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Wave number-Explicit Analysis for Galerkin Discretizations of Lossy Helmholtz Problems

Jens M. Melenk, Stefan A. Sauter, Céline Torres

TL;DR

The paper addresses stability and convergence of Galerkin discretizations for the lossy Helmholtz operator $-\Delta u+\zeta^{2}u$ under Robin boundary conditions with wave numbers $\zeta$ in the right complex half-plane ($\operatorname{Re}\zeta\ge 0$, $|\zeta|\ge 1$). It develops a frequency-explicit framework based on a continuity bound, two inf-sup estimates that cover both sectorial and non-sectorial regimes, and a regular decomposition of the Helmholtz solution to separate analytic and low-regularity parts; these ingredients enable robust analysis for $hp$-FEM discretizations via adjoint approximability and discrete inf-sup constants. The work yields quasi-optimality results under a resolution condition in non-sectorial regions and unconditional quasi-optimality in sectorial regions, with detailed estimates that are explicit in $\operatorname{Re}\zeta$ and $\operatorname{Im}\zeta$. Numerical experiments corroborate the theory, showing how pollution effects diminish with higher polynomial degree and how the method behaves across different orientations of $\zeta$ in the complex plane. Overall, the results provide a rigorous, explicit framework for stable and accurate Galerkin discretizations of lossy Helmholtz problems across a broad class of complex frequencies, informing effective $hp$-FEM strategies and numerical contour integration contexts.

Abstract

We present a stability and convergence theory for the lossy Helmholtz equation and its Galerkin discretization. The boundary conditions are of Robin type. All estimates are explicit with respect to the real and imaginary part of the complex wave number $ζ\in\mathbb{C}$, $\operatorname{Re}ζ\geq0$, $\left\vert ζ\right\vert \geq1$. For the extreme cases $ζ\in\operatorname*{i}\mathbb{R}$ and $ζ\in\mathbb{R}_{\geq0}$, the estimates coincide with the existing estimates in the literature and exhibit a seamless transition between these cases in the right complex half plane.

Wave number-Explicit Analysis for Galerkin Discretizations of Lossy Helmholtz Problems

TL;DR

The paper addresses stability and convergence of Galerkin discretizations for the lossy Helmholtz operator under Robin boundary conditions with wave numbers in the right complex half-plane (, ). It develops a frequency-explicit framework based on a continuity bound, two inf-sup estimates that cover both sectorial and non-sectorial regimes, and a regular decomposition of the Helmholtz solution to separate analytic and low-regularity parts; these ingredients enable robust analysis for -FEM discretizations via adjoint approximability and discrete inf-sup constants. The work yields quasi-optimality results under a resolution condition in non-sectorial regions and unconditional quasi-optimality in sectorial regions, with detailed estimates that are explicit in and . Numerical experiments corroborate the theory, showing how pollution effects diminish with higher polynomial degree and how the method behaves across different orientations of in the complex plane. Overall, the results provide a rigorous, explicit framework for stable and accurate Galerkin discretizations of lossy Helmholtz problems across a broad class of complex frequencies, informing effective -FEM strategies and numerical contour integration contexts.

Abstract

We present a stability and convergence theory for the lossy Helmholtz equation and its Galerkin discretization. The boundary conditions are of Robin type. All estimates are explicit with respect to the real and imaginary part of the complex wave number , , . For the extreme cases and , the estimates coincide with the existing estimates in the literature and exhibit a seamless transition between these cases in the right complex half plane.

Paper Structure

This paper contains 14 sections, 18 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. The Continuity Constant
  4. The Inf-Sup Constant of $a_{\zeta}\left( \cdot,\cdot\right)$
  5. The Inf-Sup Constant for $\operatorname{Re}\zeta>0$
  6. The Inf-Sup Constant of $a_{\zeta}\left( \cdot,\cdot\right)$ for $\zeta\in S_{\beta}^{c}$
  7. Regular Decomposition of the Helmholtz Solution
  8. The Full Space Adjoint Problem for $\zeta\in S_{\beta}^{c}$
  9. The Helmholtz Solution with Robin Boundary Conditions
  10. The Solution Operators $N_{\zeta}$, $S_{\zeta}^{\Delta}$, $S_{\zeta}^{L}$, and $S^{\zeta}$
  11. Discretization
  12. Discrete Inf-Sup Constant $\gamma_{\operatorname*{disc}}$ and Quasi-Optimality
  13. Impact on $hp$-FEM Approximation
  14. Numerical Experiments

Key Result

Theorem 3.1

The sesquilinear form $a_{\zeta}$ is continuous and with $C_{b}$ independent of $\zeta\in\mathbb{C}_{\geq0}$.

Theorems & Definitions (21)

  • Theorem 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Remark 4.3
  • Theorem 4.4
  • Lemma 5.1
  • Remark 5.2
  • Theorem 5.3
  • Lemma 5.4: Def. of lifting $G^N$
  • Lemma 5.5: properties of $L_\Gamma$ and $H_\Gamma$
  • ...and 11 more