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# On singular Galerkin discretizations for three models in high-frequency scattering

Théophile Chaumont-Frelet, Stefan Sauter

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Théophile Chaumont-Frelet, Stefan Sauter. On singular Galerkin discretizations for three models in high-frequency scattering. 2026. hal-05511410

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# On singular Galerkin discretizations for three models in high-frequency scattering

T. Chaumont-Frelet* S. Sauter†

February 15, 2026

## Abstract

We consider three common mathematical models for time-harmonic high frequency scattering: the Helmholtz equation in two and three spatial dimensions, a transverse magnetic problem in two-dimensions, and Maxwell's equation in three dimensions with dissipative boundary conditions such that the continuous problem is well posed. In this paper, we construct meshes for popular (low order) Galerkin finite element discretizations such that the discrete system matrix becomes singular and the discrete problem is not well posed. This implies that a condition "the finite element space has to be sufficiently rich" in the form of a *resolution condition* – typically imposed for discrete well-posedness – is not an artifact from the proof by a compact perturbation argument but necessary for discrete stability of the Galerkin discretization.

AMS Subject Classification: 65N12, 65N30, 35J25, 65F99

Key Words: Helmholtz equation; Maxwell equation; high wavenumber; pre-asymptotic stability; hp-finite elements

## 1 Introduction

The finite element discretization of partial differential equations (PDEs) modeling time-harmonic wave propagation is a challenging problem in numerical analysis. It is especially difficult to develop optimal stability and convergence results because the natural variational formulations of such PDE models are not coercive. In particular, this lack of coercivity leads to linear systems with indefinite matrices, which must be numerically solved. It is well-known that if the discretization space is sufficiently “rich”, the discretized problem has a unique solution if the original PDE model is well-posed. However, what happens for finite element spaces that are not large enough is less clear. In this

<footnote>* (theophile.chaumont@inria.fr), Inria, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, 59000 Lille, France</footnote>

<footnote>† (stas@math.uzh.ch), Institut für Mathematik, Universität Zürich, Winterthurerstr 190, CH-8057 Zürich, Switzerland</footnote>
work, we construct meshes such the system matrices resulting from finite element discretizations are singular for certain wavenumbers, even if the original PDE model is well-posed.

Given a polytopic domain $\Omega \subset \mathbb{R}^d$ ($d = 2$ or 3) with diameter $\ell$, a right-hand side in the dual energy space $f \in \mathcal{H}(\Omega)^*$, and a wavenumber $k > 0$, we consider time-harmonic wave propagation problems of the form: Find $u \in \mathcal{H}(\Omega)$ such that

$$ -k^2(u, v)_{\Omega} - ik(\gamma_D u, \gamma_D v)_{\partial\Omega} + (Du, Dv)_{\Omega} = \langle f, v \rangle \quad \forall v \in \mathcal{H}(\Omega). \quad (1) $$

In (1), $\mathcal{D}$ is a differential operator from the de Rham complex (i.e., the gradient, the curl or the divergence), $\gamma_D$ is the naturally associated trace operator, and $\mathcal{H}(\Omega)$ is the corresponding Sobolev space. These notation are rigorously introduced in Section 2 below.

The dissipation introduced on the boundary ensures that (1) is well-posed for all $k > 0$. It is furthermore possible to control the norm of the solution $u$ by the norm of the right-hand side $f$ explicitly in $k$, see e.g. [11, 16, 18, 23, 32]. We focus on so-called “Robin” boundary conditions for simplicity in this introduction, but the results of this work are valid for generic boundary conditions, including Dirichlet-to-Neumann operators arising from the Sommerfeld radiation condition [8, 30].

We are interested in the conforming finite element method (FEM) applied to (1) based on a finite-dimensional subspace $\mathcal{H}_h \subset \mathcal{H}(\Omega)$. In this setting, the goal is to find a discrete approximation $u_h \in \mathcal{H}_h$ such that

$$ -k^2(u_h, v_h)_{\Omega} - ik(\gamma_D u_h, \gamma_D v_h)_{\partial\Omega} + (Du_h, Dv_h)_{\Omega} = \langle f, v_h \rangle \quad \forall v_h \in \mathcal{H}_h. \quad (2) $$

This is a finite-dimensional linear system and, given a basis of $\mathcal{H}_h$, the left-hand side of (2) can be represented by the system matrix $A$.

Finite element spaces are based on a simplicial mesh $\mathcal{T}_h$ of $\Omega$ and $\mathcal{H}_h$ collects all the elements of $\mathcal{H}(\Omega)$ whose restriction to each simplex belongs to a suitable polynomial subspace $\hat{P}$. This is detailed in Section 3. The key parameters controlling the design of the finite element space are the diameter $h$ of the largest simplex in $\mathcal{T}_h$ and the maximal polynomial degree $p$ of the elements of $\hat{P}$. The quantity $kh/p$ is then inversely proportional to the number of degrees of freedom per wavelength.

Following [2, 9, 10, 15, 23, 26, 22, 29], there exist constants $\beta, c > 0$ independent of $h$ and $k$ such that if

$$ \frac{kh}{p} \le c(k\ell)^{\beta} \quad (3) $$

then the matrix $A$ is known to be regular, and the approximation $u_h$ to $u$ is uniformly accurate for all $k$. The particular value of $\beta$ depends on $p$ and the geometry of $\Omega$, as we recap in Section 3. As a general rule of thumb, the values of $\beta$ decreases with $p$, which makes high-order methods more suitable. It is numerically observed that the resolution condition in (3) is necessary to obtain
accurate approximations. In fact, under appropriate assumptions, the optimal
value of $\beta$ is $1/(2p)$.

In this work, we are interested in the properties of the matrix $A$ when no
resolution conditions as in (3) are imposed. Although such discretization does
not lead to accurate approximations, it is still of interest in the context of
preconditioning [1] and adaptivity [5]. Indeed, high accuracy is not expected
for the coarsest level of a multigrid solver nor for the initial mesh of an adaptive
algorithm performing iterative refinements. Rather, one desires that the matrix
$A$ possesses basic algebraic and analytical properties and, in particular, that it
is non-singular.

In [4, Theorem 3.1] a positive result in one space dimension has been proved: Any hp finite element Galerkin discretization of the one-dimensional Helmholtz equation with Robin boundary condition has a unique solution for all wavenumbers $k \in \mathbb{R} \setminus \{0\}$ and all one-dimensionsal meshes.

In this paper we investigate the well-posedness of the Galerkin discretiza-
tion for time-harmonic high-frequency scattering problems in two and three
dimensions. The results we provide are of a negative nature. Specifically, we
construct a mesh $\hat{T}$ for which the system matrix $A$ is singular for particular val-
ues of $k$. Our work is inspired by [4] where one mesh was presented such that the
lowest-order Lagrange finite element in two space dimensions becomes singular.
Here, we expand on [4] and construct further meshes such that the Galerkin
discretization becomes singular, more precisely we show that for the Galerkin
method on these meshes for (a) two- and three-dimensional settings, (b) La-
grange, Raviart-Thomas and Nédélec finite elements, and (c) larger polynomial
degrees the arising linear system can become singular.

Our examples do not only concern the particular mesh $\hat{T}$, but in fact, any
mesh $\mathcal{T}_h$ containing (a possibly scaled version of) $\hat{T}$ as a submesh is affected.
To be more specific, we in fact construct finite element functions $u_h \in \mathcal{H}_h \setminus \{0\}$
with local support such that $(Du_h, Dv_h)_\Omega = k^2(u_h, v_h)_\Omega$ for all $v_h \in \mathcal{H}_h$. This
is a manifestation of the fact that the so-called "unique continuation principle"
does not hold at the discrete level [14]. This is in contrast to the continuous
level, where such a principle is in fact used to establish the well-posedness of (1),
see e.g. [7, 33].

The numerical discretization of unique continuation problems is also of in-
dependent interest (see e.g. [6]), and we expect that the present results might
be of interest in this context.

The remainder of this work is organized as follows. In Section 2, three
model problems for high-frequency scattering are introduced on the continuous
level and well-posedness results from the literature are recalled. The Galerkin
discretizations for these models by popular finite element spaces are presented
in Section 3 and their well-posedness is formulated provided the finite element
space satisfies a resolution condition which is stated in an explicit way.

In Section 4 we introduce the two basic meshes, one for two and one for three
dimensions, which are used to construct the singular discretizations. They are
inspired by the work [4] and we denote them by *Bernkopf meshes*. We assemble
the system matrices by an implementation<fnref n="1" /> of our finite element methods in the symbolic mathematics software MATHEMATICA™ (see [34]). The matrix entries then are quadratic functions of the wavenumber $k$. To analyze the critical wavenumbers leading to a singular system matrix we have computed its determinant (a polynomial in $k$) and analyzed its roots. Here, we present the results by explicitly characterizing all the critical frequencies $k$ for these meshes and different examples.

In the concluding Section 5 we summarize our findings and also explain that general meshes which contain a (scaled and dilated) Bernkopf mesh as a submesh lead to singular system matrices for scaled versions of the critical wavenumbers $k$ in the basic Bernkopf meshes.

For the Maxwell equation, discretized by Nédélec elements, the analysis of the determinant of the system matrix involves a discussion of a cubic polynomial with parameter-dependent coefficients. We have shifted this tedious analysis to Appendix A.

## 2 Setting

In this section we introduce our model scattering problems and their discretization. The physical domain $\Omega \subset \mathbb{R}^d$, $d \in \{2, 3\}$ is assumed to be bounded and Lipschitz. Its boundary is denoted by $\Gamma := \partial\Omega$. and the outer normal vector field by $\mathbf{n}$.

Let $L^p(\Omega)$, $p \in [1, \infty]$, and $H^1(\Omega)$ be the standard Lebesgue and Sobolev spaces on $\Omega$ and we denote by $L^p(\Omega) := (L^p(\Omega))^d$, $H^1(\Omega) := (H^1(\Omega))^d$ their vector valued analogues. We will also need the spaces

$$ \mathbf{H}(\operatorname{div}; \Omega) := \{\mathbf{v} \in L^2(\Omega) \mid \operatorname{div} \mathbf{v} \in L^2(\Omega)\}, \\ \mathbf{H}(\operatorname{curl}; \Omega) := \{\mathbf{v} \in L^2(\Omega) \mid \operatorname{curl} \mathbf{v} \in L^2(\Omega)\} $$

and the space of tangential fields in $L^2(\Gamma)$:

$$ L_T^2(\Gamma) := \{\mathbf{v} \in L^2(\Gamma) \mid \mathbf{n} \cdot \mathbf{v} = 0 \text{ on } \Gamma\}. $$

For the spaces $H^1(\Omega)$, $\mathbf{H}(\operatorname{div}; \Omega)$, $\mathbf{H}(\operatorname{curl}; \Omega)$ we introduce norms where the $L^2$ term is weighted by the wavenumber $k \in \mathbb{R} \setminus \{0\}$

$$ \begin{align*} \|u\|_{\nabla, k} &:= \left( \|\nabla u\|_{L^2(\Omega)}^2 + k^2 \|u\|_{L^2(\Omega)}^2 \right)^{1/2} & \text{for } H^1(\Omega), \\ \|\mathbf{u}\|_{\operatorname{div}, k} &:= \left( \|\operatorname{div} \mathbf{u}\|_{L^2(\Omega)}^2 + k^2 \|\mathbf{u}\|_{L^2(\Omega)}^2 \right)^{1/2} & \text{for } \mathbf{H}(\operatorname{div}; \Omega), \\ \|\mathbf{u}\|_{\operatorname{curl}, k} &:= \left( \|\operatorname{curl} \mathbf{u}\|_{L^2(\Omega)}^2 + k^2 \|\mathbf{u}\|_{L^2(\Omega)}^2 \right)^{1/2} & \text{for } \mathbf{H}(\operatorname{curl}; \Omega). \end{align*} $$

<footnote n="1">¹The source code of the MATHEMATICA notebook and data files are online available, Link: https://drive.math.uzh.ch/index.php/s/daFi9g27CZ3eXmC</footnote>
**Problem 1 (Helmholtz problem)** The weak formulation of the Helmholtz problem with Robin boundary condition is given by

$$ \text{find } u \in H^1(\Omega) \text{ s.t. } a_k^H(u, v) = F(v) \quad \forall v \in H^1(\Omega), \quad (4) $$

for some given continuous anti-linear form $F \in (H^1(\Omega))'$ and sesquilinear form $a_k^H : H^1(\Omega) \times H^1(\Omega) \to \mathbb{C}$ defined by

$$ a_k^H(u, v) := (\nabla u, \nabla v)_{L^2(\Omega)} - k^2(u, v)_{L^2(\Omega)} - ik(u, v)_{L^2(\Gamma)}. $$

Next, we consider the transverse magnetic problem in 2D with impedance boundary condition. The energy space for this problem is chosen to be

$$ \mathbf{V}_{\text{imp}} := \{\mathbf{u} \in \mathbf{H}(\text{div}; \Omega) | \mathbf{u} \cdot \mathbf{n} \in L^2(\Gamma)\} \quad (5) $$

and furnished with the norm

$$ \|\mathbf{u}\|_{\text{div},k,+} := \left( \|\mathbf{u}\|_{\text{div},k}^2 + |k| \|\mathbf{u} \cdot \mathbf{n}\|_{L^2(\Gamma)}^2 \right)^{1/2}. $$

**Problem 2 (Transverse magnetic problem in 2D)** The weak form of the transverse magnetic problem in two dimensions is given by

$$ \text{find } \mathbf{u} \in \mathbf{V}_{\text{imp}} \text{ s.t. } a_k^{\text{M2D}}(\mathbf{u}, \mathbf{v}) = F(\mathbf{v}) \quad \forall \mathbf{v} \in \mathbf{V}_{\text{imp}}, \quad (6) $$

for some given continuous anti-linear form $F \in \mathbf{V}'_{\text{imp}}$ and sesquilinear form $a_k^{\text{M2D}} : \mathbf{V}_{\text{imp}} \times \mathbf{V}_{\text{imp}} \to \mathbb{C}$ given by

$$ a_k^{\text{M2D}}(\mathbf{u}, \mathbf{v}) := (\text{div} \, \mathbf{u}, \text{div} \, \mathbf{v})_{L^2(\Omega)} - k^2(\mathbf{u}, \mathbf{v})_{L^2(\Omega)} - ik(\mathbf{u} \cdot \mathbf{n}, \mathbf{v} \cdot \mathbf{n})_{L^2(\Gamma)}. $$

Finally, we consider the three-dimensional Maxwell equations with impedance boundary condition. The energy space is chosen to be

$$ \mathbf{X}_{\text{imp}} := \{\mathbf{u} \in \mathbf{H}(\text{curl}; \Omega) | \mathbf{u}_T \in \mathbf{L}_T^2(\Gamma)\}, \quad (7) $$

where $\mathbf{u}_T := \mathbf{n} \times (\mathbf{u}|_{\Gamma} \times \mathbf{n})$; its norm is given by

$$ \|\mathbf{u}\|_{\text{curl},k,+} := \left( \|\mathbf{u}\|_{\text{curl},k}^2 + |k| \|\mathbf{u}_T\|_{\mathbf{L}^2(\Gamma)}^2 \right)^{1/2}. $$

**Problem 3 (Maxwell's equation in 3D)** For some given continuous anti-linear form $F \in \mathbf{X}'_{\text{imp}}$, the weak form of the Maxwell equation in 3D with impedance boundary condition is given by

$$ \text{find } \mathbf{u} \in \mathbf{X}_{\text{imp}} \text{ s.t. } a_k^{\text{M3D}}(\mathbf{u}, \mathbf{v}) = F(\mathbf{v}) \quad \forall \mathbf{v} \in \mathbf{X}_{\text{imp}} \quad (8) $$

with sesquilinear form $a_k^{\text{M3D}} : \mathbf{X}_{\text{imp}} \times \mathbf{X}_{\text{imp}} \to \mathbb{C}$ defined by

$$ a_k^{\text{M3D}}(\mathbf{u}, \mathbf{v}) = (\text{curl} \, \mathbf{u}, \text{curl} \, \mathbf{v})_{L^2(\Omega)} - k^2(\mathbf{u}, \mathbf{v})_{L^2(\Omega)} - ik(\mathbf{u}_T, \mathbf{v}_T)_{L^2(\Gamma)}. $$
These problems have in common that they are well posed for all wavenumbers $k \in \mathbb{R} \setminus \{0\}$ and we cite here the relevant theorems.

**Remark 4** For the well-posedness of the Galerkin discretization of these problems an approximation property (see (14), (15), (16)) will be key which involves the continuous solution operator applied to functions in a space which is compactly embedded in the dual energy space. This is why we define in the following the solution operator for these subspaces.

**Proposition 5 ([23, Prop. 8.1.3])** Let $\Omega$ be a bounded Lipschitz domain. Then, (4) is uniquely solvable for all $F \in (H^1(\Omega))'$ and the solution depends continuously on the data.

Proposition 5 implies that the solution operator $T_k^H : L^2(\Omega) \to H^1(\Omega)$ is well defined for $f \in L^2$ by

$$a_k^H (T_k^H f, v) = (f, v)_{L^2(\Omega)} \quad \forall v \in H^1(\Omega). \qquad (9)$$

**Proposition 6 ([9, Thm. 3.1])** Let $\Omega$ be a bounded Lipschitz domain. Then, (6) is uniquely solvable for all $F \in V'_{\text{imp}}$ and the solution depends continuously on the data<fnref n="2" />.

If the assumptions in Proposition 6 are satisfied, the solution operator $T_k^{M2D} : L^2(\Omega) \to \mathbf{H}$ (div; $\Omega$) is well defined by

$$a_k^{M2D} (T_k^{M2D} \mathbf{f}, \mathbf{v}) = (\mathbf{f}, \mathbf{v})_{L^2(\Omega)} \quad \forall \mathbf{v} \in \mathbf{H}^1 \text{ (div; } \Omega)$$

for all $\mathbf{f} \in L^2$.

Well-posedness of the Maxwell problem with impedance condition is proved in [27, Thm. 4.17].

**Proposition 7** Let $\Omega \subset \mathbb{R}^3$ be a bounded Lipschitz domain with simply connected and sufficiently smooth boundary. Then, (8) is uniquely solvable for all $F \in X'_{\text{imp}}$ and the solution depends continuously on the data.

We introduce the space

$$V_{k,0} := \{ \mathbf{v} \in X_{\text{imp}} | k^2 (\mathbf{v}, \nabla \varphi)_{L^2(\Omega)} + i k (\mathbf{v}_T, (\nabla \varphi)_T)_{L^2(\Gamma)} = 0 \quad \forall \varphi \in H_{\text{imp}}^1(\Omega) \},$$

where

$$H_{\text{imp}}^1(\Omega) := \{ \varphi \in H^1(\Omega) | \varphi|_{\Gamma} \in H^1(\Gamma) \}.$$

The assumptions of Proposition 7 imply that the solution operator $T_k^{M3D} : V_{k,0} \to X_{\text{imp}}$ is well defined by

$$a_k^{M3D} (T_k^{M3D} \mathbf{f}, \mathbf{v}) = (\mathbf{f}, \mathbf{v})_{L^2(\Omega)} \quad \forall \mathbf{v} \in X_{\text{imp}}$$

for all $\mathbf{f} \in V_{k,0}$.

<footnote n="2">²Note that the general assumption that $\Omega$ is a convex polygonal domain in [9] is not needed in the proof of Theorem 3.1 therein.</footnote>
### 3 The finite element spaces

A Galerkin method is employed to discretize these equations which will be based on common finite element spaces on conforming (no hanging nodes) simplicial meshes $\mathcal{T}$ for $\Omega$ that satisfy

1. The (closed) elements $K \in \mathcal{T}$ cover $\Omega$, i.e., $\bar{\Omega} = \cup_{K \in \mathcal{T}} K$.

2. Associated with each element $K$ is the element map, a $C^1$-diffeomorphism $\phi_K: \hat{K} \to K$. The set $\hat{K}$ is the reference simplex

$$ \hat{K} := \left\{ \mathbf{x} \in \mathbb{R}_{\ge 0}^d \mid \sum_{i=1}^d x_i \le 1 \right\}. $$

3. Denoting $h_K = \text{diam } K$, there holds, with some shape-regularity constant $\gamma\tau$,

$$ h_K^{-1} \|\phi'_K\|_{L^\infty(\hat{K})} + h_K \|(\phi'_K)^{-1}\|_{L^\infty(\hat{K})} \le \gamma\tau. \quad (10) $$

4. The intersection of two elements is either empty, a vertex, an edge, a face (for $d=3$), or they coincide (here, vertices, edges, and faces are the images of the corresponding entities on the reference tetrahedron $\hat{K}$).

5. The parametrization of common inner edges (2D) and faces (3D) is compatible: their pullbacks by $\phi_K^{-1}$ and $\phi_{K'}^{-1}$ for two adjacent elements $K, K'$ are affine equivalent.

**Remark 8** The construction of simplicial finite element meshes that satisfy the above assumption can be based on patchwise structured meshes as described, for example, in [25, Ex. 5.1] or [24, Sec. 3.3.2].

Next, we recall the definition of conforming finite element spaces for the discretization of Problems 1 - 3. For a simplex $K \in \mathcal{T}$, let $P_p(K)$ denote the space of $d$-variate polynomials of total degree $\le p$.

**Definition 9 (Conforming hp finite element space)** Let $\mathcal{T}$ be a conforming finite element mesh for the domain $\Omega$. Then the conforming hp finite element space $S_p(\mathcal{T}) \subset H^1(\Omega)$ is given by

$$ S_p(\mathcal{T}) := \{u \in H^1(\Omega) \mid \forall K \in \mathcal{T}, u|_K \circ \phi_K \in P_p(K)\}. $$

We employ this space for the discretization of Problem 1:

$$ \text{find } u_S \in S_p(\mathcal{T}) \text{ s.t. } a_k^H(u_S, v) = F(v) \quad \forall v \in S_p(\mathcal{T}). \quad (11) $$

**Definition 10 (Raviart-Thomas elements)** On the reference element the Raviart-Thomas element of order $p$ is given by

$$ \mathbf{RT}_p(\hat{K}) := \left\{ \mathbf{p}(\mathbf{x}) + \mathbf{x}q(\mathbf{x}) \mid \mathbf{p} \in (\mathbb{P}_p(\hat{K}))^3, q \in \mathbb{P}_p(\hat{K}) \right\}. $$
For a conforming simplicial finite element mesh $\mathcal{T}$ of $\Omega$, the Raviart-Thomas finite element of order $p$ is the space:

$$ \mathbf{RT}_p(\mathcal{T}) := \{ \mathbf{u} \in \mathbf{H}(\mathrm{div}; \Omega) \mid (\det \phi'_K)(\phi'_K)^{-1} \mathbf{u}|_K \circ \phi_K \in \mathbf{RT}_p(\hat{\mathbf{K}}) \}. $$

The Raviart-Thomas finite element is employed for the discretization of Problem 2:

$$ \text{find } \mathbf{u}_{\mathrm{RT}} \in \mathbf{RT}_p(\mathcal{T}) \text{ s.t. } a_k^{\mathrm{M2D}}(\mathbf{u}_{\mathrm{RT}}, \mathbf{v}) = F(\mathbf{v}) \quad \forall \mathbf{v} \in \mathbf{RT}_p(\mathcal{T}). \quad (12) $$

**Definition 11 (Nédélec elements)** On the reference element $\hat{\mathbf{K}}$ the Nédélec (type I) element of order $p$ is given by

$$ \mathbf{N}_p^I(\hat{\mathbf{K}}) := \{ \mathbf{p}(\mathbf{x}) + s\mathbf{x} \times \mathbf{q}(\mathbf{x}) \mid \mathbf{p}, \mathbf{q} \in (\mathbb{P}_p(\hat{\mathbf{K}}))^3 \}. $$

For a conforming simplicial finite element mesh $\mathcal{T}$ of $\Omega$, the Nédélec (type I) finite element of order $p$ is the space:

$$ \mathbf{N}_p^I(\mathcal{T}) := \{ \mathbf{u} \in \mathbf{H}(\mathrm{curl}; \Omega) \mid (\phi'_K)^T \mathbf{u}|_K \circ \phi_K \in \mathbf{N}_p^I(\hat{\mathbf{K}}) \}. $$

We consider discretizations of Problem 3 by a Galerkin method based on the Nédélec space $\mathbf{N}_p^I(\mathcal{T})$:

$$ \text{find } \mathbf{u}_N \in \mathbf{N}_p^I(\mathcal{T}) \text{ s.t. } a_k^{\mathrm{M3D}}(\mathbf{u}_N, \mathbf{v}) = F(\mathbf{v}) \quad \forall \mathbf{v} \in \mathbf{N}_p^I(\mathcal{T}). \quad (13) $$

Since the original variational problems (Prob. 1-3) are not coercive the well-posedness of the Galerkin discretization is not simply inherited. The common theory of well-posedness for the discretizations is based on a compact perturbation argument which is known as the “Schatz argument” (see [29]); for Maxwell’s equation the argument is more subtle since $\mathbf{H}(\mathrm{curl}, \Omega)$ is not compactly embedded in $\mathbf{L}^2(\Omega)$. In any case the well-posedness theorem for conforming Galerkin discretizations of the original variational problem is typically of the form: if the finite element space is “rich enough”, then the discrete equations are well-posed. We cite here the relevant theorems but first introduce some notation.

**Proposition 12 ([26, Thm. 3.2])** Let the simplicial finite element mesh $\mathcal{T}$ and the polynomial degree be chosen such that the adjoint approximation property (cf. (9))

$$ \eta(S_p(\mathcal{T})) := \sup_{f \in L^2(\Omega) \setminus \{0\}} \inf_{v \in S_p(\mathcal{T})} \frac{\|T_{-k}^H f - v\|_{\mathcal{H}}}{\|f\|_{L^2(\Omega)}} \quad (14) $$

satisfies

$$ C_c \eta(S_p(\mathcal{T})) < \frac{1}{2}. $$

Then the Galerkin discretization (11) of Problem 1 is well-posed.
In summary, we have illustrated that for some high frequency scattering problems which are well-posed on the continuous level, the conforming Galerkin discretization is well-posed if the finite element space is “rich enough” such that an adjoint approximation property is sufficiently small.

Hence, it is a natural question whether the conforming Galerkin discretization can be singular if the resolution condition is not satisfied or whether this is an artefact of the theory based on the Schatz perturbation argument. Interestingly, for the Helmholtz equation with Robin boundary condition on a one-dimensional interval $\Omega = (a, b)$ it has been proven in [4] that the conforming Galerkin discretization always admits a unique solution without any restriction to the mesh and/or the polynomial degree.

The resolution condition is sufficient for discrete well-posedness but leaves the question open whether there exist meshes such that the Galerkin discretization becomes singular. This is a motivation for the development of various modifications of the conforming Galerkin discretization which are always well-posed; among them is the first order system least squares (FOSLS) finite element method (see [13], [19], [3]), stabilized methods [17], [35], [12], and Discontinuous Petrov Galerkin methods [28]. These methods have in common that the resulting discretization always have a unique solution. However, the number of degrees of freedom can be substantially larger than for the conforming Galerkin FEM and their implementation is more subtle. This was one motivation to investigate the question:

> Can the conforming Galerkin FEM lead to singular system matrices for problems (11), (12), (13)?

# 4 Bernkopf-type meshes

In [4] an example (Bernkopf mesh) is presented for the Helmholtz problem 1 in 2D for lowest order elements such that the conforming Galerkin discretization has a non-trivial homogeneous solution. In this section, we investigate whether also for higher polynomial degrees, 3D problems, and Maxwell's equation such examples exist.

**Remark 16** *We have realized the computations for all examples in the symbolic mathematics program MATHEMATICA™ (cf. [34]) and hence no rounding errors occur in our computation.*

First, we introduce the domain and the meshes and start with the two-dimensional Bernkopf mesh.

**Definition 17 (2D Bernkopf mesh) (see [4, §3.2])** Let $\Omega = (-1, 1)^2$. The vertices in the triangulation are $\mathbf{P}_1^\Gamma = (-1, -1)^T$, $\mathbf{P}_2^\Gamma = (1, -1)^T$, $\mathbf{P}_3^\Gamma = (1, 1)^T$, $\mathbf{P}_4^\Gamma = (-1, 1)^T$, $\mathbf{P}_1^\Omega = (-\alpha, 0)^T$, $\mathbf{P}_2^\Omega = (0, -\alpha)^T$, $\mathbf{P}_3^\Omega = (\alpha, 0)^T$, $\mathbf{P}_4^\Omega = (0, \alpha)^T$, $\mathbf{P}_5^\Omega = (0, 0)^T$ for a parameter $\alpha \in ]0, 1[$. The triangulation $\mathcal{T}_2$ consists of 12 triangles $T_j$, $j \in \{1, 2, \dots, 12\}$ as depicted in Figure 1.
<figure><img src="image_3.png" /><figcaption>Figure 1: Bernkopf mesh in 2D for $\alpha = 1/2$.</figcaption></figure>

**Definition 18 (3D Bernkopf mesh)** Let $V_i^\Gamma = (P_i^\Gamma, 0)$ for $i \in \{1, 2, 3, 4\}$. Set $V_5^\Gamma := (0, 0, 1)^T$, and $V_6^\Gamma := (0, 0, -1)^T$. The domain $\Omega$ is the convex hull of $V_i^\Gamma$, $i \in \{1, 2, \dots, 6\}$ and the inner vertices are given by $V_i^\Omega = (P_i^\Omega, 0)^T$, $i \in \{1, \dots, 5\}$. The mesh $T_3$ (see 2) consists of 24 tetrahedra of the form $K_j = \text{conv}(T_j, V_5^\Gamma)$, $1 \le j \le 12$ and $K_{12+j} = \text{conv}(T_j, V_6^\Gamma)$, $1 \le j \le 12$.

## 4.1 Helmholtz Problem 1, $d=2$, $p=1,2$

We explain our conceptual approach for finding critical wavenumbers $k \in \mathbb{R} \setminus \{0\}$ such that the system matrix for the Helmholtz problem is singular, while the procedure for Problems 2 and 3 is completely analogue. The system matrix is singular iff there is a non-trivial $u \in S \setminus \{0\}$ to the equation

$$a^H(u, v) = 0 \quad \forall v \in S. \qquad (17)$$

By choosing $v = u$ and considering the imaginary part of the sesquilinear form for $k \in \mathbb{R} \setminus \{0\}$ leads to $\|u\|_{L^2(\Gamma)} = 0$ and, in turn to $u|_{\Gamma} = 0$. For such a function $u$ the boundary part of the sesquilinear form is zero and we have

$$a^H(u, v) = (\nabla u, \nabla v)_{L^2(\Omega)} - k^2(u, v)_{L^2(\Omega)}.$$

Let $(b_y)_{y \in \Sigma}$ denote the standard local nodal basis for the space $S$ with the set $\Sigma$ of degrees of freedom. The degrees of freedom for the inner nodes define the subset $\Sigma^\circ$. We compute the matrix

$$A_\alpha(k) = (a_{y,z}(k))_{y \in \Sigma} \quad \text{with} \quad a_{y,z} := (\nabla b_y, \nabla b_z)_{L^2(\Omega)} - k^2(b_y, b_z)_{L^2(\Omega)} \quad (18)$$
<figure><img src="image_4.png" /><figcaption>Figure 2: Bernkopf mesh in 3D for the choice α = 1/2.</figcaption></figure>
which is a rectangular matrix with more rows than columns. From basic linear algebra we know that problem (17) has a non-trivial solution $u \in S \setminus \{0\}$ iff $k \in \mathbb{R} \setminus \{0\}$, we are interested in positive roots of $Q_{\alpha}(\kappa)$. Our MATHEMATICA implementation computes first the system matrix $A_{\alpha}(k)$, then the matrix $B_{\alpha}(k)$, and finally its determinant $\det B_{\alpha}(k)$ as well as the polynomial $Q_{\alpha}$. We discuss $Q_{\alpha}$ and finally compute positive roots which are the (squares of the) critical wave numbers $k$. Since $Q_{\alpha}$ is a polynomial the number of roots is always finite. We derive by symbolic calculus that, for any $\alpha \in ]0, 1[$, there exists at least one positive root $\kappa$ of $Q_{\alpha}$ and hence, at least two spurious frequencies. Since the $\alpha$-dependent coefficients in $Q_{\alpha}$ are rather complicated we do not attempt to determine all roots of $Q_{\alpha}$ but some of them.

For any $\alpha \in ]0, 1[$ and $p = 1$ our MATHEMATICA implementation shows that $Q_{\alpha}(\kappa)$ can be factorized into three polynomials $\eta_1 \in \mathbb{P}_1$, $\eta_2 \in \mathbb{P}_2$, $\eta_3 \in \mathbb{P}_4$: $Q_{\alpha}(\kappa) = \frac{\eta_1^2 \eta_2^2 \eta_3}{f_{\alpha}}$, where $f_{\alpha}$ is a positive polynomial for $\alpha \in ]0, 1[$. The root of $\eta_1$ is given by $\frac{6(2-\alpha)}{\alpha(1-\alpha)}$ (positive for $\alpha \in ]0, 1[$) while the two roots of $\eta_2$ have non-zero imaginary part and hence are not relevant for critical wavenumbers $k \in \mathbb{R} \setminus \{0\}$. We do not analyze the polynomial $\eta_3$ furthermore but simply conclude from the analysis of $\eta_1$ that for any $\alpha \in ]0, 1[$ two critical wavenumbers exist<fnref n="3" />

$$k(\alpha)^2 = \frac{6(2-\alpha)}{\alpha(1-\alpha)}$$

and the conforming Galerkin discretization with finite element space $S_1(\mathcal{T}_2)$, has a spurious mode, i.e., there exists an element $u_p \in S_p(\mathcal{T}_2) \setminus \{0\}$ such that

$$a_{k(\alpha)}^{\mathrm{H}}(u_p, v) = 0 \quad \forall v \in S_p(\mathcal{T}_2).$$

For any $\alpha \in ]0, 1[$ and $p = 2$, the MATHEMATICA implementation shows that $Q_{\alpha}(\kappa)$ can be factorized into five polynomials $\eta_1 \in \mathbb{P}_1$, $\eta_2 \in \mathbb{P}_2$, $\eta_3 \in \mathbb{P}_3$, $\eta_4, \eta_5 \in \mathbb{P}_{10}$: $Q_{\alpha}(\kappa) = f_{\alpha}\eta_1^2\eta_2\eta_3\eta_4\eta_5^2$, where $f_{\alpha}$ is a positive, rational function for $\alpha \in ]0, 1[$. The root of $\eta_1$ is given by

$$k(\alpha)^2 = \frac{15(\alpha^2 + (2-\alpha)^2)}{\alpha(1-\alpha)(2-\alpha)}$$

(positive for $\alpha \in ]0, 1[$) while the roots of the other polynomials are not discussed here. In this way, we have shown that for any $\alpha \in ]0, 1[$ two critical wavenumbers exist. The spurious modes for $p = 1, 2$ are depicted in Figure 3.

<footnote n="3">³Note that this example for $p = 1$ is the one treated in [4, Lem. 3.2] and an elementary proof is given without symbolic computation.</footnote>
<figure><img src="image_5.png" /><figcaption>Figure 3: Spurious solutions for the $P_1$ and $P_2$ Galerkin discretization of the two-dimensional Helmholtz equation for $\alpha = 1/2$. Left: $p = 1$ and $k(\alpha) = \pm 6$, right: $p = 2$ and $k(\alpha) = \pm 10$.</figcaption></figure>

## 4.2 Helmholtz Problem 1, $d = 3$

For any $\alpha \in ]0, 1[$ and $p = 1$, symbolic manipulation by the MATHEMATICA implementation shows that $Q_\alpha(\kappa)$ can be factorized into three polynomials $\eta_1 \in P_1$, $\eta_2 \in P_2$, $\eta_3 \in P_4$: $Q_\alpha(\kappa) = \frac{\eta_1^2 \eta_2^2 \eta_3}{f_\alpha}$ with a positive polynomial $f_\alpha$ for all $\alpha \in ]0, 1[$. The root of $\eta_1$ is given by $\kappa = \frac{20}{\alpha(1-\alpha)}$ (positive for $\alpha \in ]0, 1[$) while the roots of the other polynomials are not discussed here. In this way, we have shown that for any $\alpha \in ]0, 1[$ two critical wavenumbers exist

$$k(\alpha)^2 = \frac{20}{\alpha(1-\alpha)}.$$

For any $\alpha \in ]0, 1[$ and $p = 2$ symbolic manipulation shows that $Q_\alpha(\kappa)$ can be factorized into eight polynomials $\eta_1, \eta_2 \in P_1, \eta_3 \in P_2, \eta_4, \eta_5 \in P_4, \eta_6 \in P_8, \eta_7 \in P_{12}, \eta_8 = P_{14}$: $Q_\alpha(\kappa) = f_\alpha \eta_1^2 \eta_2^2 \eta_3^2 \eta_4 \eta_5 \eta_6 \eta_7^2 \eta_8$ with a positive rational function $f_\alpha$ for all $\alpha \in ]0, 1[$. The root of $\eta_1$ is given by $\kappa = \frac{42}{\alpha(1-\alpha)}$ (positive for $\alpha \in ]0, 1[$), the root of $\eta_2$ by $\frac{21(\alpha+1)\alpha^2+2(2-\alpha)}{(2-\alpha)(1-\alpha)\alpha}$ (positive for $\alpha \in ]0, 1[$), the roots of $\eta_3$ have non-zero imaginary part for any $\alpha \in ]0, 1[$ and not relevant for critical frequencies $k \in \mathbb{R} \setminus \{0\}$. The functions $\eta_4, \dots, \eta_8$ are not discussed. In this way, we have shown that for any $\alpha \in ]0, 1[$ four critical wavenumbers exist

$$k(\alpha)^2 \in \left\{ \frac{42}{\alpha(1-\alpha)}, \frac{21(\alpha+1)\alpha^2+2(2-\alpha)}{(2-\alpha)(1-\alpha)\alpha} \right\}.$$

The spurious solution for $p = 1$ is shown in Figure 4 while each of the two cases for $p = 2$ are depicted in Figure 5.
<figure><img src="image_6.png" /><figcaption>Figure 4: Spurious solution for the $P_1$ Galerkin discretization of the 3D Helmholtz problem for the Bernkopf mesh with $\alpha = 1/2$ and critical wavenumber $k_1 (1/2) = \pm\sqrt{80}$.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 5: Spurious solutions for the $P_2$ Galerkin discretization for the Helmholtz problem in 3D for $\alpha = 1/2$. Left: $k_2 (1/2)^2 = 168$, right: $k_2 (1/2)^2 = 189$.</figcaption></figure>
## 4.3 Maxwell 2D, RT1:

For any $\alpha \in ]0, 1[$ and $p = 1$, symbolic manipulation by the MATHEMATICA implementation shows that $Q_{\alpha}(\kappa)$ for this problem can be factorized into five polynomials $\eta_1(\kappa) = \kappa^{10}$, $\eta_2, \eta_3 \in \mathbb{P}_2$, $\eta_4 \in \mathbb{P}_4$, $\eta_5 \in \mathbb{P}_6: Q_{\alpha}(\kappa) = f_{\alpha}\eta_1\eta_2^2\eta_3\eta_4\eta_5^2$ with a positive rational function $f_{\alpha}$ for all $\alpha \in ]0, 1[$. Clearly, the only root of $\eta_1$ is $\kappa = 0$ which is not relevant for critical wavenumbers $k \in \mathbb{R} \setminus \{0\}$; $\eta_2$ has two positive roots$^4$

$$ \kappa_{\pm}(\alpha) = \frac{12\alpha^2(2+7\alpha+3\alpha^2-3\alpha^3)\pm\sqrt{3\alpha^6+3(5-4\alpha)\alpha^4+(37-30\alpha)\alpha^2+28\alpha+4}}{2+6\alpha-\alpha^2-\alpha^3}, $$

while the imaginary parts of the roots of $\eta_3$ are always non-zero for $\alpha \in ]0, 1[$. We do not investigate the roots of $\eta_4$ and $\eta_5$. It follows that the conforming Galerkin discretization of the transverse magnetic problem in 2D by Raviart-Thomas finite elements $RT_1(\mathcal{T}_2)$ has a spurious mode for any $\alpha \in ]0, 1[$ and $k^2 = \kappa_{\pm}(\alpha)$.

## 4.4 Maxwell 3D, $N_p^I$ for p=1:

For any $\alpha \in ]0, 1[$ and $p = 1$ symbolic manipulation by the MATHEMATICA implementation shows that $Q_{\alpha}(\kappa)$ for this problem can be factorized into nine polynomials $\eta_1(\kappa) = \kappa^{10}$, $\eta_2 \in \mathbb{P}_1$, $\eta_3, \eta_4 \in \mathbb{P}_2$, $\eta_5 \in \mathbb{P}_3$, $\eta_6, \eta_7, \eta_8 \in \mathbb{P}_4$, $\eta_9 \in \mathbb{P}_8$: $Q_{\alpha}(\kappa) = \frac{\eta_1\eta_2^2\eta_3\eta_4^2\eta_5^2\eta_6\eta_7\eta_8\eta_9^2}{f_{\alpha}}$ with a positive polynomial $f_{\alpha}$ for all $\alpha \in ]0, 1[$. Clearly, the only root of $\eta_1$ is $\kappa = 0$ which is not relevant for critical wavenumbers $k \in \mathbb{R} \setminus \{0\}$; $\eta_2$ has a positive root

$$ \kappa(\alpha) = 20 \frac{2 - \alpha}{1 + \alpha(1 - \alpha)}, $$

while the imaginary parts of the roots of $\eta_3, \eta_4$ are always non-zero for $\alpha \in ]0, 1[$. We do not investigate the roots of $\eta_6, \dots, \eta_9$.

The polynomial $\eta_5$ requires a more subtle analysis. Define coefficients

$$ c_0(\alpha) := 1024000(2\alpha^2 + \alpha + 2), \qquad (19) $$

$$ c_1(\alpha) := -1600(\alpha^3(60 - 19\alpha) + 47\alpha^2 + 136\alpha + 36), $$

$$ c_2(\alpha) := 20(\alpha^3(222 - 35\alpha^3) + (16 - 15\alpha^5) + 12\alpha^4 + 188\alpha^2 + 112\alpha) $$

$$ c_3(\alpha) := -(\alpha^4(26 - 21\alpha) + \alpha^2(41 - 31\alpha) + 52\alpha + 8)\alpha^2 \qquad (20) $$

and the polynomial

$$ p_{\alpha}(x) = \sum_{\ell=0}^{3} c_{\ell}(\alpha) x^{\ell}. \qquad (21) $$

<footnote>$^4$It is easy to see that, by expanding the fraction for the case $((\cdot) - \sqrt{\cdot})$ by $((\cdot) + \sqrt{\cdot})$, the numerator becomes $6\alpha^2(4 + \alpha(10 - 8\alpha - \alpha^2 + \alpha^3))$, i.e., positive for any $\alpha \in ]0, 1[$ while the positivity of the denominator is obvious. For the case $((\cdot) + \sqrt{\cdot})$ the positivity is obvious.</footnote>
<figure><img src="image_8.png" /><figcaption>Figure 6: Discriminant $\Delta$ of the cubic polynomial $\eta_5$ as a function of $\alpha \in [0, 1]$.</figcaption></figure>

All three roots of the polynomial $p_\alpha$ are real, positive, and distinct. Since the proof is somehow tedious but elementary we have postponed it to Appendix A. Here we depict the graph of the discriminant:

$$ \Delta := -18c_0c_1c_2c_3 + 4c_2^3c_0 - c_1^2c_2^2 + 4c_3c_1^3 + 27c_3^2c_0^2 $$

motivated by the following reasoning: It holds (see, e.g., [31, Chap. IX, p.122])

$$ \Delta(\alpha) < 0 \iff \text{all three roots of } p_{\alpha} \text{ are distinct and real.} $$

Note that the coefficients $c_\ell(\alpha)$ in $p_\alpha(x)$ satisfy sign properties which directly follow from their representations (19)-(20): for $\alpha \in [0, 1]$ it holds:

$$ c_0(\alpha) > 0, \quad c_1(\alpha) < 0, \quad c_2(\alpha) > 0, \quad c_3(\alpha) < 0 $$

and $p_\alpha(x) > 0$ for all $x \le 0$ follows. Consequently: if $\Delta < 0$ in $]0, 1[$ then all three roots of $p_\alpha$ are distinct, real, and positive. The graph of $\Delta$ is shown in Figure 6.

In summary, the conforming Galerkin discretization by the Nédélec space $N_1^I(\mathcal{T}_3)$ has eight spurious wavenumbers

$$ k(\alpha) \in \left\{ \pm \sqrt{20 \frac{2 - \alpha}{1 + \alpha(1 - \alpha)}}, \pm \sqrt{x_1(\alpha)}, \pm \sqrt{x_2(\alpha)}, \pm \sqrt{x_3(\alpha)} \right\}, $$

where $x_m(\alpha)$, $1 \le m \le 3$, are the three distinct and positive roots of the equation

$$ p_{\alpha}(x) = 0. \qquad (22) $$

# 5 Conclusion

In this paper we considered three common models in two and three spatial dimensions for high-frequency scattering problems, all of which are well posed
on the continuous level. We constructed simplicial meshes for popular, low-order, conforming Galerkin finite element discretizations for these variational problems such that the discrete system is singular for certain discrete values of $k$. These meshes are highly symmetric and for all constructed examples, the number of degrees of freedom is relatively low on the domain boundary.

Even for these specially constructed meshes the system matrix is singular only for finitely many spurious frequencies $k > 0$ indicating that these cases are very rare events.

Next, we comment on the resolution condition applied to periodic copies of the Bernkopf meshes. Let $N \in \mathbb{N}$ and $h = 1/(N+1)$. Let $\mathcal{T}_3^+$ denote some conforming tetrahedral mesh of $(-1, 1)^3$ which a) contains $\mathcal{T}_3$ as a submesh and b) the four congruent triangles, which arise by inserting the two diagonals into each facet of the cube $(-1, 1)^3$, are facets of some tetrahedra in $\mathcal{T}_3^+$. For a uniform notation we set $\mathcal{T}_2^+ := \mathcal{T}_2$.

A scaling of the mesh $\mathcal{T}_d^+$, $d = 2, 3$, by a factor $h/2$ leads to a mesh $\mathcal{T}_{d,h}^+$ of $\Omega_h := (-\frac{h}{2}, \frac{h}{2})^d$ which contains the scaled version $\mathcal{T}_{d,h}$ of $\mathcal{T}_d$ as a submesh. The translation $\Phi_\mu : \Omega_h \to \Omega_h(\mu)$ is given for

$$ \mu \in \mathcal{J}_N := \{ \nu = (\nu_j)_{j=1}^d \in \mathbb{N}_0^d \mid \|\mu\|_{\max} \le N \} \quad \text{and} \quad 1 = (1)_{i=1}^d $$

by

$$ \Phi_{\mu}(\mathbf{x}) = \mathbf{x} + h \left( \mu + \frac{1}{2} \mathbf{1} \right). $$

The simplices in $\Omega_h(\mu)$, $\mu \in \mathcal{J}_N$, define a conforming simplicial mesh for $(0, 1)^d$ where each cell $\Omega_h(\mu)$ contains a copy of the scaled structured mesh $\mathcal{T}_{d,h}$. By a scaling argument (which is detailed in [4, Lem. 3.3]) one obtains that there are non-trivial homogeneous solutions of the discrete problem which are localized in one of the cells if

$$ kh = k(\alpha) $$

for the spurious wavenumbers $k(\alpha)$ defined for the Galerkin discretization of our model problems on the constructed meshes. On the other hand, this observation can also be formulated in a reversed way: since $k(\alpha)$ is independent of $h$ a condition

$$ kh < k(\alpha) $$

ensures that no non-trivial discrete homogeneous solutions exist which are localizing in one cell. Such a condition is substantially weaker than a typical condition “for some fixed $\nu > 1$, the product $k^\nu h$ must be sufficiently small” as it appears, e.g., with $\nu = 2$ in (15).

# A Analysis of the roots of $p_\alpha$ in (21)

In this appendix we prove the following lemma.
**Lemma 19** The cubic polynomial $p_\alpha$ in (21) has three distinct positive roots for any $\alpha \in [0, 1]$.

**Proof.** The coefficients $c_m(\alpha)$ in $p_\alpha(x) = \sum_{\ell=0}^3 c_\ell(\alpha) x^\alpha$ satisfy the following sign properties which directly follow from their representations (19)-(20): for $\alpha \in [0, 1]$ it holds:

$$c_0(\alpha) > 0, \quad c_1(\alpha) < 0, \quad c_2(\alpha) > 0, \quad c_3(\alpha) < 0.$$

This implies

$$\operatorname{sign} p_{\alpha}(0) = \operatorname{sign} c_{0}(\alpha) = 1, \qquad (23)$$

$$\operatorname{sign} p_{\alpha}(+\infty) = \operatorname{sign} c_{3}(\alpha) = -1. \qquad (24)$$

Next we prove that

$$\operatorname{sign} p_{\alpha}(50) = -1. \qquad (25)$$

**Case 1.** $\alpha \in [\frac{1}{2}, 1]$ and we set $b = 1 - \alpha \in [0, \frac{1}{2}]$. For $x = 50$ it holds $p_\alpha(50) = -\frac{1000}{\alpha^7}w(b)$ for

$$w(b) = 13375b^5(1-b) + 2625b^7 + b^2\tilde{w}_3(b) + 500b + 55 \qquad (26)$$

with

$$\tilde{w}_3(b) := 14875b^3 - 29495b^2 + 10905b + 567.$$

Next we show $\tilde{w}_3(b) \ge 0$ in the range $b \in [0, \frac{1}{2}]$, i.e., $\alpha \in [\frac{1}{2}, 1]$. A straightforward discussion of $\tilde{w}_3$ shows that it has a local minimum at $\frac{347}{525} + \frac{2}{8925}\sqrt{3833194} > 1$, i.e., outside the interval $[0, \frac{1}{2}]$. Hence, $\tilde{w}_3$ has no local minimum in this interval and

$$\min_{0 \le b \le \frac{1}{2}} \tilde{w}_3(b) = \min\left\{\tilde{w}_3(0), \tilde{w}_3\left(\frac{1}{2}\right)\right\} = \frac{4041}{8} > 0.$$

This directly implies $w(b) \ge 55$ and, in turn, $p_\alpha(50) < 0$ for $\alpha \in [\frac{1}{2}, 1]$.

**Case 2.** $\alpha \in [0, \frac{1}{2}]$. We write $p_\alpha(50) = -\frac{1000}{\alpha^7}v(\alpha)$ for

$$v(\alpha) := 2625\alpha^6(1-\alpha) + \alpha^4v_2(\alpha) + \alpha v_3(\alpha) + 32 \qquad (27)$$

with

$$v_2(\alpha) := 2375\alpha^2 - 3125\alpha + 3005 \quad \text{and} \quad v_3(\alpha) := 200\alpha^2 - 6688\alpha + 4256.$$

Again, a straightforward discussion yields, that $v_2$ has a local minimum at $\frac{25}{38}$ with value $\frac{150255}{76} > 0$ and hence is positive for all $\alpha \in \mathbb{R}$. The function $v_3$ has a local minimum at $\frac{418}{25} > 1$. Hence there is no local minimum in $[0, \frac{1}{2}]$ and

$$\min_{\alpha \in [0, \frac{1}{2}]} v_3(\alpha) = \min\left\{v_3(0), v_3\left(\frac{1}{2}\right)\right\} = 962.$$
We use these positivity estimates in (27) to get

$$ \min_{\alpha \in [0, \frac{1}{2}]} v(\alpha) \geq 32. $$

In turn we have proved that

$$ \forall \alpha \in [0, 1] \quad p_{\alpha}(50) < 0. $$

Finally we prove

$$ \forall \alpha \in [0, 1] \quad \exists y_{\alpha} > 50 \quad \text{sign} p_{\alpha}(y_{\alpha}) = 1 \quad (28) $$

and distinguish between two cases:

**Case 1.** For $\alpha \in [0, \frac{1}{2}]$ we prove that $p_{\alpha}(50) > 0$. It holds $p_{\alpha}(50) = \frac{1000}{\alpha^7}z(\alpha)$ with

$$ z(\alpha) := \alpha^5 z_2(\alpha) + \alpha z_3(\alpha) + 608 \quad (29) $$

with

$$ z_2(\alpha) := 70875\alpha^2 - 103500\alpha + 97875, $$

$$ z_3(\alpha) := -128415\alpha^3 - 90000\alpha^2 + 48368\alpha + 18784. $$

A straightforward analysis yields that $z_2$ has a local minimum at $\alpha = \frac{46}{63}$ with positive value $\frac{420625}{7}$ and hence $z_2(\alpha) > 0$ for all $\alpha \in \mathbb{R}$. A straightforward discussion of the function $z_3$ shows that it has a local minimum at $\alpha = -\frac{2000}{8561} - \frac{4}{128415}\sqrt{185649515} < 0$. Consequently if has no local minimum in $[0, \frac{1}{2}]$ and

$$ \min_{\alpha \in [0, 1/2]} z_3(\alpha) = \min \left\{ z_3(0), z_3\left(\frac{1}{2}\right) \right\} = \frac{35329}{8} > 0. $$

From (29) it follows $z(\alpha) \geq 608$ for all $\alpha \in [0, 1/2]$ and in turn $p_{\alpha}(50) > 0$ in this range.

**Case 2.** For $\alpha \in [\frac{1}{2}, 1]$ we prove that $p_{\alpha}(y_{\alpha}) > 0$ for $y_{\alpha} = 105 - 50\alpha$. Note that $y_{\alpha} > 50$ in this range. We obtain $p_{\alpha}(105 - 50\alpha) = \frac{125}{\alpha^7}\zeta(b)$ for $b = 1 - \alpha \in [0, \frac{1}{2}]$ and

$$ \zeta(b) = 100b^9\zeta_1(b) + b^5\zeta_3(b) + b\zeta_3(b) + 95 \quad (30) $$

with

$$ \zeta_1(b) := 517 - 210b, $$

$$ \zeta_3(b) := 12494 + 165531b - 113321b^2 - 6930b^3, $$

$$ \zeta_3(b) := 4588 + 42097b + 14067b^2 - 153097b^3. $$

Clearly, the first factor $\zeta_1$ is positive for $b \in [0, \frac{1}{2}]$. The function $\zeta_3(b)$ has a local minimum at $\alpha = -\frac{1}{20790}\sqrt{16283038531} - \frac{113321}{20790} < 0$. Hence, there is no local minimum in $[0, \frac{1}{2}]$ and

$$ \min_{b \in [0, 1/2]} \zeta_3(b) = \min \left\{ \zeta_3(0), \zeta_3\left(\frac{1}{2}\right) \right\} = 12494. $$
The function $\xi_3$ has a local minimum at $\frac{4689}{153097} - \frac{2}{459291}\sqrt{4883163429} < 0$. Again there is no local minimum in $[0, \frac{1}{2}]$ and

$$ \min_{b \in [0,1/2]} \xi_3(b) = \min \left\{ \xi_3(0), \xi_3\left(\frac{1}{2}\right) \right\} = 4588. $$

From (30) it follows that $\min_{0 \le b \le 1/2} \zeta(b) \ge 95$ and in turn $p_\alpha(105 - 50\alpha) > 0$ for any $\alpha \in [0, 1/2]$.

By using the meanvalue theorem the sign properties (23), (24), (25), (28) of $p_\alpha$ imply that all three roots of $p_\alpha$ are real, positive and distinct. ■

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