint $v_h \in (iu)^{\perp}$ cannot change the (locally) minimal energy value, because for any $v_h \in V_h$ there exists some $\omega \in [-\pi, \pi)$ such that $\exp(i\omega)v_h \in (iu)^{\perp}$ and $E(\exp(i\omega)v_h) = E(v_h)$. For that reason, $u_h$ must be a local minimizer both in $V_h \cap (iu)^{\perp}$ and $V_h$. By the first order condition for local minimizers we therefore have $\langle E'(u_h), v_h \rangle = 0$ for all $v_h \in V_h$. $\square$

## 5.4 Proof of Theorem 3.2

We are now ready to combine the results of the previous subsections to prove Theorem 3.2. We first establish the main result in a more abstract form.

**Theorem 5.9.** Consider a (local) minimizer $u \in H^1(\Omega)$ of $E$ satisfying (A1). There exist constants $r_{\star} \gtrsim 1$, $\eta_{\star} \gtrsim 1$ and $M_{\star} \lesssim 1$ such that if $\eta_h \le \eta_{\star}$ and

$$ \kappa^{d/2} \rho(\kappa) \min_{v_h \in V_h} \|u - v_h\|_{H_k^1(\Omega)} \le r_{\star}, \quad (43) $$

then there exists a unique discrete minimizer $u_h \in V_h \cap (iu)^{\perp}$ in the neighborhood

$$ K := \{v \in H^1(\Omega) \mid \rho(\kappa) \|u - v\|_{H_k^1(\Omega)} \le M_{\star} \text{ and } \kappa^{d/4} \rho(\kappa) \|u - v\|_{L^4(\Omega)} \le M_{\star}\}, $$
and we have

$$ \|u - u_h\|_{H_{\kappa}^{1}(\Omega)} \lesssim \min_{v_h \in V_h} \|u - v_h\|_{H_{\kappa}^{1}(\Omega)}. \quad (44) $$

In addition, the L²-error estimate holds true

$$ \|u - u_h\|_{L^2(\Omega)} \lesssim \eta_h \min_{v_h \in V_h} \|u - v_h\|_{H_{\kappa}^1(\Omega)} + \rho(\kappa) \kappa^{d/2} \left( \min_{v_h \in V_h} \|u - v_h\|_{H_{\kappa}^1(\Omega)} \right)^2. \quad (45) $$

*Proof.* We apply Theorem 5.8 with a small enough $\tau_\dagger \le \tau_\star$ with $\tau_\dagger \gtrsim 1$ to be fixed. Then, under the assumption that

$$ \kappa^{d/2} \rho(\kappa) \|u - R_h(u)\|_{H_{\kappa}^1(\Omega)} \le c_{\star} \tau_{\dagger}, \quad (46) $$

there exists a unique local minimizer $u_h \in V_h \cap (iu)^\perp$ with $E'(u_h)|_{V_h} = 0$ and with

$$ \kappa^{d/4} \rho(\kappa) \|u_h - u\|_{L^4(\Omega)} \le \tau_{\dagger}. \quad (47) $$

We split

$$ \|u_h - u\|_{H_{\kappa}^1(\Omega)} \le \|u - R_h(u)\|_{H_{\kappa}^1(\Omega)} + \|u_h - R_h(u)\|_{H_{\kappa}^1(\Omega)}, $$

and work on the second term. Conclusion 5.4 implies that

$$ \begin{aligned} \|u_h - R_h(u)\|_{H_{\kappa}^1(\Omega)} &\lesssim \rho(\kappa) \left( \kappa^{d/4} \|u - u_h\|_{L^4(\Omega)} + \kappa^{d/2} \|u - u_h\|_{L^4(\Omega)}^2 \right) \|u - u_h\|_{H_{\kappa}^1(\Omega)} \\ &\lesssim \rho(\kappa) \left( \tau_{\dagger} \kappa^{d/4} \kappa^{-d/4} \rho(\kappa)^{-1} + \tau_{\dagger}^2 \kappa^{d/2} \kappa^{-d/2} \rho(\kappa)^{-2} \right) \|u - u_h\|_{H_{\kappa}^1(\Omega)} \\ &\lesssim \tau_{\dagger} \|u - u_h\|_{H_{\kappa}^1(\Omega)} \ll \|u - u_h\|_{H_{\kappa}^1(\Omega)} \end{aligned} $$

for a suitable choice of $\tau_\dagger \gtrsim 1$. Combining the previous two estimates yields $\|u_h - u\|_{H_{\kappa}^1(\Omega)} \lesssim \|u - R_h(u)\|_{H_{\kappa}^1(\Omega)}$ whenever (46) is met. Furthermore, with the assumption $\eta_h \le \eta_\star$, we have proved in Lemma 5.2 that there is a constant $C_\star > 0$ such that

$$ \|u - R_h(u)\|_{H_{\kappa}^1(\Omega)} \le C_{\star} \min_{v_h \in V_h} \|u - v_h\|_{H_{\kappa}^1(\Omega)}. $$

Consequently, (46) is satisfied if we select $r_\star$ in (43) as $r_\star = c_\star \tau_\dagger / C_\star$. In conclusion, a combination of the estimates yields (44). We can set $M_\star := \tau_\dagger$.

For the L²-error estimate, to shorten notation, we introduce a minimizer $v_h^\star \in V_h$ such that

$$ \|u - v_h^\star\|_{H_{\kappa}^1(\Omega)} = \inf_{v_h \in V_h} \|u - v_h\|_{H_{\kappa}^1(\Omega)}. $$

Then, we first use again Lemma 5.2 for $\eta_h \le \eta_\star$ to obtain

$$ \|u - R_h(u)\|_{L^2(\Omega)} \lesssim \eta_h \|u - v_h^\star\|_{H_{\kappa}^1(\Omega)}. $$

In total we obtain with the suboptimal estimate $\|u_h - R_h(u)\|_{L^2(\Omega)} \lesssim \|u_h - R_h(u)\|_{H_{\kappa}^1(\Omega)}$ and Lemma 5.3 that

$$ \begin{aligned} \|u - u_h\|_{L^2(\Omega)} &\lesssim \eta_h \|u - v_h^\star\|_{H_{\kappa}^1(\Omega)} + \|u_h - R_h(u)\|_{H_{\kappa}^1(\Omega)} \\ &\lesssim \eta_h \|u - v_h^\star\|_{H_{\kappa}^1(\Omega)} + \eta_h \|R_h(u) - u_h\|_{L^2(\Omega)} + \rho(\kappa) \left( \|u - u_h\|_{L^4(\Omega)}^2 + \kappa^{d/4} \|u - u_h\|_{L^4(\Omega)}^3 \right). \end{aligned} $$

At that point, we split

$$ \|R_h(u) - u_h\|_{L^2(\Omega)} \le \|u - u_h\|_{L^2(\Omega)} + \|u - R_h(u)\|_{L^2(\Omega)} \lesssim \|u - u_h\|_{L^2(\Omega)} + \|u - v_h^\star\|_{H_{\kappa}^1(\Omega)}, $$
and for $\eta_*$ sufficiently small, we have

$$ \|u - u_h\|_{L^2(\Omega)} \lesssim \eta_h \|u - v_h^*\|_{H^1(\Omega)} + \rho(\kappa) \left( \|u - u_h\|_{L^4(\Omega)}^2 + \kappa^{d/4} \|u - u_h\|_{L^4(\Omega)}^3 \right). $$

For the last term, we employ the Gagliardo-Nirenberg estimate from (4) which gives

$$ \begin{aligned} \|u - u_h\|_{L^4(\Omega)}^2 + \kappa^{d/4} \|u - u_h\|_{L^4(\Omega)}^3 & \lesssim \kappa^{d/2} \|u - u_h\|_{H^1(\Omega)}^2 + \kappa^d \|u - u_h\|_{H^1(\Omega)}^3 \\ & \lesssim (1 + \kappa^{d/2} \|u - u_h\|_{H^1(\Omega)}) \kappa^{d/2} \|u - u_h\|_{H^1(\Omega)}^2 \\ & \lesssim (1 + \kappa^{d/2} \|u - v_h^*\|_{H^1(\Omega)}) \kappa^{d/2} \|u - v_h^*\|_{H^1(\Omega)}^2 \\ & \lesssim \kappa^{d/2} \|u - v_h^*\|_{H^1(\Omega)}^2 \end{aligned} $$

due to the resolution condition in the assumptions.

<figure><img src="image_4.png" /></figure>

Now, Theorem 3.2 simply follows from applying the explicit expressions we previously obtained from $\eta_h$ and the best-approximatino error.

*Proof of Theorem 3.2.* From (33) in Theorem 5.1 we recall that

$$ \eta_h \lesssim \kappa^{\varepsilon} (h\kappa + \rho(\kappa)(k\kappa)^p) \quad (48) $$

and

$$ \min_{v_h \in V_h} \|u - v_h\|_{H^1(\Omega)} \lesssim (h\kappa)^p. \quad (49) $$

We now observe that for $K_*$ small enough, the condition that

$$ \kappa^{d/2} \rho(\kappa) (h\kappa)^p \leq K_* $$

ensures that $\eta_h \leq \eta_*$ as well as

$$ \kappa^{d/2} \rho(\kappa) \min_{v_h \in V_h} \|u - v_h\|_{H^1(\Omega)} \leq r_* $$

for the constants $\eta_*, r_* \gtrsim 1$ of Theorem 5.9. Here, we used the fact that $\kappa^{\varepsilon} \leq \kappa^{d/2}$.

We are therefore in the setting of Theorem 5.9, and the error estimates (21) and (22) explicitly expressed in terms of $h$ and $\kappa$ simply follow by plugging (48) and (49) in their abstract versions (44) and (45).

<figure><img src="image_5.png" /></figure>

# 6 Numerical experiments

In this section we present numerical experiments to support our theoretical results for local minimizers that are presumably also global minimizers. For that we consider a setting previously studied in [6, 15] where the computational domain is given by $\Omega = [0, 1]^2 \subset \mathbb{R}^2$ and the divergence-free magnetic potential (with a vanishing normal trace) by

$$ A(x, y) := \sqrt{2} \begin{pmatrix} \sin(\pi x) \cos(\pi y) \\ -\cos(\pi x) \sin(\pi y) \end{pmatrix}. $$

The Ginzburg-Landau parameter is kept variable, where we consider the cases $\kappa = 8, 16, 24, 32, 40$. Note that $\kappa = 8, 24$ were previously studied in [15] (for P1-FEM) and $\kappa = 8, 16, 32$ were studied in [6] (for P1-FEM and LOD spaces). The case $\kappa = 40$ in this setting is not yet documented
in the literature. Our reference minimizers $u_{\text{ref}}$ for the different $\kappa$ values were computed with a nonlinear conjugate gradient method and P2 finite elements on a uniform triangular mesh with mesh size $h_{\text{ref}} = 2^{-10}$.

As a tolerance for the nonlinear conjugate gradient method we used that the difference in two successive energy approximations should be smaller than $10^{-15}$. Furthermore, after a converged state $\tilde{u}$ was found, we computed $E''(\tilde{u})$ and checked that the spectrum was indeed non-negative (which we recall as the necessary condition for minimizers). If there was a negative eigenvalue, we took one step (of optimal step length) in the direction of the corresponding eigenfunction, noting that it must be a descent direction. From the obtained result, we then restarted the conjugate gradient method. This avoids that the iterative solver gets stuck in a saddle point. It is worth to mention that such a step was indeed necessary for most $\kappa$ values.

The densities $|u_{\text{ref}}|$ of the corresponding minimizers that we found are depicted in Figure 1 and the corresponding energy values in Table 1. In all our test cases, we found lower energy states than the ones documented in [6, 15]. We also note that our results are consistent with the findings of [6, 15] as we could identify the states from the aforementioned references (for $\kappa = 8, 16, 24, 32$) as local minimizers of $E$. Hence, there is no contradiction to previous results. For $\kappa = 40$ we do not have any reference comparison. We verified that the computed states are indeed (local) minimizers, by checking the spectrum of $E''(u_{\text{ref}})$ and verifying that the smallest eigenvalue is zero (up to numerical precision) and the remaining eigenvalues are positive. For brevity we will write in the following $u$ instead of $u_{\text{ref}}$.

<figure><img src="image_6.png" /><figcaption>Figure 1: Vortex patterns of the reference solution $|u_{\text{ref}}|$ for $\kappa = 8, 16, 24, 32, 40$ (from the upper left to the lower right picture).</figcaption></figure>

<table><caption>Table 1: Approximative minimal energy values $E(u_{\text{ref}})$ for the reference solution $u_{\text{ref}}$ for the values $\kappa = 8, 16, 24, 32, 40$.</caption><tr><td>κ</td><td>8</td><td>16</td><td>24</td><td>32</td><td>40</td></tr><tr><td>E(u<sub>ref</sub>)</td><td>1.164614e-01</td><td>8.314057e-02</td><td>6.599864e-02</td><td>5.530416e-02</td><td>4.789870e-02</td></tr></table>

When comparing a reference minimizer $u$ with a discrete FE minimizer $u_h$ (for given mesh size $h$, polynomial order $p$ and fixed $\kappa$), the two states are ensured to be in the same phase so that the errors are not polluted by potential phase differences. For that, we compute the
phase factors

$$ \alpha_h := \frac{\int_{\Omega} u \overline{u_h} dx}{|\int_{\Omega} u \overline{u_h} dx|}, $$

which guarantee $|\alpha_h u_h| = |u_h|$ and $\int_{\Omega} u \overline{\alpha_h u_h} dx = |\alpha_h|^2 = 1$. Hence, $u$ and $\alpha_h u_h$ are in the same phase (that is $\alpha_h u_h \in (iu)^{\perp}$) and we can compute the relevant errors $u - \alpha_h u_h$ as covered by our theory. For readability, we shall only write $u - u_h$ when presenting the results.

## 6.1 Comparison of energy errors

<figure><img src="image_7.png" /><figcaption>Figure 2: Error in energy $|E(u) - E(u_h)|$ for $\kappa = 8, 16, 24, 32, 40$ between the reference minimal energy $E(u)$ and the minimal energy $E(u_h)$ in the finite element spaces for P1 (solid lines) and P2 (dashed lines) respectively.</figcaption></figure>

We start with comparing the minimal energy level $E(u)$ with the discrete minimal energy levels $E(u_h)$ for P1 and P2 respectively. The corresponding results for $\kappa = 8, 16, 24, 32, 40$ are depicted in Figure 2. As predicted by Proposition 3.1, we can clearly see the second order convergence in $h$ for the P1 energy and fourth order convergence for the P2 energy. Furthermore, we can confirm that the resolution condition $h\kappa \lesssim 1$ is sufficient to observe the correct asymptotic convergence rates. As expected, the pre-asymptotic regime is still barely visible for $\kappa = 8$ and it becomes slightly more pronounced with growing $\kappa$. For fixed $\kappa$, the asymptotic convergence regime starts essentially simultaneously for the P1 and P2 energy approximation. All in all, the experiments fully confirm the behavior of the energy error as stated in Proposition 3.1.

## 6.2 Comparison of $L^2$- and $H_\kappa^1$-errors

Next, we compare the actual $L^2$- and $H_\kappa^1$-errors for discrete minimizers $u_h$ for P1 and P2 respectively. The results are depicted in Figure 3. As predicted by Theorem 3.2, the resolution condition $h\kappa \lesssim 1$ is no longer sufficient to guarantee reasonable approximations and we can identify a very pronounced pre-asymptotic regime. Postulating an algebraic growth of $\rho(\kappa)$ in $\kappa$, the analytical results of Theorem 3.2 imply an effective resolution condition of the form $\kappa^m(h\kappa)^p \lesssim 1$ for some unknown power $m \ge 1$. This is fully consistent with the numerical
<figure><img src="image_8.png" /><figcaption>Figure 3: Comparison of the errors $\|u - u_h\|_{L^2(\Omega)}$ and $\frac{1}{\kappa}\|\nabla u - \nabla u_h\|_{L^2(\Omega)}$ for the discrete minimizers $u_h$ in the FEM spaces $V_{h,1}$ (P1, solid lines) and $V_{h,2}$ (P2, dashed lines), where $\kappa = 8, 16, 24, 32, 40$.</figcaption></figure>

results in Figure 3. On the one hand, the pre-asymptotic regime is significantly larger than what we observed for the energy error in Figure 2 (where $h\kappa \lesssim 1$) and on the other hand, we can also observe that the pre-asymptotic regime is considerably reduced for the P2-FEM approximations compared to P1-FEM. This confirms that the usage of higher order finite elements can reduce the pollution effect significantly.

Asymptotically, we observe for the P1-FEM approximation a convergence of order $O(h)$ for the $H^1$-error and $O(h^2)$ for the $L^2$-error. For P2-FEM we observe the asymptotic convergence of order $O(h^2)$ for the $H^1$-error and $O(h^3)$ for the $L^2$-error. These are the natural expected rates.

It is also worth to mention that the errors for $\kappa = 40$ look asymptotically slightly better than the errors for $\kappa = 32$. This might indicate that the state that we found for $\kappa = 40$ is only a local minimizer, but not a global minimizer. However, even extensive tests with numerous starting values and path-tracking strategies did not lead to a state of smaller energy. Hence, it remains open why the errors for $\kappa = 40$ are asymptotically smaller than for $\kappa = 32$.

To complete the investigation of the pre-asymptotic effect, we also study the behaviour of the $H^1$-best-approximation $P_h(u)$ of $u$. Here, $P_h(u)$ is given by the $H^1$-projection $P_h : H^1(\Omega) \to V_h$ with

$$ \frac{1}{\kappa^2}(\nabla P_h(u) - \nabla u, \nabla v_h)_{L^2(\Omega)} + (P_h(u) - u, v_h)_{L^2(\Omega)} = 0 \quad \text{for all } v_h \in V_h. \quad (50) $$

The corresponding results are presented in Figure 4, where we observe the same convergence regime as for the energy error. This is exactly what we expect since the best-approximation is not affected by the pollution effect and convergence is obtained as soon as $h\kappa \lesssim 1$. Indeed, after a short initial phase, both the $H^1$- and $L^2$-errors (for P1 and P2) are converging with the expected rate.

In conclusion, our numerical experiments fully support our theoretical predictions and in particular the existence of a significant numerical pollution effect that is decreased by the use of higher order finite elements.
<figure><img src="image_9.png" /><figcaption>Figure 4: Comparison of the errors $\|u - P_h(u)\|_{L^2(\Omega)}$ and $\frac{1}{\kappa}\|\nabla u - \nabla P_h(u)\|_{L^2(\Omega)}$ for the $H_\kappa^1$-best-approximation $P_h(u)$ of $u$ given by (50) for $V_{h,1}$ (P1, solid lines) and $V_{h,2}$ (P2, dashed lines). The errors include the cases $\kappa = 8, 16, 24, 32, 40$.</figcaption></figure>

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# A Quasi-interpolation on curved meshes

## A.1 Setting

In this appendix, we consider a domain $\Omega \subset \mathbb{R}^d$, with $d=2$ or $3$. We denote by $L$ the diameter of $\Omega$. $\Gamma_D \subset \partial\Omega$ is a relatively open subset of the boundary that is assumed to have a (possible empty) Lipschitz boundary $\partial\Gamma_D$.
### A.1.1 Reference simplicies

In this section we fix a (closed) reference $d$ simplex $\hat{K}$. We also fix a reference $d-1$ simplex $\hat{F}_{d-1} \subset \mathbb{R}^{d-1}$. For simplicity, we assume that $|\hat{K}| \le 1$, which is the case for the standard choice of the convex hull formed from the canonical basis vectors of $\mathbb{R}^d$. We denote by $\hat{\mathcal{F}}$ the set of faces of $\hat{K}$. For each face $\hat{F} \in \hat{\mathcal{F}}$, there exists a finite number of bijective affine mappings $A: \hat{F} \to \hat{F}_{d-1}$. A specific affine mapping is selected by specifying on which vertex of $\hat{F}_{d-1}$ is mapped each vertex of $\hat{F}$.

### A.1.2 Curved meshes

**Definition A.1** (Curved matching mesh). A curved matching mesh $\mathcal{T}_h$ of $\Omega$ is a finite collection of elements $K$ with associated mappings $\mathcal{F}_K: \hat{K} \to K$ having the following properties. We first ask that the elements are non-degenerate, meaning that for each $K \in \mathcal{T}_h$ the mapping $\mathcal{F}_K: \hat{K} \to K$ is bilipschitz. We further demand that $\mathcal{T}_h$ is non-overlapping, i.e., that for any two distinct elements $K_{\pm} \in \mathcal{T}_h$, the intersection $K_{-} \cap K_{+}$ is either empty, a single point, or an $\ell$-manifold with boundary for some $\ell \in \{1, \dots, d-1\}$. We also require that the mappings are compatible, meaning that if $F := K_{-} \cap K_{+}$ is a $d-1$ manifold for some $K_{\pm} \in \mathcal{T}_h$ then $\hat{F}_{\pm} := \mathcal{F}_{K_{\pm}}^{-1}(F) \in \hat{\mathcal{F}}$ and there exist bijective affine mappings $A_{\pm}: \hat{F}_{d-1} \to \hat{F}_{\pm}$ such that

$$ \mathcal{F}_{K_{-}} \circ A_{-} = \mathcal{F}_{K_{+}} \circ A_{+} \qquad (51) $$

as maps from $\hat{F}_{d-1}$ to $F$. Finally, we demand that the mesh covers the domain, i.e.,

$$ \bar{\Omega} = \bigcup \mathcal{T}_h, $$

and that the mesh is compatibly with the boundary condition, meaning that every boundary face either entirely belongs to $\bar{\Gamma}_D$ or $\partial\Omega \setminus \Gamma_D$.

In what follows we consider such a curved matching mesh $\mathcal{T}_h$ of $\Omega$. Notice that Definition A.1 implies that the elements $K \in \mathcal{T}_h$ are closed sets. It is also clear that if the maps $\mathcal{F}_K$ are affine, then our definition is the standard definition of a matching mesh (see e.g. [20, Definition 6.1.1]).

The notion of “patch” will be important in what follows. For $K \in \mathcal{T}_h$, we finally employ the notation

$$ \mathcal{T}_h^K := \{ K' \in \mathcal{T}_h \mid K' \cap K \neq \emptyset \} $$

for the patch of elements around $K$.

We now introduce a few metrics that measure the fineness and regularity of the mesh. Classically, the fineness of the mesh is quantified with the elements diameters

$$ h_K := \max_{\mathbf{x}, \mathbf{y} \in K} |\mathbf{x} - \mathbf{y}| \quad \forall K \in \mathcal{T}_h, $$

and $h := \max_{K \in \mathcal{T}_h} h_K$ is known as the mesh size. The “shape regularity” of each element $K \in \mathcal{T}_h$, is measured by the quantity

$$ M_K := \max(h_K \|(J\mathcal{F}_K)^{-1}\|_{L^\infty(\hat{K})}, h_K^{-1} \|(J\mathcal{F}_K)\|_{L^\infty(\hat{K})}), $$

where $J\mathcal{F}_K$ denotes the Jacobian of $\mathcal{F}_K$ and where, for each $\hat{x} \in \hat{K}$, the matrix norm associated with the Euclidean metric is considered. For $\mathcal{T} \subset \mathcal{T}_h$, we also use the notation $M_{\mathcal{T}} := \max_{K \in \mathcal{T}} M_K$. These metrics are the natural extensions of the ones used for straight elements. The fact that $M_{\mathcal{T}_h}$ is well-defined and finite is a consequence of Definition A.1. We will
sometimes need more regularity on the element mappings. We will say that the element $K \in \mathcal{T}_h$ is $r$-regular if $\mathcal{F}_K \in C^r(\hat{K}, K)$. In this case, we will write

$$M_K(r) := \max_{|\alpha| \le r} \left( \frac{h_K}{L} \right)^{|\alpha|} \frac{\| \partial^\alpha \mathcal{F}_K \|_{L^\infty(\hat{K})}}{L}. \qquad (52)$$

The mesh $\mathcal{T}_h$ is $r$-regular if all its elements are, and we then write $M_{\mathcal{T}}(r) := \max_{K \in \mathcal{T}} M_K(r)$ for all $\mathcal{T} \subset \mathcal{T}_h$.

For later reference, we note that for all $K \in \mathcal{T}_h$, there exists a constant $\mu_K \ge 1$ solely depending on $M_K$ such that

$$\mu_K^{-1} h_K^d \le |(\det J \mathcal{F}_K)(\hat{\mathbf{x}})| \le \mu_K h_K^d \quad \forall \hat{\mathbf{x}} \in \hat{K}. \qquad (53)$$

### A.1.3 Finite element space

In the remainder, we fix $p \ge 1$, and consider a finite element space

$$V_h := \{ v_h \in H_{\Gamma_D}^1(\Omega) \mid v_h|_K \circ \mathcal{F}_K \in \mathcal{P}_p(\hat{K}) \}$$

where $\mathcal{P}_p(\hat{K})$ is the space of (complex-valued) polynomials of degree at most $p$ defined on the reference simplex $\hat{K}$. The notation

$$\mathcal{P}(K) := \{ v \in H^1(K) \mid v \circ \mathcal{F}_K \in \mathcal{P}_p(\hat{K}), \}$$

will sometimes be useful. We also denote by

$$V_h^{\flat} := \{ v_h \in L^2(\Omega) \mid v_h|_K \in \mathcal{P}(K) \forall K \in \mathcal{T}_h \}.$$

the “broken” counterpart of $V_h$.

### A.1.4 Sobolev spaces

For a submesh $\mathcal{T} \subset \mathcal{T}_h$ covering a domain $D$, if $s \ge 1$ and $1 \le q \le +\infty$, we introduce the broken Sobolev space

$$W^{s,q}(\mathcal{T}) := \{ v \in L^q(D) \mid v|_K \in W^{s,q}(K) \forall K \in \mathcal{T} \}$$

equipped with the semi-norm

$$|v|_{W^{s,q}(\mathcal{T})} := \left( \sum_{K \in \mathcal{T}} \sum_{|\alpha| \le s} \| \partial^\alpha v \|_{L^q(K)}^q \right)^{1/q}$$

and the norm

$$\|v\|_{W^{s,q}(\mathcal{T})} = \sum_{t=0}^{s} L^t |v|_{W^{t,q}(\mathcal{T})}.$$

As usual, we employ the notation $H^1(\mathcal{T}) = W^{1,2}(\mathcal{T})$, for the Hilbert space and for the subscript in the associated norm and semi-norm.
## A.2 Main result

**Theorem A.2 (Quasi-interpolation).** If $\mathcal{T}_h$ is a curved matching mesh of $\Omega$, then there exists a projection $I_h: H_{\Gamma_D}^1(\Omega) \to V_h$ with the following properties. If $v \in H_{\Gamma_D}^1(\Omega) \cap W^{s+1,q}(\mathcal{T}_h)$ for some $2 \le q \le +\infty$ and $0 \le s \le p$ and if $\mathcal{T}_h$ is $s+1$ regular, then for each $K \in \mathcal{T}_h$, we have

$$h_K^{-1} \|v - I_h(v)\|_{L^r(K)} + |v - I_h(v)|_{W^{1,r}(K)} \lesssim h_K^{d/r-d/q} \left(\frac{h_K}{L}\right)^s \sum_{t=0}^{s} L^t |v|_{W^{t+1,q}(\mathcal{T}_h^K)}. \quad (54)$$

for all $1 \le r \le q$, with a hidden constant depending on $p, r, q, M_{\mathcal{T}_h^K}$ and $M_{\mathcal{T}_h^K}(s+1)$. It follows in particular that

$$h^{-1} \|v - I_h(v)\|_{L^r(\Omega)} + |v - I_h(v)|_{W^{1,r}(\Omega)} \lesssim h^{d/r-d/q} h^s \|v\|_{W^{s+1,q}(\mathcal{T}_h)} \quad (55)$$

with a hidden constant depending on $p, r, q, M_{\mathcal{T}_h}$ and $M_{\mathcal{T}_h}(s+1)$.

For easy reference, we highlight the following direct consequence of Theorem A.2. The proof simply follows by selecting $r = q = 2$ and $s = 0$, and applying a triangular inequality.

**Corollary A.3 (Stability in energy norm).** Let $v \in H_{\Gamma_D}^1(\Omega)$. For all $K \in \mathcal{T}_h$ we have

$$|I_h(v)|_{H^1(K)} \lesssim |v|_{H^1(\mathcal{T}_h^K)},$$

In particular,

$$|I_h(v)|_{H^1(\Omega)} \lesssim |v|_{H^1(\Omega)}.$$

Here, $I_h$ is the projection from Theorem A.2 and the hidden constants only depend on $M_{\mathcal{T}_h}$.

## A.3 Consequences of mesh regularity

Here, we show that the number of elements in a patch is bounded by a constant that only depends on $M_{\mathcal{T}_h}$, and that they all have similar diameters. These results are standard for straight elements and are extended here to curved ones.

**Lemma A.4 (Element patches).** There exists a constant $N_{\mathcal{T}_h}$ only depending on $M_{\mathcal{T}_h}$ such that the cardinality of $\mathcal{T}_h^K$ is less than $N_{\mathcal{T}_h}$ for all $K \in \mathcal{T}_h$. In addition, we have

$$h_{K'} \approx h_K \quad (56)$$

with hidden constants only depending on $M_{\mathcal{T}_h}$.

*Proof.* Consider an element $K \in \mathcal{T}_h$ and one of its vertices $\mathbf{a}$ (i.e. the image of a vertex of $\hat{K}$ through $\mathcal{F}_K$). We can then consider, for each $K' \in \mathcal{T}_h$ with $\mathbf{a} \in K'$ the affine map

$$F_{K'}: \hat{K} \ni \hat{\mathbf{x}} \to \mathbf{a} + (J\mathcal{F}_{K'})(\mathbf{a}) \hat{\mathbf{x}} \in K'_{\uparrow},$$

where $K'_{\uparrow}$ is a straight tetrahedron with shape-regularity measure only depends on $M_{K'}$. We now observe that because the mappings are matching, the collection of all straight elements $K'_{\uparrow}$ obtained from $K' \in \mathcal{T}_h$ with $\mathbf{a} \in K'$ is a standard shape-regular vertex patch of straight elements, with shape-regularity measure only controlled by $M_{\mathcal{T}_h}$. For a straight vertex patch, it is a standard result that the number of elements is bounded by a constant that only depends on the shape-regularity measure, which shows that the number of elements $K$ sharing $\mathbf{a}$ is bounded. Since $K$ has only $d+1$ vertices and all $K' \in \mathcal{T}_h^K$, must have at least one of those vertices, this concludes the existence of the constant $N_{\mathcal{T}_h}$ controlling the cardinality of $\mathcal{T}_h^K$.
To prove (56), we first consider two elements $K_{\pm} \in \mathcal{T}_h$ sharing a face $F$. We let $\hat{F}_{\pm} := \mathcal{F}_{K_{\pm}}^{-1}(F)$, and denote by $A_{\pm} : \hat{F}_{d-1} \to \hat{F}_{\pm}$ the affine maps from (51). We now pick two distinct points $\hat{x}_{d-1}, \hat{y}_{d-1} \in \hat{F}_{d-1}$. We let $\hat{x}_{\pm} := A_{\pm}(\hat{x}_{d-1})$ and $\hat{y}_{\pm} := A_{\pm}(\hat{y}_{d-1})$. Since the $A_{\pm}$ is a bijective affine mapping, we have

$$|\hat{x}_{\pm} - \hat{y}_{\pm}| \approx |\hat{x} - \hat{y}|.$$

We then let $\boldsymbol{x} := \mathcal{F}_{K_{\pm}}(\hat{x}_{\pm})$ and $\boldsymbol{y} := \mathcal{F}_{K_{\pm}}(\hat{y}_{\pm})$, where the definition is not ambiguous due to (51). We can then write on the one hand that

$$|\boldsymbol{x} - \boldsymbol{y}| = |\mathcal{F}_{K_+}(\hat{\boldsymbol{x}}_+) - \mathcal{F}_+(\hat{\boldsymbol{y}}_+)| \lesssim h_{K_+}|\hat{\boldsymbol{x}}_+ - \hat{\boldsymbol{y}}_+| \lesssim h_{K_+}|\hat{\boldsymbol{x}}_{d-1} - \hat{\boldsymbol{y}}_{d-1}|,$$

where the hidden constant only depends on $M_{K_+}$ and on the other hand that

$$|\hat{\boldsymbol{x}}_{d-1} - \hat{\boldsymbol{y}}_{d-1}| \lesssim |\hat{\boldsymbol{x}}_{-} - \hat{\boldsymbol{y}}_{-}| = |\mathcal{F}_{K_{-}}^{-1}(\boldsymbol{x}) - \mathcal{F}_{K_{-}}^{-1}(\boldsymbol{y})| \lesssim h_{K_{-}}^{-1}|\boldsymbol{x} - \boldsymbol{y}|,$$

with a hidden constant solely depending on $M_{K_-}$. This is eventually leads to $h_{K_-} \lesssim h_{K_+}$, and we can conclude that

$$h_{K_{-}} \approx h_{K_{+}} \qquad (57)$$

by flipping the roles of $K_-$ and $K_+$, where the hidden constant depends only on $M_{\mathcal{T}_h}$.

Now, if $K \in \mathcal{T}_h$ and $K' \in \mathcal{T}_h^K$, then we can join $K$ to $K'$ by a chain $K = K_1, \dots, K_n = K'$ of $n \le N_{\mathcal{T}_h}$ elements connected through a face. And we obtain (56) by applying (57) $n$. This is indeed concludes the proof since the constants in (57) are then exponentiated at most to power $N_{\mathcal{T}_h}$. $\square$

## A.4 Change of variables properties

As per usual, our error estimates will rely on scaling arguments, which ultimately hinge on changes of variables. Such reasoning is completely standard for straight elements, and is extended to curved elements with an effort of using minimal assumptions.

**Lemma A.5 (Change of variable for Lebesgue spaces).** Let $K \in \mathcal{T}_h$. We have

$$\mu_K^{-1/r} h_K^{d/r} \|v \circ \mathcal{F}_K\|_{L^r(\hat{K})} \le \|v\|_{L^r(K)} \le \mu_K^{1/r} h_K^{d/r} \|v \circ \mathcal{F}_K\|_{L^r(\hat{K})} \qquad (58)$$

for all $1 \le r \le +\infty$ and all $v \in L^r(K)$.

*Proof.* The statement is obvious for $r = +\infty$. If $1 \le r < +\infty$, we introduce the short-hand notation $\hat{v} := v \circ \mathcal{F}_K \in L^q(\hat{K})$. We have

$$\|v\|_{L^r(K)}^r = \int_K |v(\boldsymbol{x})|^r d\boldsymbol{x} = \int_{\hat{K}} |\hat{v}(\hat{\boldsymbol{x}})|^r \det J\mathcal{F}_K(\hat{\boldsymbol{x}}) |d\hat{\boldsymbol{x}}| \le \mu_K h_K^d \|\hat{v}\|_{L^r(\hat{K})}^r$$

which is the upper bound in (58). The lower bound is obtained similarly, after observing that (53) implies that

$$|\det(J\mathcal{F}_K^{-1})(\boldsymbol{x})| \le \mu_K h_K^{-d}$$

for all $\boldsymbol{x} \in K$.

**Lemma A.6 (Change of variables for Sobolev spaces).** Let $K \in \mathcal{T}_h$ and $v \in W^{1,q}(K)$. Then, we have

$$h_K |v|_{W^{1,q}(K)} \le M_K \mu_K^{1/q} h_K^{d/q} |v \circ \mathcal{F}_K|_{W^{1,q}(\hat{K})} \qquad (59)$$
In addition, if (52) holds true with $r = s$ and $v \in W^{s,q}(K)$ for some $s \ge 1$, then

$$h_K^{d/q} |v \circ \mathcal{F}_K|_{W^{s,q}(\hat{K})} \lesssim \left(\frac{h_K}{L}\right)^s \sum_{t=1}^s L^t |v|_{W^{t,q}(K)} \quad (60)$$

where the hidden constant depends on $M_K$, $M_K(s)$, $s$ and $q$.

**Proof.** Let us first establish (59). For that, we let $g := \nabla v \in L^q(K)$, $\hat{v} := v \circ \mathcal{F}_K \in W^{1,q}(\hat{K})$ and $\hat{g} := \nabla \hat{v} \in L^q(\hat{K})$, so that $\|g\|_{L^q(K)} = |v|_{W^{1,q}(K)}$ and $\|\hat{g}\|_{L^q(\hat{K})} = |\hat{v}|_{W^{1,q}(\hat{K})}$. Due to the chain rule, we have

$$\hat{g} = (J\mathcal{F}_K)^T \cdot (g \circ \mathcal{F}_K)$$

and it follows that

$$\|g \circ \mathcal{F}_K\|_{L^q(\hat{K})} = \|J\mathcal{F}_K^{-T} \cdot \hat{g}\|_{L^q(\hat{K})} \le M_K h_K^{-1} \|\hat{g}\|_{L^q(\hat{K})}.$$

We then conclude the proof with the estimate

$$\|g\|_{L^q(K)} \le \mu_K^{1/q} h_K^{d/q} \|g \circ \mathcal{F}_K\|_{L^q(\hat{K})}$$

which we obtain by applying (58) component wise.

The estimate in (60) is established with similar arguments, though more technical. For shortness, we do not repeat those here, and follow the proof of [9, Lemma B.3] where the estimate

$$|\partial^\alpha (v \circ \mathcal{F}_K)| \lesssim \sum_{\substack{\beta \le \alpha \\ |\beta| \ge 1}} L^{|\beta|} \left(\frac{h}{L}\right)^{|\alpha|} |(\partial^\beta v) \circ \mathcal{F}_K|$$

is established a.e. in $\hat{K}$ for all multi-indices $\alpha$ with $|\alpha| = s$ with hidden constant depending on $\alpha$ and $M_K(s)$. It also follows from the triangular inequality that

$$\|\partial^\alpha (v \circ \mathcal{F}_K)\|_{L^q(\hat{K})} \lesssim \sum_{\substack{\beta \le \alpha \\ |\beta| \ge 1}} L^{|\beta|} \left(\frac{h}{L}\right)^{|\alpha|} \|(\partial^\beta v) \circ \mathcal{F}_K\|_{L^q(\hat{K})}$$

and (60) follows from (58). □

**Lemma A.7 (Change of exponent).** Assume that $1 \le r \le q \le +\infty$. Then, for all $v \in L^q(K)$, we have

$$\|v\|_{L^r(K)} \le \mu_K^{1/r+1/q} h_K^{d/r-d/q} \|v\|_{L^q(K)}. \quad (61)$$

**Proof.** Here too, we introduce the short-hand notation $\hat{v} := v \circ \mathcal{F}_K \in L^q(\hat{K})$. We are first going to show that

$$\|\hat{v}\|_{L^r(\hat{K})} \le \|\hat{v}\|_{L^q(\hat{K})} \quad (62)$$

on the reference element. Let us first observe that (62) clearly holds true if $r = q$. We may therefore assume that $1 \le r < q$. Then, since $q/r > 1$, Hölder inequality gives that

$$\|\hat{v}\|_{L^r(\hat{K})}^r = \|\hat{v}\|^r_{L^1(\hat{K})} \le \|1\|_{L^{q/(q-r)}(\hat{K})} \|\hat{v}\|^r_{L^{q/r}(\hat{K})} = |\hat{K}|^{(q-r)/q} \|\hat{v}\|_{L^q(\hat{K})}^r,$$

which we indeed gives (62) since $|\hat{K}| \le 1$ by assumption and $(q-r)/(qr) \ge 0$. Now, (61) simply follows by applying the upper bound in (58), (62) and the lower bound in (58). □
## A.5 Properties of the element-wise projection

Our quasi-interpolation operator hinges on an element-wise $H^1$ orthogonal projection operator. Specifically, for $K \in \mathcal{T}_h$ and $v \in H^1(K)$ we define the elementwise projection $\Pi_K v \in \mathcal{P}(K)$ by requiring that

$$ (v - \Pi_K v, 1)_K = 0, \quad (\nabla(v - \Pi_K v)), \nabla w_K) = 0 \quad \forall w_K \in \mathcal{P}(K). $$

In this section, we established some key properties of this elementwise projection. These results are elementary for straight elements $K$ with affine mappings $\mathcal{F}_K$. The extension to curved elements under suitable regularity assumptions for the mapping is straightforward, but we include it here for completeness.

**Lemma A.8 (Poincaré inequality).** For all $K \in \mathcal{T}_h$ and $w \in H^1(K)$ with $(w, 1)_K = 0$, we have

$$ \|w\|_{L^2(K)} \lesssim h_K |w|_{H^1(K)} \quad (63) $$

where the hidden constant only depends on $M_K$. It follows in particular that

$$ \|v - \Pi_K v\|_{L^2(K)} \lesssim h_K |v - \Pi_K v|_{H^1(K)} \quad (64) $$

for all $v \in H^1(K)$.

*Proof.* We first observe that since $(w, 1)_K = 0$, we have

$$ \|w\|_{L^2(K)} \le \|w - k\|_{L^2(K)} $$

for any choice of constant $k \in \mathbb{C}$. We now apply (58), which gives

$$ \|w - k\|_{L^2(K)} \lesssim h_K^{d/2} \|w \circ \mathcal{F}_K - k\|_{L^2(\hat{K})}. $$

We can now choose $k := (w \circ \mathcal{F}_K, 1)_{\hat{K}}$ so that

$$ \|w \circ \mathcal{F}_K - k\|_{L^2(\hat{K})} \lesssim |w \circ \mathcal{F}_K|_{H^1(\hat{K})}, $$

where the hidden constant only depends on the choice of the reference element $\hat{K}$. Then, (63) follows from (60). □

**Lemma A.9 (Stability).** Let $K \in \mathcal{T}_h$. For all $q \ge 2$, we have

$$ \|\Pi_K v\|_{L^q(K)} \lesssim \|v\|_{L^q(K)} + h_K |v|_{W^{1,q}(K)} \quad (65) $$

and

$$ |\Pi_K v|_{W^{1,q}(K)} \lesssim |v|_{W^{1,q}(K)} \quad (66) $$

for all $v \in W^{1,q}(K)$, where the hidden constants depend on $p, q$ and $M_K$.

*Proof.* We start with (65). By the change of exponent formula in (58), we first observe that

$$ \|\Pi_K v\|_{L^q(K)} \lesssim h_K^{d/q} \|(\Pi_K v) \circ \mathcal{F}_K\|_{L^q(\hat{K})}. $$

Since $(\Pi_K v) \circ \mathcal{F}_K \in \mathcal{P}_p(\hat{K})$ by assumption, we have

$$ \|(\Pi_K v) \circ \mathcal{F}_K\|_{L^q(\hat{K})} \lesssim \|(\Pi_K v) \circ \mathcal{F}_K\|_{L^2(\hat{K})} $$
since $\mathcal{P}_p(\hat{K})$ is a finite dimensional space. We apply again the change of variable formula from (58), leading to

$$ \begin{gathered} \|(\Pi_K v) \circ \mathcal{F}_K\|_{L^2(\hat{K})} \lesssim h_K^{-d/2} \|\Pi_K v\|_{L^2(\hat{K})}, \\ \|\Pi_K v\|_{L^q(\hat{K})} \lesssim h^{d/q-d/2} \|\Pi_K v\|_{L^2(\hat{K})}. \end{gathered} \tag{67} $$

and all in all

We continue with the Poincaré inequality in (64), giving us that

$$ \|\Pi_K v\|_{L^2(\hat{K})} \le \|v\|_{L^2(\hat{K})} + \|v - \Pi_K v\|_{L^2(\hat{K})} \lesssim \|v\|_{L^2(\hat{K})} + h_K |v|_{H^1(\hat{K})}. $$

We now apply (61) to $v$ and each $\partial^\alpha v$ with $|\alpha| = 1$, yielding

$$ \|\Pi_K v\|_{L^2(\hat{K})} \le C_q h_K^{d/2-d/q} (\|v\|_{L^q(\hat{K})} + h_K |v|_{W^{1,q}(\hat{K})}), \tag{68} $$

which is possible since $q \ge 2$. Combining (67) and (68) proves (65).

We now turn to (66). For simplicity, we introduce the notation $\hat{g} := \nabla(\Pi_K v \circ \mathcal{F}_K) \in L^q(\hat{K})$. Due to (59), we have

$$ |\Pi_K v|_{W^{1,q}(\hat{K})} \lesssim h_K^{-1} h_K^{d/q} |\Pi_K v \circ \mathcal{F}_K|_{W^{1,q}(\hat{K})} = h_K^{-1} h_K^{d/q} \|\hat{g}\|_{L^q(\hat{K})}. $$

Since each component $\hat{g}$ is a polynomial of degree $p-1$, we can again apply a finite-dimensional argument to show that

$$ |\Pi_K v|_{W^{1,q}(\hat{K})} \lesssim h_K^{-1} h_K^{d/q} \|\hat{g}\|_{L^2(\hat{K})} = h_K^{-1} h_K^{d/q} |\Pi_K v \circ \mathcal{F}_K|_{H^1(\hat{K})} \lesssim h_K^{d/q-d/2} |\Pi_K v|_{H^1(\hat{K})} $$

where we employed (60) in the last estimate. Since $\Pi_K$ is the $H^1(\hat{K})$ orthogonal projection onto $\mathcal{P}(\hat{K})$, we now have

$$ |\Pi_K v|_{H^1(\hat{K})} \le |v|_{H^1(\hat{K})}, $$

and we conclude by applying (61) to each $\partial^\alpha v$, with $|\alpha| = 1$. $\square$

**Theorem A.10 (Error estimate).** Let $1 \le r \le +\infty$, $\max(2, r) \le q \le +\infty$ and $0 \le s \le p$. Assume that $K \in \mathcal{T}_h$ is $(s+1)$-regular, then

$$ h_K^{-1} \|v - \Pi_K v\|_{L^r(\hat{K})} + |v - \Pi_K v|_{W^{1,r}(\hat{K})} \lesssim h_K^{d/r-d/q} \left( \frac{h_K}{L} \right)^s \sum_{t=0}^s L^t |v|_{W^{t+1,q}(\hat{K})} \tag{69} $$

for all $v \in W^{s+1,q}(\hat{K})$, where the hidden constants depend on $p, r, q, s, M_K$ and $M_K(s+1)$. Proof. Since $\Pi_K$ is a projection onto $\mathcal{P}(\hat{K})$, we have

$$ \|v - \Pi_K v\|_{L^r(\hat{K})} = \|v - v_h - \Pi_K(v - v_h)\|_{L^r(\hat{K})} \le \|v - v_h\|_{L^r(\hat{K})} + \|\Pi_K(v - v_h)\|_{L^r(\hat{K})}, $$

and similarly

$$ |v - \Pi_K v|_{W^{1,r}(\hat{K})} \le |v - v_h|_{W^{1,r}(\hat{K})} + |\Pi_K(v - v_h)|_{W^{1,r}(\hat{K})}, $$

for all $v_h \in \mathcal{P}(\hat{K})$. It then follows from (65) and (66) that

$$ h_K^{-1} \|v - \Pi_K v\|_{L^r(\hat{K})} + |v - \Pi_K v|_{W^{1,r}(\hat{K})} \lesssim h_K^{-1} \|v - v_h\|_{L^r(\hat{K})} + |v - v_h|_{W^{1,r}(\hat{K})}. $$

We next apply (58) and (59) which give

$$ \|v - \Pi_K v\|_{L^r(\hat{K})} \lesssim h^{d/r} h_K^{-1} \left( \|\hat{v} - v_h \circ \mathcal{F}_K\|_{L^r(\hat{K})} + |\hat{v} - v_h \circ \mathcal{F}_K|_{W^{1,r}(\hat{K})} \right) $$
where $\hat{v} := v \circ \mathcal{F}_K$. Since $v_h \in \mathcal{P}_K$ was arbitrary, it follows that

$$ \|v - \Pi_K v\|_{L^r(K)} \lesssim h^{d/r} h_K^{-1} \min_{\hat{v}_h \in \mathcal{P}_p(\hat{K})} \left( \|\hat{v} - \hat{v}_h\|_{L^r(\hat{K})} + |\hat{v} - \hat{v}_h|_{W^{1,r}(\hat{K})} \right) $$

and it follows from [13, Theorem 3.1.1] that

$$ \|v - \Pi_K v\|_{L^r(K)} \lesssim h^{d/r} h_K^{-1} |\hat{v}|_{W^{s+1,q}(\hat{K})}. $$

At that point, (69) follows from (60). □

## A.6 Construction of the quasi-interpolation operator

We are almost ready to introduce a quasi-interpolation operator satisfying Theorem A.2. Before that, we need to recall some standard facts about the nodal basis of $V_h$.

There exists a finite set of nodes $\mathcal{D}_h \subset \bar{\Omega} \setminus \Gamma_D$ and an associated basis $\{\phi_a\}_{a \in \mathcal{D}_h}$ of $V_h$ such that

$$ \phi_a(\mathbf{b}) = \delta_{a,b} $$

for all $a, b \in \mathcal{D}_h$. These basis functions are obtained in each physical element from reference basis functions, meaning that there exists a finite set $\hat{\Phi} \subset \mathcal{P}_p(\hat{K})$ such that for each $a \in \mathcal{D}_h$ and $K \in \mathcal{T}_h$, we have

$$ \phi_a|_K = \hat{\phi} \circ \mathcal{F}_K \qquad (70) $$

for some $\hat{\phi} \in \hat{\Phi}$. The nodes in $\mathcal{D}_h$ are associated with vertices, edges or interiors of cells of the mesh $\mathcal{T}_h$, and the basis functions have local support.

We denote by $\mathcal{D}_h(K) := \mathcal{D}_h \cap K$ the set of nodes $a \in \mathcal{D}_h$ with non-vanishing basis function $\phi_a$ on $K$. In addition, for each $a \in \mathcal{D}_h$

$$ \mathcal{T}_h^a := \{K \in \mathcal{T}_h \mid a \in \mathcal{D}_h(K)\} $$

is the set of elements in the support of $\phi_a$, and we use the short hand notation $\sharp a$ for the cardinal of $\mathcal{T}_h^a$.

We are now ready to introduce our quasi-interpolation operator. It is an “Oswald” interpolation operator in the spirit of the seminal contribution [35]. Our quasi-interpolation operator $I_h: H_{\Gamma_D}^1(\Omega) \to V_h$, is defined by requiring that

$$ I_h(v)(a) = \frac{1}{\sharp a} \sum_{K \in \mathcal{T}_h^a} (\Pi_K v)(a) $$

for all degree of freedom nodes $a \in \mathcal{D}_h$. In fact, as we already alluded to, our analysis essentially extends the one in [19].

The following shorthand notation will be useful. We introduce the broken projection $I_h^b: H_{\Gamma_D}^1(\Omega) \to V_h^b$ by setting $I_h^b(v)|_K = \Pi_K(v)$ for all $K \in \mathcal{T}_h$.

**Lemma A.11 (Dofs mismatch).** Let $a \in \mathcal{D}_h$. Then, for all pairs $K_{\pm} \in \mathcal{T}_h^a$ of elements sharing a face $F = \partial K_+ \cap \partial K_-$, we have

$$ |(\Pi_{K_+} v)(a) - (\Pi_{K_-} v)(a)| \lesssim h_{K_+}^{1-d/2} |v - \Pi_{K_+} v|_{H^1(K_+)} + h_{K_-}^{1-d/2} |v - \Pi_{K_-} v|_{H^1(K_-)} \quad (71) $$

for all $v \in H_{\Gamma_D}^1(\Omega)$ with hidden constants depending on $M_{K_{\pm}}$. Besides, if $K \in \mathcal{T}_h$ and $a \in \mathcal{D}_h(K)$, we have

$$ |(\Pi_K v)(a) - (\Pi_{K'} v)(a)| \lesssim h_K^{1-d/2} |v - I_h^b(v)|_{H^1(\mathcal{T}_h^K)} \quad (72) $$

for all $K' \in \mathcal{T}_h^a$. Here, the hidden constant only depends on $M_{\mathcal{T}_h^K}$.
**Proof.** We first focus on (71), and the following notation will be useful. We let $\hat{F}_{\pm} := \mathcal{F}_{K_{\pm}}^{-1}(F) \in \hat{\mathcal{F}}$ and denote by $A_{\pm}$ the bijective affine maps between $\hat{F}_{d-1}$ and $\hat{F}_{\pm}$ used in (51). We further write $v_{\pm} := \Pi_{K_{\pm}}v \in \mathcal{P}(K_{\pm})$, $\hat{v}_{\pm} := v_{\pm} \circ \mathcal{F}_{K_{\pm}} \circ A_{\pm} \in L^2(\hat{F}_{d-1})$ and $\hat{\mathbf{a}} = (\mathcal{F}_{K_{\pm}} \circ A_{\pm})^{-1}(\mathbf{a}) \in \hat{F}_{d-1}$. We further point out that $\hat{v}_{\pm} \in \mathcal{P}_p(\hat{F}_{d-1})$ since $A_{\pm}$ are affine maps, and that the definition of $\hat{\mathbf{a}}$ is not ambiguous due to (51).

We start by writing that

$$|v_{+}(\mathbf{a}) - v_{-}(\mathbf{a})| = |\hat{v}_{+}(\hat{\mathbf{a}}) - \hat{v}_{-}(\hat{\mathbf{a}})| \leq \|\hat{v}_{+} - \hat{v}_{-}\|_{L^{\infty}(\hat{F}_{d-1})} \lesssim \|\hat{v}_{+} - \hat{v}_{-}\|_{L^2(\hat{F}_{d-1})}$$

where we used a standard inverse inequality on the face $\hat{F}_{d-1}$, which is possible since $\hat{v}_{\pm}$ belong to the finite-dimensional space $\mathcal{P}_p(\hat{F}_{d-1})$. We then introduce $\hat{v} := v \circ \mathcal{F}_{K_{\pm}} \circ A_{\pm} \in L^2(\hat{F}_{d-1})$ and we write that

$$\|\hat{v}_{+} - \hat{v}_{-}\|_{L^2(\hat{F}_{d-1})} = \|\hat{v}_{+} - \hat{v} + (\hat{v} - \hat{v}_{-})\|_{L^2(\hat{F}_{d-1})} \leq \|\hat{v}_{+} - \hat{v}\|_{L^2(\hat{F}_{d-1})} + \|\hat{v} - \hat{v}_{-}\|_{L^2(\hat{F}_{d-1})}.$$

We can now write

$$\|\hat{v} - \hat{v}_{\pm}\|_{L^2(\hat{F}_{d-1})} \lesssim \|(v - v_{\pm}) \circ \mathcal{F}_{K_{\pm}}\|_{L^2(\hat{F}_{\pm})}$$

since the map $A_{\pm}$ is affine and invertible and only depends on the choice of reference simplices. A trace inequality on the reference $d$-simplex then gives that

$$\|\hat{v} - \hat{v}_{\pm}\|_{L^2(\hat{F}_{d-1})} \lesssim \|(v - v_{\pm}) \circ \mathcal{F}_{K_{\pm}}\|_{L^2(\hat{K})} + |(v - v_{\pm}) \circ \mathcal{F}_{K_{\pm}}|_{H^1(\hat{K})},$$

and we conclude with (58), (60) and (64).

We now prove (72). If $K = K'$, there is nothing to show, and if $K$ and $K'$ share a face, then (72) is a direct consequence of (71). In the general case, however, we might only have $K \cap K' = \mathbf{a}$ if $\mathbf{a}$ is vertex of $\mathcal{T}_h$. Nevertheless, in this case, there exists a chain of $n$ elements $K = K_1, \dots, K_n = K'$ that are all connected by a face, and (72) then follows from the triangular inequality

$$|\Pi_K v(\mathbf{a}) - (\Pi_{K'} v)(\mathbf{a})| \leq \sum_{j=1}^{n-1} |(\Pi_{K_{j+1}} v)(\mathbf{a}) - (\Pi_{K_j} v)(\mathbf{a})|$$

by applying (71) to each summand and observing that $h_{K_j} \lesssim h_K$ for $j = 1, \dots, n$ due to (56)

<figure><img src="image_10.png" /></figure>

**Lemma A.12 (Basis function scaling).** For all $\mathbf{a} \in \mathcal{D}_h$ and $K \in \mathcal{T}_h$, we have

$$\|\phi_{\mathbf{a}}\|_{L^r(K)} + h_K |\phi_{\mathbf{a}}|_{W^{1,r}(K)} \lesssim h_K^{d/r} \quad (73)$$

where the hidden constant depends on $p$ and $M_K$.

**Proof.** It is a simple combination of (70) and the change of coordinates formulas (58) and (59).

<figure><img src="image_11.png" /></figure>

**Theorem A.13 (Error in conformity enforcement).** Let $v \in H_{\Gamma_D}^1(\Omega)$. Then, for all $K \in \mathcal{T}_h$, we have

$$h_K^{-1} \| I_h^{\mathfrak{b}}(v) - I_h(v) \|_{L^r(K)} + |I_h^{\mathfrak{b}}(v) - I_h(v)|_{W^{1,r}(K)} \lesssim h_K^{d/r-d/2} |v - I_h^{\mathfrak{b}}(v)|_{H^1(\mathcal{T}_h^K)} \quad (74)$$

with a hidden constant solely depending on $M_{\mathcal{T}_h^K}$.
Proof. Let us first observe that for a fixed $K \in \mathcal{T}_h$, we have

$$ \begin{aligned} \left. (I^\flat(v) - I_h(v)) \right|_K &= \sum_{\mathbf{a} \in \mathcal{D}_h(K)} \left\{ (\Pi_K v)(\mathbf{a}) - \frac{1}{\sharp \mathbf{a}} \sum_{K' \in \mathcal{T}_h^\mathbf{a}} (\Pi_{K'} v)(\mathbf{a}) \right\} \phi_{\mathbf{a}}|_K \\ &= \sum_{\mathbf{a} \in \mathcal{D}_h(K)} \frac{1}{\sharp \mathbf{a}} \sum_{K' \in \mathcal{T}_h^\mathbf{a}} \{ (\Pi_K v)(\mathbf{a}) - (\Pi_{K'} v)(\mathbf{a}) \} \phi_{\mathbf{a}}|_K, \end{aligned} $$

so that

$$ \| I_h^\flat(v) - I_h(v) \|_{L^r(K)} \le \sum_{\mathbf{a} \in \mathcal{D}_h(K)} \sum_{K' \in \mathcal{T}_h^\mathbf{a}} |(\Pi_K v)(\mathbf{a}) - (\Pi_{K'} v)(\mathbf{a})| \|\phi_{\mathbf{a}}\|_{L^r(K)} $$

and

$$ |I_h^\flat(v) - I_h(v)|_{W^{1,r}(K)} \le \sum_{\mathbf{a} \in \mathcal{D}_h(K)} \sum_{K' \in \mathcal{T}_h^\mathbf{a}} |(\Pi_K v)(\mathbf{a}) - (\Pi_{K'} v)(\mathbf{a})| |\phi_{\mathbf{a}}|_{W^{1,r}(K)}. $$

It then follows from (73) that

$$ h_K^{-1} \| I_h^\flat(v) - I_h(v) \|_{L^r(K)} + |I_h^\flat(v) - I_h(v)|_{W^{1,r}(K)} \lesssim h_K^{d/r-1} \sum_{\mathbf{a} \in \mathcal{D}_h(K)} \sum_{K' \in \mathcal{T}_h^\mathbf{a}} |(\Pi_K v)(\mathbf{a}) - (\Pi_{K'} v)(\mathbf{a})|, $$

and (74) follows from (72) and Lemma A.4. □

We are now ready to establish our main result.

Proof of Theorem A.2. We first observe that (55) follows from (54) due to Lemma A.4 by observing that

$$ \begin{aligned} \sum_{K \in \mathcal{T}_h} |\phi|_{W^{t+1,q}(\mathcal{T}_h^K)}^q &= \sum_{K \in \mathcal{T}_h} \sum_{\substack{K' \in \mathcal{T}_h \\ K' \cap K \neq \emptyset}} |\phi|_{W^{t+1,q}(K')}^q \\ &= \sum_{K' \in \mathcal{T}_h} \sum_{\substack{K \in \mathcal{T}_h \\ K \cap K' \neq \emptyset}} |\phi|_{W^{t+1,q}(K')}^q \le N |\phi|_{W^{t+1,q}(\mathcal{T}_h)}^q \lesssim |\phi|_{W^{t+1,q}(\mathcal{T}_h)}^q \end{aligned} $$

for all $t \ge 0$, $q \ge 2$ and $\phi \in W^{t+1,q}(\mathcal{T}_h)$.

We therefore fix $K \in \mathcal{T}_h$ and establish (54). We start with the triangular inequality

$$ h_K^{-1} \|v - I_h(v)\|_{L^r(K)} + |v - I_h(v)|_{W^{1,r}(K)} \le h_K^{-1} \|v - I_h^\flat(v)\|_{L^r(K)} + |v - I_h^\flat(v)|_{W^{1,r}(K)} \\ + h_K^{-1} \|I_h^\flat(v) - I_h(v)\|_{L^r(K)} + |I_h^\flat(v) - I_h(v)|_{W^{1,r}(K)}. $$

Since $v - I_h^\flat(v)|_K = v - \Pi_K v$, we can immediately apply (69) for the corresponding terms. For the other terms, we involve (74), to write

$$ h_K^{-1} \| I_h^\flat(v) - I_h(v) \|_{L^r(K)} + |I_h^\flat(v) - I_h(v)|_{W^{1,r}(K)} \lesssim h_K^{d/r-d/2} |v - I_h^\flat(v)|_{H^1(\mathcal{T}_h^K)}. $$

At that point, (54) follows from (69). □