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Analysis of the $hp$-version of a first order system least squares method for the Helmholtz equation

Maximilian Bernkopf, Jens Markus Melenk

TL;DR

This work analyzes the $L^2$ convergence of a first-order system least squares discretization for the Helmholtz equation with wavenumber $k$ on analytic domains. By refining a duality argument and developing $H(\operatorname{div})$-conforming $RT_p$ and $BDM_p$ approximation operators, the authors obtain an $L^2$ a priori error bound under the scale-resolution conditions $\frac{kh}{p}$ small and $\frac{p}{\log k}$ large, with improved $p$-dependence. The approach combines a detailed dual decomposition into analytic and highly regular components with $p$-optimal polynomial approximations on reference elements and Piola-transformed $H(\operatorname{div})$-conforming operators. Numerical experiments corroborate the theoretical hp-convergence behavior, illustrating the method’s potential for high-frequency Helmholtz problems and highlighting the trade-offs in degrees of freedom compared to standard $h$-FEM.

Abstract

Extending the wavenumber-explicit analysis of [Chen & Qiu, J. Comput. Appl. Math. 309 (2017)], we analyze the $L^2$-convergence of a least squares method for the Helmholtz equation with wavenumber $k$. For domains with an analytic boundary, we obtain improved rates in the mesh size $h$ and the polynomial degree $p$ under the scale resolution condition that $hk/p$ is sufficiently small and $p/\log k$ is sufficiently large.

Analysis of the $hp$-version of a first order system least squares method for the Helmholtz equation

TL;DR

This work analyzes the convergence of a first-order system least squares discretization for the Helmholtz equation with wavenumber on analytic domains. By refining a duality argument and developing -conforming and approximation operators, the authors obtain an a priori error bound under the scale-resolution conditions small and large, with improved -dependence. The approach combines a detailed dual decomposition into analytic and highly regular components with -optimal polynomial approximations on reference elements and Piola-transformed -conforming operators. Numerical experiments corroborate the theoretical hp-convergence behavior, illustrating the method’s potential for high-frequency Helmholtz problems and highlighting the trade-offs in degrees of freedom compared to standard -FEM.

Abstract

Extending the wavenumber-explicit analysis of [Chen & Qiu, J. Comput. Appl. Math. 309 (2017)], we analyze the -convergence of a least squares method for the Helmholtz equation with wavenumber . For domains with an analytic boundary, we obtain improved rates in the mesh size and the polynomial degree under the scale resolution condition that is sufficiently small and is sufficiently large.

Paper Structure

This paper contains 11 sections, 12 theorems, 91 equations, 4 figures.

Table of Contents

  1. Introduction
  2. First order system least squares method and useful results
  3. First order system least squares
  4. Auxiliary results
  5. Duality Argument
  6. Approximation properties of Raviart-Thomas and Brezzi-Douglas-Marini spaces
  7. Preliminaries
  8. Polynomial approximation on the reference element
  9. $\pmb{H}(\operatorname{div}, \Omega)$-conforming approximation operators
  10. A priori estimate
  11. Numerical examples

Key Result

proposition 1

Let $\Omega \subset \mathbb{R}^d, d \in \left\{ 2, 3 \right\}$, be a bounded Lipschitz domain with an analytic boundary. Fix $s \in \mathbb{N}_0$. Then there exist constants $C, \gamma > 0$ independent of $k$ such that for every $f \in H^s(\Omega)$ and $g \in H^{s+1/2}(\partial \Omega)$ the solution

Figures (4)

  • Figure 1: Comparison between the $h$-FEM (left) and $h$-FOSLS (right) for $p = 1$, $2$, $3$, $4$ as described in Example \ref{['bernkopf_melenk_example:smooth_solution']}. The reference line in black corresponds to $h^{p+1}$.
  • Figure 2: Comparison between the $p$-FEM (left) and $p$-FOSLS (right) for $kh = 5$ as described in Example \ref{['bernkopf_melenk_example:singular_solution_corner_domain']}. We include the reference lines $p^{-4 \cdot 2/3} = p^{-8/3}$ and $p^{-(2 \cdot 2/3 + 1)} = p^{-7/3}$.
  • Figure 3: Comparison between the $h$-FEM (left) and $h$-FOSLS (right) for $p = 1,\ldots,5$ as described in Example \ref{['bernkopf_melenk_example:singular_solution_singular_f']}. The reference line in black corresponds to $h^{\min(2.5, p+1)}$.
  • Figure 4: Comparison between the error terms $e_1 \coloneqq \left\lVert i k \pmb{e^\varphi} + \nabla e^u\right\rVert_{L^2(\Omega)}$ (left) and $e_2 \coloneqq \left\lVert i k e^u + \nabla \cdot \pmb{e^\varphi}\right\rVert_{L^2(\Omega)}$ (right) for $p = 1,\ldots,5$ as described in Remark \ref{['bernkopf_melenk_remark:error_terms_explaining_imporved_convergence']} and Example \ref{['bernkopf_melenk_example:singular_solution_singular_f']}. The reference line on the left corresponds to $h^1$ for $p = 1$ and $h^{1.5}$ for $p > 1$. The reference line on the right corresponds to $h^{1/2}$.

Theorems & Definitions (29)

  • remark 1
  • remark 2
  • proposition 1: melenk-parsania-sauter13 combined with baskin-spence-wunsch16
  • remark 3
  • proposition 2: melenk05
  • lemma 1
  • proof
  • definition 1
  • definition 2: restriction property
  • remark 4
  • ...and 19 more