# Guaranteed stability bounds for second-order PDE problems satisfying a Garding inequality

Théophile Chaumont-Frelet

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Théophile Chaumont-Frelet. Guaranteed stability bounds for second-order PDE problems satisfying a Garding inequality. 2026. hal-05440306

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# GUARANTEED STABILITY BOUNDS FOR SECOND-ORDER PDE PROBLEMS SATISFYING A GÅRDING INEQUALITY*

T. CHAUMONT-FRELET*

**ABSTRACT.** We propose an algorithm to numerically determine whether a second-order linear PDE problem satisfying a Gårding inequality is well-posed. This algorithm further provides a lower bound to the inf-sup constant of the weak formulation, which may in turn be used for a posteriori error estimation purposes. Our numerical lower bound is based on two discrete singular value problems involving a Lagrange finite element discretization coupled with an a posteriori error estimator based on flux reconstruction techniques. We show that if the finite element discretization is sufficiently rich, our lower bound underestimates the optimal constant only by a factor roughly equal to two.

## 1. INTRODUCTION

Linear boundary value problems with indefinite weak formulations arise in many important applications including convection-dominated diffusion and time-harmonic wave propagation problems. In such cases, it is not always known whether the problem is well-posed. Besides, even in cases where well-posedness is guaranteed, the magnitude of the stability constant controlling the norm of the solution in terms of the norm of the right-hand side is often unknown. In this work, we provide a numerical algorithm that can certify that the boundary value problem under consideration is well-posed, and provide a guaranteed upper bound on its stability constant.

We focus on second-order PDE problems of the form: Given $f: \Omega \to \mathbb{C}$, find $u: \Omega \to \mathbb{C}$ such that

$$ (1.1) \qquad \left\{ \begin{array}{ll} -k^2 du + ikc \cdot \nabla u - \nabla \cdot (ikbu + \underline{\mathbf{A}}\nabla u) & = f \quad \text{in } \Omega, \\ u & = 0 \quad \text{on } \Gamma_D, \\ (ikbu + \underline{\mathbf{A}}\nabla u) \cdot \mathbf{n} & = 0 \quad \text{on } \Gamma_N, \end{array} \right. $$

where $\underline{\mathbf{A}}, \mathbf{b}, \mathbf{c}$ and $d$ are piecewise constant complex-valued coefficients, and $\Omega \subset \mathbb{R}^n$ is a bounded domain with $n = 2$ or 3. The real number $k > 0$ and the complex unit $i$ are conventionally introduced to make the PDE coefficients physically dimensionless, whereby the dimension of $k$ is the reciprocal of a length. This convention is especially natural for time-harmonic wave propagation problems where $k$ is the wavenumber, and the coefficients describe the material properties of the propagation medium. For convection-dominated diffusion problems only involving real-valued coefficients, the proposed algorithm may be run employing only real (floating point) numbers.

We demand that the weak formulation of (1.1) satisfies a Gårding inequality as stated precisely in (2.4) below. This is for instance always true if the matrix-coefficient $\underline{\mathbf{A}}$ satisfies the positivity property

$$ \mathrm{Re} \underline{\mathbf{A}}(\mathbf{x}) e \cdot \bar{e} \geq \alpha_{\star} > 0 $$

<footnote>*Inria, Univ. Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé</footnote>
for a.e. $\boldsymbol{x}$ in $\Omega$ and all unit vector $e \in \mathbb{C}^d$. Under this assumption, we propose an algo-
rithm that provides a guaranteed lower bound $\gamma_h$ to the inf-sup constant of the sesquilinear
form $\beta(\cdot, \cdot)$ associated with (1.1). If (1.1) is well-posed, we show that $\gamma_h > 0$ whenever the fi-
nite element space employed in the algorithm is sufficiently rich. This numerically guarantees
the well-posedness of (1.1), and leads to upper bounds for the norm of the operator mapping
$f$ to $u$ in natural norms.

For simplicity, we assume that the coefficients are piecewise constant onto a polytopal partition and that the domain and the boundary partition are polytopal. However, we do not make any regularity assumptions, meaning that the geometry described by the domain and coefficients can include sharp edges and corners.

The algorithm is based on two discrete singular value problems arising from a finite element discretization. More specifically, a Lagrange finite element discretization of (1.1) is combined with an a posteriori error estimator based on a flux reconstruction technique [6, 13, 15]. If the problem under consideration is well-posed, it is guaranteed that the algorithm provides an upper bound for the stability constant, provided that the finite element space is sufficiently rich. In fact, we show that as soon as the finite element space provides reasonable approximate solutions to (1.1), the overestimation on the stability constant does not exceed roughly a factor two. Furthermore, the overestimation is independent of the polynomial degree of the finite element space. This is key for time-harmonic wave propagation problems, where high-order discretization are often drastically more performant [2, 10, 11, 23].

Besides their independent interest, guaranteed estimations of the inf-sup constant are cru-
cial in error certification, as they enter a posteriori error estimates [8, 14, 28]. As a result,
the present result may be combined with existing error estimators to provide fully-guaranteed
error bounds when (1.1) is discretized by finite elements.

The problem under consideration here has already been tackled in the literature with related ideas, see [30] and the references therein. However, to the best of the author’s knowledge, these works all require explicit regularity shifts for the principle part of the PDE operator. In practice, this restricts the setting to convex domains with $\mathbf{A} = \mathbf{I}$, or to domains with smooth boundaries [24, Section 6.2.7]. Furthermore, the bounds obtained are not necessarily efficient, especially for high-order finite element discretizations. In contrast, we employ here a polynomial-degree-robust a posteriori error estimator which allows us to work in a general setting where regularity shifts are not available or not explicit, and to fully exploit the power of high-order finite elements.

Another recent work similar to the present one is [22], where a discrete eigenvalue problem involving an a posteriori error estimator based on flux reconstruction techniques is employed. However, [22] only focuses on self-adjoint problems, and does not show that the proposed lower bound is efficient. Besides, polynomial-degree-robustness properties have not been analyzed in [22].

We finally mention that for self-adjoint problems, cheaper algorithms based on non-conforming or mixed finite element discretization are available, see e.g. [7, 19]. However, it is not clear that such techniques may be bridged to the present context.

The remainder of this work is organized as follows. Section 2 introduces key notation, make the assumptions on (1.1) precise, and collects useful results from the literature. In Section 3 we present our computational algorithm and establish our guaranteed lower bound. Finally, Section 4 is dedicated to the efficiency of the algorithm, whereby we show that our numerical inf-sup lower bound cannot arbitrarily underestimate the optimal one.
## 2. NOTATION, ASSUMPTIONS AND TOOLS

**2.1. Complex numbers.** Classically, we denote by $\mathbb{R}$ and $\mathbb{C}$ the fields of real and complex numbers. The notation $\mathbb{R}^d$ (resp. $\mathbb{C}^d$) and $\mathbb{R}^{d \times d}$ (resp. $\mathbb{C}^{d \times d}$) are used for vectors and matrices with real (resp. complex) coefficients. If $z \in \mathbb{C}$, $z_r = \operatorname{Re} z$ and $z_i = \operatorname{Im} z$ respectively denote the real and imaginary parts of $z$. $z_\dagger$ is its complex conjugate and $|z|$ its modulus. For a vector $z \in \mathbb{C}^d$, $z_\dagger$ is its component-wise complex conjugate, and $|z|$ is its $\ell^2(\mathbb{C}^3)$ norm. Finally, if $\underline{Z} \in \mathbb{C}^{d \times d}$ is a matrix, $\underline{Z}_r$ and $\underline{Z}_i$ are its component-wise real and imaginary parts. Finally $\underline{Z}_\dagger$ is the adjoint of $\underline{Z}$, i.e., the entries of $\underline{Z}_\dagger$ are the complex conjugate of the ones of the transpose of $\underline{Z}$.

**2.2. Domain and coefficients.** Throughout this work, $\Omega \subset \mathbb{R}^d$ is a weakly Lipschitz polytopal domain. The boundary $\partial\Omega$ of $\Omega$ is split into two disjoint relatively open polytopal subsets $\Gamma_D$ and $\Gamma_N$ in such way that $\partial\Omega = \bar{\Gamma}_D \cup \bar{\Gamma}_N$.

We consider coefficients $\underline{A} : \Omega \to \mathbb{C}^{d \times d}$, $b, c : \Omega \to \mathbb{C}^d$ and $d : \Omega \to \mathbb{C}$ that are piecewise constant on a polytopal partition of $\Omega$. Specifically, there exists a finite set $\mathcal{Q}$ of disjoint open polytopal subsets of $\Omega$ with $\bar{\Omega} = \cup_{Q \in \mathcal{Q}} \bar{Q} \bar{Q}$ such that for all $Q \in \mathcal{Q}$, there exist constants $\underline{A}_Q \in \mathbb{C}^{d \times d}$, $b_Q, c_Q \in \mathbb{C}^d$ and $d_Q \in \mathbb{C}$ such that

$$ (2.1) \qquad \underline{A}(\boldsymbol{x}) = \underline{A}_Q, \quad \boldsymbol{b}(\boldsymbol{x}) = \boldsymbol{b}_Q, \quad \boldsymbol{c}(\boldsymbol{x}) = \boldsymbol{c}_Q, \quad \boldsymbol{d}(\boldsymbol{x}) = \boldsymbol{d}_Q $$

for all $\boldsymbol{x} \in \mathcal{Q}$.

We denote by $\ell$ the diameter of $\Omega$ and let $\hat{\Omega} := (1/\ell)\Omega$. Throughout the manuscript, $c(\hat{\Omega})$ denote a constant, that can change from one occurrence to the other, that only depend on $\hat{\Omega}$.

**2.3. Function spaces.** For an open set $U \subset \Omega$ with Lipschitz boundary, we denote by $L^2(U)$ the Lebesgue space of (complex-valued) square-integrable functions defined on $U$, and we let $L^2(U) := [L^2(U)]^d$. The inner products of both spaces are denoted by $(\cdot, \cdot)_U$. For measurable weights $w : U \to \mathbb{R}$ and $\underline{W} : U \to \mathbb{R}^{d \times d}$, we introduce $\|v\|_{w,U} := \sqrt{(wv, v)_U}$ and $\|v\|_{\underline{W},U} := \sqrt{(\underline{W}v, v)_U}$ for all $v \in L^2(U)$ and $v \in L^2(U)$. When $w$ is uniformly away bounded from 0 and $+\infty$, $\| \cdot \|_{w,U}$ is equivalent to the standard norm on $L^2(U)$. Similarly, if $\underline{W}$ is symmetric and uniformly bounded from above and below in the sense of quadratic forms, then $\| \cdot \|_{\underline{W},U}$ is equivalent to standard norm of $L^2(U)$.

The notation $\underline{H}^1(U)$ is used for the standard Sobolev space of functions $v \in L^2(U)$ such that $\nabla v \in L^2(U)$, where $\nabla v$ is the gradient defined in the sense of distributions. If $\gamma \subset \partial U$ is a relatively open subset of the boundary, then $H_\gamma^1(U)$ collects functions of $H^1(U)$ with vanishing traces on $\gamma$.

We refer the reader to [1] for more details on Lebesgue and Sobolev spaces.

We will finally employ the vector Sobolev space $\underline{H}(\operatorname{div}, U)$ of vector fields $v \in L^2(U)$ with weak divergence $\nabla \cdot v \in L^2(U)$, see e.g. [20]. As above, $\underline{H}_\gamma(\operatorname{div}, U)$ is the subset of $\underline{H}(\operatorname{div}, U)$ consisting of vector fields with vanishing normal trace on $\gamma$, as per [17].

**2.4. Sesquilinear form.** We use the notation $\beta : H_{\Gamma_D}^1(\Omega) \times H_{\Gamma_D}^1(\Omega) \to \mathbb{C}$ for the sesquilinear form associated with the weak formulation of (1.1). It is given by

$$ (2.2) \qquad \beta(u, v) := (-k^2 du + ikc \cdot \nabla u, v)_\Omega + (ikbu + \underline{A}\nabla u, \nabla v)_\Omega $$

for all $u, v \in H_{\Gamma_D}^1(\Omega)$. For simplicity, we record here that we equivalently write that

$$ (2.3) \qquad \beta(u, v) = (u, -k^2 d_\dagger v - ikb_\dagger \cdot \nabla v)_\Omega + (\nabla u, \underline{A}_\dagger \nabla v - ikc_\dagger v)_\Omega. $$
For Helmholtz problems without convection, we have $b = c = o$. In addition, $d$ and $\mathbf{A}$ are real-valued and positive in the majority of the domain. These coefficients can have a non-zero imaginary part in parts of the domain containing absorbing materials, or if a radiation condition has been approximated by a perfectly matched layer [4].

**2.5. Gårding inequality.** The key assumption we make throughout this work is that the sesquilinear form $\beta$ is coercive up to compact perturbation. Specifically, we assume that there exist weights $\mathbf{m}, \mathbf{p}: \Omega \to \mathbb{R}$ and $\mathfrak{A}: \Omega \to \mathbb{R}^{d \times d}$ such that the Gårding inequality

$$ (2.4) \qquad \mathrm{Re} \beta(u, u) \geq \|u\|_{\Omega}^2 - 2k^2 \|u\|_{\mathbf{p}, \Omega}^2 $$

holds true with

$$ (2.5) \qquad \|u\|_{U}^{2} := k^{2}\|u\|_{\mathbf{m},U}^{2} + \|\nabla u\|_{\mathfrak{A},U}^{2}, \quad U \subset \Omega. $$

Here, it is assumed that the three weights are piecewise constant on the partition $\mathcal{Q}$ as per (2.1), that $\mathbf{p} \ge 0$, that $\mathbf{m} > 0$ and that $\mathfrak{A} > 0$ in the sense of quadratic forms. We also assume for simplicity that $\mathbf{p} \neq 0$.

For Helmholtz problems, we can take $\mathbf{p} = \mathbf{m} = d_r$ and $\mathfrak{A} = \mathbf{A}_r$.

**2.6. Computational mesh.** We consider a mesh $\mathcal{T}_h$ of the domain $\Omega$ consisting of (open) simplicial elements $K$. We assume that the mesh is matching, meaning that the intersection $\bar{K}_+ \cap \bar{K}_-$ of two distinct elements $K_\pm \in \mathcal{T}_h$ is either empty, or a full subsimplex (vertex, edge or face) of both elements. We demand that the mesh is conforming, meaning that the union of the elements cover the domain. We further require that the coefficients are constant in each element. We also finally denote by $\mathcal{F}_h$ the set of mesh faces, and require that every boundary face either entirely belong to $\Gamma_D$ or to $\Gamma_N$.

For an element $K \in \mathcal{T}_h$, $h_K$ is a diameter of $K$ and $\rho_K$ is the diameter of the largest ball contained in $\bar{K}$. Then, $\kappa_K := h_K/\rho_K \ge 1$ denote the shape regularity parameter of $K$, and $\kappa := \max_{K \in \mathcal{T}_h} \kappa_K$.

We will often employ the notation $c(\kappa)$ for a constant, which may differ at each occurrence, only depending only on $\kappa$.

**2.7. Wavespeed.** For $K \in \mathcal{T}_h$, we denote by $m_K := \mathbf{m}|_K$ and $p_K := \mathbf{p}|_K$ the (constant) restrictions to $\mathbf{m}$ and $\mathbf{p}$ to $K$. Similarly, $\alpha_K^b$ and $\alpha_K^\sharp$ denote the smallest and largest eigenvalues of $\mathfrak{A}|_K$. We finally write

$$ (2.6) \qquad \vartheta_K := \sqrt{\frac{p_K}{\alpha_K^\sharp}} $$

for the “wavespeed” in the element $K$.

**2.8. Polynomial spaces.** If $K \in \mathcal{T}_h$ is simplex and $r \ge 0$, we denote by $\mathcal{P}_r(K)$ the set of (complex-valued) polynomials defined on $K$ of degree less than or equal to $r$, and we set $\mathcal{P}_r(K) := [\mathcal{P}_r(K)]^d$. We will also need the Raviart–Thomas polynomial space defined by $RT_r(K) := \mathcal{P}_r(K) + x\mathcal{P}_r(K)$, see [25, 27]. If $\mathcal{T} \subset \mathcal{T}_h$ is a set of elements, we write $\mathcal{P}_r(\mathcal{T})$, $\mathcal{P}_r(\mathcal{T})$ and $RT_r(\mathcal{T})$ for functions whose restriction to each $K \in \mathcal{T}$ respectively belong to $\mathcal{P}_r(K)$, $\mathcal{P}_r(K)$ and $RT_r(K)$. Note that these spaces do not embed any compatibility conditions.
2.9. **Finite element spaces.** Throughout, we fix a polynomial degree $p \ge 1$ and consider the Lagrange finite element space $V_h := \mathcal{P}_p(\mathcal{T}_h) \cap H_\Gamma^1(\Omega)$. We will also need an auxiliary space of (discontinuous) piecewise polynomials. Specifically, we fix $q \ge 0$ and let $Q_h := \mathcal{P}_q(\mathcal{T}_h)$. In practice, we could build $Q_h$ and $V_h$ on different partitions of the mesh, but for simplicity, we do not. We also note that most of the proposed analysis is carried out with the case $q = 0$ in mind, irrespectively of the value of $p$.

2.10. **Projection.** For $\theta \in L^2(\Omega)$, we denote by $\pi_h\theta \in Q_h$ the orthogonal projection defined by

$$ (\pi_h\theta, r_h)_\Omega = (\theta, r_h)_\Omega $$

for all $r_h \in Q_h$. Classically, this projection is in fact defined elementwise, and we have that

$$ (2.7) \qquad \|\theta - \pi_h\theta\|_K \le \frac{h_K}{\pi} \|\nabla\theta\|_K, $$

whenever $\theta \in H^1(K)$, see e.g. [3]. Applying (2.7) elementwise then gives

$$ (2.8) \qquad k\|u - \pi_h u\|_{\mathfrak{P},\Omega} \le \frac{k\mathfrak{h}}{\pi\mathfrak{v}} \|\nabla u\|_{\mathfrak{A},\Omega} $$

for all $u \in H^1(\Omega)$, where $\mathfrak{h} := h_{K_*}$ and $\mathfrak{v} := \vartheta_{K_*}$ for (one of) the element(s) $K_* \in \mathcal{T}_h$ such that

$$ \frac{h_{K_*}}{\vartheta_{K_*}} = \max_{K \in \mathcal{T}_h} \frac{h_K}{\vartheta_K}. $$

Finally, because $\mathfrak{m}$ is piecewise constant, $\pi_h$ is also an orthogonal projection in the $\mathfrak{m}$-weighted $L^2(\Omega)$ inner-product, and we have

$$ (2.9) \qquad \|\pi_h\theta\|_{\mathfrak{m},\Omega} \le \|\theta\|_{\mathfrak{m},\Omega}. $$

### 3. GUARANTEED INF-SUP LOWER BOUND

We are now ready to describe our algorithm. It relies on the fact that the finite element discretization with the space $V_h$ to (1.1) is well-posed, and is based on two discrete singular value problems involving the space $Q_h$.

3.1. **Discrete Solution operator.** We assume that for all $\theta_h \in Q_h$, there exists a unique $\mathcal{P}_h\theta_h \in V_h$ such that

$$ (3.1) \qquad \beta(w_h, \mathcal{P}_h(\theta_h)) = k^2(\mathfrak{p}w_h, \theta_h)_\Omega $$

for all $w_h \in V_h$. We then introduce

$$ \Theta_h := \max_{\substack{\theta_h \in Q_h \\ k\|\theta_h\|_{\mathfrak{m}}=1}} \|\mathcal{P}_h(\theta_h)\|_{\Omega}. $$

The constant $\Theta_h$ can be computed as the solution to matrix singular value problem. In practice, $\Theta_h$ is not exactly computable, but guaranteed upper bound of arbitrary accuracy may be numerically evaluated, see [24, Chapter 12]. As we will see $\Theta_h$ is the key ingredient of our inf-sup lower bound. Specifically, $1/(1 + 2\Theta_h)$ is a satisfactory bound if the mesh is sufficiently fine.
3.2. **Error estimator.** We rely on a posteriori error estimation to detect whether the mesh is sufficiently fine to trust the bound based on $\Theta_h$. We will call a flux reconstruction any linear map $\mathcal{F}_h: Q_h \to H_{\Gamma_N}(\mathrm{div}, \Omega)$ such that

$$ (3.2) \qquad \nabla \cdot \mathcal{F}_h(\theta_h) = k^2 \mathfrak{p} \theta_h + k^2 d_\dagger \mathcal{P}_h(\theta_h) + ik \mathbf{b}_\dagger \cdot \nabla \mathcal{P}_h(\theta_h) $$

for all $\theta_h \in Q_h$. For shortness, we also introduce

$$ \mathcal{R}_h(\theta_h) := \underline{\mathbf{A}}_\dagger \nabla \mathcal{P}_h(\theta_h) - ik \mathbf{c}_\dagger \mathcal{P}_h(\theta_h) + \mathcal{F}_h(\theta_h), $$

and

$$ (3.3) \qquad \rho_h := \max_{\substack{\theta_h \in Q_h \\ \|\theta_h\|_{\mathfrak{P}}=1}} \|\mathcal{R}_h(\theta_h)\|_{\mathfrak{A}^{-1}, \Omega}. $$

As for $\Theta_h$, the constant $\rho_h$ can be computed (or at least, rigorously estimated from above) via the numerical solution of a discrete singular value problem.

3.3. **Lower bound.** Our numerical algorithm simply amounts to computing $\Theta_h$ and $\rho_h$. As we now establish, these two constants may be combined in a simple algebraic expression to provide a lower bound to the inf-sup constant of $\beta$. The proposed algorithm works for any choice of flux reconstruction $\mathcal{F}$ satisfying (3.2). A possible construction will be given in Section 4.4 below.

We start with a Prager-Synge type estimate. This result is standard, see e.g. [15, 26, 29], but have not been established for the particular setting considered here, in particular since the matrix coefficient $\underline{\mathbf{A}}$ is complex-valued. We therefore include a proof for completeness.

**Lemma 3.1 (Control of the residual).** For all $\theta_h \in Q_h$, the estimate

$$ \max_{\substack{w \in H_{\Gamma_D}^1(\Omega) \\ \|\nabla w\|_{\mathfrak{A}, \Omega}=1}} |k^2(\mathfrak{p}w, \theta_h)_{\Omega} - \beta(w, \mathcal{P}_h(\theta_h))| \le \|\mathcal{R}(\theta_h)\|_{\mathfrak{A}^{-1}, \Omega} $$

holds true. In particular, we have

$$ (3.4) \qquad \max_{\substack{\theta_h \in Q_h \\ k\|\theta_h\|_{\mathfrak{P},\Omega}=1}} \max_{\substack{w \in H_{\Gamma_D}^1(\Omega) \\ \|\nabla w\|_{\mathfrak{A},\Omega}=1}} |k^2(\mathfrak{p}w, \theta_h)_{\Omega} - \beta(w, \mathcal{P}_h(\theta_h))| \le \rho_h. $$

*Proof.* Fix $\theta_h \in Q_h$ and let $u_h := \mathcal{P}_h(\theta_h)$, $\sigma_h := \mathcal{F}_h(\theta_h)$. In view of (2.3) and (3.2), we have

$$ \begin{aligned} k^2(\mathfrak{p}w, \theta_h)_{\Omega} - \beta(w, u_h) &= (w, k^2 \mathfrak{p} \theta_h + k^2 d_\dagger u_h + ik \mathbf{b}_\dagger \cdot \nabla u_h)_{\Omega} - (\nabla w, \underline{\mathbf{A}}_\dagger \nabla u_h - ik \mathbf{c}_\dagger u_h)_{\Omega} \\ &= (w, \nabla \cdot \sigma_h)_{\Omega} - (\nabla w, \underline{\mathbf{A}}_\dagger \nabla u_h - ik \mathbf{c}_\dagger u_h)_{\Omega} \\ &= -(\nabla w, \underline{\mathbf{A}}_\dagger \nabla u_h - ik \mathbf{c}_\dagger u_h + \sigma_h)_{\Omega}, \end{aligned} $$

where use integration by part in the last identity. We conclude with a Cauchy-Schwarz inequality that

$$ \begin{aligned} |k^2(\mathfrak{p}w, \theta_h)_{\Omega} - \beta(w, u_h)| &= |(\mathfrak{A}\nabla w, \mathfrak{A}^{-1}(\underline{\mathbf{A}}_\dagger \nabla u_h - ik\mathbf{c}_\dagger u_h + \sigma_h))_{\Omega}| \\ &\le \|\nabla w\|_{\mathfrak{A},\Omega} \|\mathfrak{A}^{-1}(\underline{\mathbf{A}}_\dagger \nabla u_h - ik\mathbf{c}_\dagger u_h + \sigma_h)\|_{\mathfrak{A},\Omega} \\ &= \|\nabla w\|_{\mathfrak{A},\Omega} \|\underline{\mathbf{A}}_\dagger \nabla u_h - ik\mathbf{c}_\dagger u_h + \sigma_h\|_{\mathfrak{A}^{-1},\Omega}, \end{aligned} $$

from which the conclusion follows. $\square$

We now establish our guaranteed lower bound for the inf-sup constant of $\beta$.
**Theorem 3.2** (Guaranteed bounds). *The lower bound*

$$
(3.5) \qquad \operatorname{Re} \beta(u, u + 2 \mathcal{P}_h(\pi_h u)) \ge \left\{ 1 - 2 \left( \frac{kh}{v} \right)^2 - 2\rho_h \right\} \|u\|_{\Omega}^2
$$

holds true for all $u \in H_{\Gamma_D}^1(\Omega)$. In particular

$$
(3.6) \qquad \min_{\substack{u \in H_{\Gamma_D}^1(\Omega) \\ \|u\|_{\Omega}=1}} \max_{\substack{v \in H_{\Gamma_D}^1(\Omega) \\ \|v\|_{\Omega}=1}} \operatorname{Re} \beta(u, v) \ge \gamma_h,
$$

with

$$
(3.7) \qquad \gamma_h := \left\{ 1 - 2 \left( \frac{kh}{v} \right)^2 - 2\rho_h \right\} \frac{1}{1 + 2\Theta_h}.
$$

*Proof.* Considering an arbitrary $u \in H_{\Gamma_D}^1(\Omega)$, we start with the Gårding inequality stated in (2.4), namely

$$
\operatorname{Re} \beta(u, u) \geq \|u\|_{\Omega}^{2} - 2k^{2} \|u\|_{p, \Omega}^{2}.
$$

We then use (2.8), showing that

$$
k^2 \|u\|_{\mathbf{p}, \Omega}^2 = k^2 \|\pi_h u\|_{\mathbf{p}, \Omega}^2 + k^2 \|u - \pi_h u\|_{\mathbf{p}, \Omega}^2 \le \left(\frac{kh}{\pi v}\right)^2 \|u\|_{\Omega}^2 + k^2 \|\pi_h u\|_{\mathbf{p}, \Omega}^2
$$

from which we infer

$$
(3.8) \qquad \operatorname{Re} \beta(u, u) \ge \left\{ 1 - 2 \left( \frac{kh}{\pi v} \right)^2 \right\} \|u\|_{\Omega}^2 - 2k^2 \|\pi_h u\|_{\mathbf{p}, \Omega}^2.
$$

We now invoke (3.4), which allows to write that

$$
\begin{align*}
|k^2(\mathfrak{p}w, \pi_h u)_\Omega - \beta(w, \mathcal{P}_h(\pi_h u))| &\le \rho_h \|\nabla w\|_{\mathfrak{a},\Omega} k \|\pi_h u\|_{\mathfrak{m},\Omega} \\
&\le \rho_h \|\nabla w\|_{\mathfrak{a},\Omega} k \|u\|_{\mathfrak{m},\Omega} \le \rho_h \|w\|_{\Omega} \|u\|_{\Omega},
\end{align*}
$$

for all $w \in H_{\Gamma_D}^1(\Omega)$ and from which we deduce that

$$
(3.9) \qquad \operatorname{Re} \beta(u, \mathcal{P}_h(\pi_h u)) \geq k^2 \|\pi_h u\|_{\mathbf{p},\Omega}^2 - \rho_h \|u\|_{\Omega}^2.
$$

At this point (3.5) follows by adding twice (3.9) to (3.8), since these estimates holds for all
$u \in H_{\Gamma_D}^1(\Omega)$.

To establish (3.6) from (3.5), we first fix $u \in H_{\Gamma_D}^1(\Omega)$ and observe that picking $v^* := u + 2\mathcal{P}_h(\pi_h u)$, we have

$$
\max_{\substack{v \in H_{\Gamma_D}^1(\Omega) \\ \|v\|_{\Omega}=1}} \operatorname{Re} \beta(u, v) \ge \frac{1}{\|v^*\|_{\Omega}} \operatorname{Re} \beta(u, v^*) \ge \frac{1}{\|v^*\|_{\Omega}} \left\{ 1 - 2 \left( \frac{kh}{v} \right)^2 - 2\rho_h \right\} \|u\|_{\Omega}^2
$$

Then the desired estimate follows from (3.5) together with the fact that

$$
\|v^*\|_{\Omega} \le \|u\|_{\Omega} + 2\Theta_h k \|\pi_h u\|_{\mathfrak{m},\Omega} \le \|u\|_{\Omega} + 2\Theta_h k \|u\|_{\mathfrak{m},\Omega} \le (1+2\Theta_h) \|u\|_{\Omega},
$$

where we employed (2.9).

□
## 4. EFFICIENCY

In this section, we show that the lower bound proposed in (3.6) is efficient. By that, we mean that if $\beta$ is indeed inf-sup stable, the numerical lower bound is be positive and does not arbitrarily underestimate the optimal inf-sup constant, provided the finite element space $V_h$ is sufficiently rich and that the flux reconstruction $\mathcal{F}$ is suitably designed.

From here on, we therefore assume that $\beta$ is inf-sup stable. We will also denote by

$$ (4.1) \qquad M := \max_{\substack{u \in H_{\Gamma_D}^1(\Omega) \\ \|u\|_{\Omega}=1}} \max_{\substack{v \in H_{\Gamma_D}^1(\Omega) \\ \|v\|_{\Omega}=1}} |\beta(u, v)| $$

the continuity constant of $\beta$ in the chosen energy norm. We can then introduce the continuous solution operator

$$ (4.2) \qquad b(w, \mathcal{P}(\theta_h)) = k^2(\mathfrak{p}w, \theta_h) $$

for all $\theta_h$ and $w \in H_{\Gamma_D}^1(\Omega)$.

For Helmholtz problems, $M$ is bounded from above by a generic $k$-independent constant. In the absence of dissipation, we usually have $M = 1$. Otherwise, it depends on the strength of the absorption, or on the parameters of the perfectly matched layers when they are employed.

**4.1. Vertex patches.** In this section, we denote by $V_h$ the set of vertices of the mesh $\mathcal{T}_h$. For each $a \in V_h$, we denote by $\psi^a \in \mathcal{P}_1(\mathcal{T}_h) \cap H^1(\Omega)$ its hat function, i.e., this only continuous piecewise affine function such that $\psi^a(\mathbf{b}) = \delta_{a,b}$ for all $\mathbf{b} \in V_h$, where $\delta$ stands for the Kronecker symbol. We denote by $\mathcal{T}_h^a \subset \mathcal{T}_h$ the set of elements having $a$ as a vertex. Then, the open domain covered by the elements of $\mathcal{T}_h^a$ is denoted by $\omega^a$, and corresponds to the support of $\psi^a$.

**4.2. Local wavespeed and contrast.** For all $a \in V_h$, we let

$$ \vartheta_{\omega^a} := \sqrt{\frac{\min_{K \in \mathcal{T}_h^a} p_K}{\max_{K \in \mathcal{T}_h^a} \alpha_K^{\sharp}}}, \qquad \mathcal{H}_{\omega^a} := \sqrt{\frac{\max_{K \in \mathcal{T}_h^a} \alpha_K^{\sharp}}{\min_{K \in \mathcal{T}_h^a} \alpha_K^{\flat}}} $$

We also denote by $h_{\omega^a}$ the diameter of $\omega^a$.

**4.3. Local function spaces.** The following spaces associated to vertex patches will be useful. For $a \in V_h$, we let $\gamma_a^c \subset \partial\omega^a$ be the set covered by the faces $F \in \mathcal{F}_h$ that share the vertex $a$ such that $F \subset \Gamma_N$. We note that for interior vertices $\gamma_a^c = \emptyset$. We also let $\gamma_a := \partial\omega^a \setminus \gamma_a^c$. We then let $H_0(\text{div}, \omega^a) := H_{\gamma_a}(\text{div}, \omega^a)$. We further let $L_0^2(\omega^a) := \nabla \cdot H_0(\text{div}, \omega^a)$. This space coincides with $L^2(\omega^a)$ if $\gamma_a^c \neq \emptyset$, and consists of zero mean value functions otherwise.

**4.4. Localized flux reconstruction.** We are now in place to propose a concrete strategy to compute a flux reconstruction $\mathcal{F}_h : Q_h \to \mathcal{R}T_{p+2}(\mathcal{T}_h) \cap \mathcal{H}_{\Gamma_N}(\text{div}, \Omega)$ satisfying (3.2). It is defined through the solve of vertex patch mixed finite element problems.

Given $\theta_h \in Q_h$, for all vertices $a \in \mathcal{T}_h$, we introduce the divergence constraint

$$ \mathcal{V}^a(\theta_h) := \psi^a(k^2 \mathfrak{p}\theta_h + k^2 d_\dagger \mathcal{P}_h(\theta_h) + ik\mathbf{b}_\dagger \cdot \nabla \mathcal{P}_h(\theta_h)) \\ - \nabla \psi^a \cdot (-ik\mathbf{c}_\dagger \mathcal{P}_h(\theta_h) + \underline{\mathbf{A}}_\dagger \nabla \mathcal{P}_h(\theta_h)) \in \mathcal{P}_{p+2}(\mathcal{T}_h^a) $$

and the target

$$ \mathcal{T}^a(\theta_h) := \psi^a(\underline{\mathbf{A}}_\dagger \nabla \mathcal{P}_h(\theta_h) - ik\mathbf{c}_\dagger \mathcal{P}_h(\theta_h)) \in \mathcal{P}_{p+1}(\mathcal{T}_h^a). $$
These data enter the construction of $\mathcal{F}_h$ as follows. For all $\mathbf{a} \in \mathcal{V}_h$, we will see below that

$$ (4.3a) \qquad \mathcal{F}_h^a(\theta_h) := \arg \min_{\substack{\sigma_h \in \mathcal{R}T_{p+2}(\mathcal{T}_h^a) \cap \mathcal{H}_0(\text{div}, \omega^a) \\ \nabla \cdot \sigma_h(\theta_h) = \mathfrak{D}^a}} \|\sigma_h + \mathbf{t}^a(\theta_h)\|_{\mathfrak{A}^{-1}, \omega^a} $$

is a sound definition. Whenever useful, we will also implicitly extend $\mathcal{F}_h^a(\theta_h)$ by $\mathbf{o}$ to $\Omega$, which produces an element of $\mathcal{H}_{\Gamma_N}(\text{div}, \Omega)$. We then let

$$ (4.3b) \qquad \mathcal{F}_h(\theta_h) := \sum_{\mathbf{a} \in \mathcal{V}_h} \mathcal{F}_h^a(\theta_h) \in \mathcal{H}_{\Gamma_N}(\text{div}, \Omega). $$

Before deriving key properties of $\mathcal{F}_h$, we immediately make a remark useful at different places.

**Lemma 4.1 (Data identity).** For all $\theta_h \in Q_h$, $\mathbf{a} \in \mathcal{V}_h$ and $w \in H^1(\omega^a)$, we have

$$ (4.4) \qquad b(\psi^a w, \mathcal{P}(\theta_h) - \mathcal{P}_h(\theta_h)) = (\nabla w, \mathbf{t}^a)_{\omega^a} - (w, \mathfrak{D}^a(\theta_h))_{\omega^a}. $$

*Proof.* For shortness, we let $u_h := \mathcal{P}_h(\theta_h)$. Then, we have

$$ (\nabla w, \mathbf{t}^a(\theta_h))_{\omega^a} = (\nabla w, \psi^a(\underline{\mathbf{A}}_\dagger \nabla u_h - ik\mathbf{c}_\dagger u_h))_{\omega^a} = (\psi^a \nabla w, \underline{\mathbf{A}}_\dagger \nabla u_h - ik\mathbf{c}_\dagger u_h) $$

and

$$ \begin{aligned} (w, \mathfrak{D}^a(\theta_h))_{\omega^a} &= (w, \psi^a(k^2 \mathfrak{p}\theta_h + k^2 d_\dagger u_h + ik\mathbf{b}_\dagger \cdot \nabla u_h))_{\omega^a} - (w, \nabla \psi^a \cdot (-ik\mathbf{c}_\dagger u_h + \underline{\mathbf{A}}_\dagger \nabla u_h))_{\omega^a} \\ &= (\psi^a w, k^2 \mathfrak{p}\theta_h) + (\psi^a w, k^2 d_\dagger u_h + ik\mathbf{b}_\dagger \cdot \nabla u_h) - (w \nabla \psi^a, -ik\mathbf{c}_\dagger u_h + \underline{\mathbf{A}}_\dagger \nabla u_h). \end{aligned} $$

Using the product rule $\nabla(\psi^a w) = \psi^a \nabla w + w \nabla \psi^a$, we have

$$ \begin{aligned} (\nabla w, \mathbf{t}^a)_{\omega^a} - (w, \mathfrak{D}^a)_{\omega^a} &= (\psi^a w, k^2 \mathfrak{p} \theta_h) \\ &\quad - \{(\psi^a w, -k^2 d_\dagger u_h - ik\mathbf{b}_\dagger \cdot \nabla u_h) + (\nabla(\psi^a w), \underline{\mathbf{A}}_\dagger \nabla u_h - ik\mathbf{c}_\dagger u_h)\} \\ &= k^2(\mathfrak{p}\psi^a w, \theta_h) - b(\psi^a w, \mathcal{P}_h(\theta_h)) \\ &= b(\psi^a w, \mathcal{P}(\theta_h) - \mathcal{P}_h(\theta_h)), \end{aligned} $$

where we used the expression for $\beta$ in (2.3). $\square$

**4.5. Efficiency of the flux reconstruction.** We can now show that the flux reconstruction in (4.3) lead to a small residual $\mathcal{R}(\theta_h)$ whenever the finite element error $(\mathcal{P} - \mathcal{P}_h)(\theta_h)$ is small.

**Lemma 4.2 (Discrete stable minimization).** For all $\theta_h \in Q_h$, the definition of $\mathcal{F}_h^a(\theta_h)$ in (4.3a) is well-posed. $\mathcal{F}_h^a(\theta_h)$ depends linearly on $\theta_h$, and we have

$$ (4.5) \qquad \|\mathcal{F}_h^a(\theta_h) + \mathbf{t}^a(\theta_h)\|_{\mathfrak{A}^{-1}, \omega^a} \le c(\kappa) \min_{\substack{\sigma \in \mathcal{H}_0(\text{div}, \omega^a) \\ \nabla \cdot \sigma = \mathfrak{D}^a(\theta_h)}} \|\sigma + \mathbf{t}^a(\theta_h)\|_{\mathfrak{A}^{-1}, \omega^a}. $$

*Proof.* Following [5, 12, 16], the well-posedness of (4.3a) and the estimate in (4.5) follow if we can show that the compatibility condition

$$ (1, \mathfrak{D}^a(\theta_h))_{\omega^a} = 0 $$

holds true for all vertices $\mathbf{a} \in \mathcal{V}_h \setminus \bar{\Gamma}_D$. To do so, we simply invoke (4.4), giving

$$ (1, \mathfrak{D}^a(\theta_h))_{\omega^a} = -b(\psi^a, \mathcal{P}(\theta_h) - \mathcal{P}_h(\theta_h)). $$

Due to the respective definitions of $\mathcal{P}(\theta_h)$ and $\mathcal{P}_h(\theta_h)$ in (4.2) and (3.1), the right-hand side vanishes since $\psi^a \in V_h$. This concludes the proof. $\square$
**Lemma 4.3 (Local efficiency).** For all $\theta_h \in Q_h$ and $\boldsymbol{a} \in \mathcal{V}_h$, we have

$$ (4.6) \quad \min_{\substack{\boldsymbol{\sigma} \in \mathcal{H}_0(\text{div}, \omega^a) \\ \nabla \cdot \boldsymbol{\sigma} = \mathfrak{D}^a}} \|\boldsymbol{\sigma} + \boldsymbol{t}^a\|_{\mathfrak{A}^{-1}, \omega^a} \le c(\kappa) M \left( \frac{kh_{\omega^a}}{p\vartheta_{\omega^a}} + \mathcal{H}_{\omega^a} \right) \|\mathcal{P}(\theta_h) - \mathcal{P}_h(\theta_h)\|_{\omega^a}. $$

*Proof.* The Euler-Lagrange equations defining the minimizer in (4.6) consists in finding $\boldsymbol{\sigma} \in \mathcal{H}_0(\text{div}, \omega^a)$ and $\xi \in L_0^2(\omega^a)$ such that

$$ \left\{ \begin{array}{l} (\mathfrak{A}^{-1}\boldsymbol{v}, \boldsymbol{\sigma})_{\omega^a} - (\nabla \cdot \boldsymbol{v}, \xi)_{\omega^a} = -(\mathfrak{A}^{-1}\boldsymbol{v}, \boldsymbol{t}^a)_{\omega^a} \quad \forall \boldsymbol{v} \in \mathcal{H}_0(\text{div}, \omega^a), \\ (w, \nabla \cdot \boldsymbol{\sigma})_{\omega^a} = (q, \mathfrak{D}^a)_{\omega^a} \quad \forall w \in L_0^2(\omega^a). \end{array} \right. $$

From the first equation, we infer that $\xi \in H_{\gamma_a^c}^1(\omega^a)$ with $\nabla \xi = \mathfrak{A}^{-1}(\boldsymbol{\sigma} + \boldsymbol{t}^a)$, and therefore

$$ \|\boldsymbol{\sigma} + \boldsymbol{t}^a\|_{\mathfrak{A}^{-1}, \omega^a} = \|\nabla \xi\|_{\mathfrak{A}, \omega^a}. $$

By using a test function $w \in H_{\gamma_a^c}^1(\omega^a) \cap L_0^2(\omega^a)$ in the second equation, we have

$$ (4.7) \quad (\mathfrak{A}\nabla\xi, \nabla w)_{\omega^a} = (\boldsymbol{t}^a, \nabla w)_{\omega^a} + (\boldsymbol{\sigma}, \nabla w)_{\omega^a} = (\boldsymbol{t}^a, \nabla w)_{\omega^a} - (\mathfrak{D}^a, w)_{\omega^a}. $$

Recalling (4.7) and the Galerkin orthogonality property satisfied by $\mathcal{P}_h(\theta_h)$, it follows that

$$ \begin{aligned} \|\nabla \xi\|_{\mathfrak{A}, \omega^a}^2 &= b(\psi^a \xi, \mathcal{P}(\theta_h) - \mathcal{P}_h(\theta_h)), \\ &= b(\psi^a \xi - \psi^a J_h \xi, \mathcal{P}(\theta_h) - \mathcal{P}_h(\theta_h)), \\ &\le M \|\mathcal{P}(\theta_h) - \mathcal{P}_h(\theta_h)\|_{\omega^a} \|\psi^a \xi - \psi^a J_h \xi\|_{\omega^a}. \end{aligned} $$

where $J_h: H_{\gamma_a^c}^1(\omega^a) \to H_{\gamma_a^c}^1(\omega^a) \cap \mathcal{P}_{p-1}(\mathcal{T}_h^a)$ is the quasi-interpolation operator from [21]. We can then write on the one hand that

$$ k \|\psi^a \xi - \psi^a J_h \xi\|_{\mathfrak{P}, \omega^a} \le k \|\xi - J_h \xi\|_{\mathfrak{P}, \omega^a} \le c(\kappa) \frac{kh_{\omega^a}}{p\vartheta_{\omega^a}} \|\nabla \xi\|_{\mathfrak{A}, \omega^a} $$

and on the other hand that

$$ \|\nabla(\psi^a \xi - \psi^a J_h \xi)\|_{\mathfrak{A}, \omega^a} \le c(\kappa) \max_{K \in \mathcal{T}_h^a} \sqrt{\alpha_K^{\dagger}} (h_{\omega^a}^{-1} \|\xi - J_h \xi\|_{\omega^a} + \|\nabla(\xi - J_h \xi)\|_{\omega^a}) \le c(\kappa) \mathcal{H}_{\omega^a} \|\nabla \xi\|_{\mathfrak{A}, \omega^a}. $$

Combining these bounds gives (4.6). $\square$

**Theorem 4.4 (Efficiency of the residual control).** For all $\theta_h \in Q_h$, we have

$$ (4.8) \quad \|\mathcal{R}(\theta_h)\|_{\mathfrak{A}^{-1}, \Omega} \le c(\kappa) M \max_{\boldsymbol{a} \in \mathcal{V}_h} \left( \mathcal{H}_{\omega^a} + \frac{kh_{\omega^a}}{p\vartheta_{\omega^a}} \right) \|(\mathcal{P} - \mathcal{P}_h)(\theta_h)\|_{\Omega}. $$

In addition, the estimate

$$ (4.9) \quad \rho_h \le c(\kappa) M \max_{\boldsymbol{a} \in \mathcal{V}_h} \left( \mathcal{H}_{\omega^a} + \frac{kh_{\omega^a}}{p\vartheta_{\omega^a}} \right) \varepsilon_h $$

holds true, where

$$ (4.10) \quad \varepsilon_h := \max_{\substack{\theta_h \in Q_h \\ k \|\theta_h\|_{\infty} = 1}} \|(\mathcal{P} - \mathcal{P}_h)(\theta_h)\|_{\Omega}. $$

*Proof.* Let $\theta_h \in Q_h$. By the definition of $\mathcal{P}$ in (4.2) and invoking the continuity of $\beta$ in (4.1), we have

$$ |(\mathfrak{P}w, \theta_h) - \beta(w, \mathcal{P}_h(\theta_h))| = |\beta(w, (\mathcal{P} - \mathcal{P}_h)(\theta_h))| \le M \|w\| \|(\mathcal{P} - \mathcal{P}_h)(\theta_h)\| \le M \varepsilon_h \|w\|, $$

and the conclusion follows from the definition of $\rho_h$ in (3.3). $\square$

<figure><img src="image_3.png" /></figure>
**4.6. Upper bound.** We introduce

$$ (4.11) \qquad \Theta := \max_{\substack{\theta \in L^2(\Omega) \\ k \|\theta\|_m = 1}} \|\mathcal{P}(\theta)\|_{\Omega}, $$

the continuous counterpart to $\Theta_h$. From the definition of $\varepsilon_h$ in (4.10), it is immediate that

$$ (4.12) \qquad \Theta_h \le \Theta + \varepsilon_h. $$

For Helmholtz problems, it is known that $\Theta$ grows linearly with the wavenumber, see e.g. [9, 18], so that this constant is expected to be large in the cases of interest.

**Lemma 4.5 (Inf-sup upper bound).** Assume that $\beta$ is symmetric in the sense that

$$ (4.13) \qquad \beta(u, v) = \beta(\bar{v}, \bar{u}) $$

for all $u, v \in H_{\Gamma_D}^1(\Omega)$. Then, we have

$$ (4.14) \qquad \min_{\substack{u \in H_{\Gamma_D}^1(\Omega) \\ \|u\|_{\Omega} = 1}} \max_{\substack{v \in H_{\Gamma_D}^1(\Omega) \\ \|v\|_{\Omega} = 1}} \operatorname{Re} \beta(u, v) \le \frac{\mathcal{K}}{\Theta} $$

where

$$ \mathcal{K} := \max_{K \in \mathcal{T}_h} \sqrt{\frac{p_K}{m_K}}. $$

**Proof.** Let $\theta \in L^2(\Omega)$ denote a maximizer in (4.11). We can then write that

$$ \operatorname{Re} \beta(w, \mathcal{P}(\theta)) = \operatorname{Re} k^2(\mathfrak{p}v, \theta) \le k^2 \|w\|_{\mathfrak{p},\Omega} \|\theta\|_{\mathfrak{p},\Omega} \le \mathcal{K} \|w\|_{\Omega}. $$

Using (4.13) and defining $u := \overline{\mathcal{P}(\theta)} / \|\mathcal{P}(\theta)\|_{\Omega} = \overline{\mathcal{P}(\theta)} / \Theta$, we have

$$ \operatorname{Re} \beta(u, v) = \frac{1}{\Theta} \operatorname{Re} b(\bar{v}, \mathcal{P}(\theta)) \le \frac{\mathcal{K}}{\Theta} \|v\|_{\Omega}, $$

for all $v \in H_{\Gamma_D}^1(\Omega)$ and (4.14) follows. $\square$

The assumption that $\beta$ is symmetric holds true for Helmholtz problems. We could also lift this assumption at the price of also analyzing adjoint problems. We refrain from doing so here for simplicity. We also recall that for Helmholtz problem, $\mathcal{K} = 1$.

**Theorem 4.6 (Efficiency of the inf-sup bound).** Assume that $\beta$ is symmetric as per (4.13).

Then, we have

$$ (4.15) \qquad \min_{\substack{u \in H_{\Gamma_D}^1(\Omega) \\ \|u\|_{\Omega} = 1}} \max_{\substack{v \in H_{\Gamma_D}^1(\Omega) \\ \|v\|_{\Omega} = 1}} \operatorname{Re} \beta(u, v) \le 2\mathcal{K}\iota_h\gamma_h $$

with

$$ (4.16) \qquad \iota_h := \frac{1}{1 - 2\left(\frac{kh}{v}\right)^2 - 2\rho_h} \left(1 + \frac{1 + 2\varepsilon_h}{2\Theta}\right). $$

**Proof.** For shortness, we set

$$ \gamma := \min_{\substack{u \in H_{\Gamma_D}^1(\Omega) \\ \|u\|_{\Omega} = 1}} \max_{\substack{v \in H_{\Gamma_D}^1(\Omega) \\ \|v\|_{\Omega} = 1}} \operatorname{Re} \beta(u, v). $$
Then, (4.14) ensures that

$$ \gamma \le \frac{\mathfrak{R}}{\Theta} \le \frac{\mathfrak{R}}{1 + 2\Theta + 2\varepsilon_h} \frac{1 + 2\Theta + 2\varepsilon_h}{\Theta} \le \frac{2\mathfrak{R}}{1 + 2(\Theta + \varepsilon_h)} \left(1 + \frac{1 + 2\varepsilon_h}{2\Theta}\right) $$

and it follows from (4.12) that

$$ \gamma \le \left(1 + \frac{1 + 2\varepsilon_h}{2\Theta}\right) \frac{2\mathfrak{R}}{1 + 2\Theta_h}. $$

At that point, (4.15) follows from the definitions of $\iota_h$ in (4.16) and $\gamma_h$ in (3.7). $\square$

**Remark 4.7** (Efficiency for Helmholtz problems). For Helmholtz problems M is generically bounded and $\Theta \ge c(\hat{\Omega})k\ell/\vartheta$, where $\vartheta := \min_{K \in \mathcal{T}_h} \vartheta_K$ is the minimal wavespeed. Hence, under the assumptions that

$$ \frac{k\ell}{\vartheta} \gg 1, \quad \frac{k\mathfrak{h}}{\mathfrak{v}} \ll 1, \quad \varepsilon_h \ll 1, $$

we have

$$ \iota_h \le 1 + c(\hat{\Omega}) \left(\frac{k\ell}{\vartheta}\right)^{-1} + c(\kappa)\mathcal{K} \left\{ \left(\frac{k\mathfrak{h}}{\mathfrak{v}}\right)^2 + \varepsilon_h \right\}, $$

where $\mathcal{K} := \max_{a \in \mathcal{V}_h} \mathcal{K}_{\omega a}$ is the maximal contrast. Since we also have $\mathfrak{R} = 1$, the lower bound provided by the proposed algorithm is expected to be sharp up to factor 2 for reasonable discretization settings. Indeed (3.6) and (4.15) can then be simplified into

$$ \gamma_h \le \min_{\substack{u \in H_{\Gamma_D}^1(\Omega) \\ \|u\|_{\Omega}=1}} \max_{\substack{v \in H_{\Gamma_D}^1(\Omega) \\ \|v\|_{\Omega}=1}} \mathrm{Re} \, \beta(u, v) \le 2 \left( 1 + c(\hat{\Omega}) \left(\frac{k\ell}{\vartheta}\right)^{-1} + c(\kappa) \mathcal{K} \left\{ \left(\frac{k\mathfrak{h}}{\mathfrak{v}}\right)^2 + \varepsilon_h \right\} \right) \gamma_h. $$

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