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Wavenumber-explicit stability and convergence analysis of hp finite element discretizations of Helmholtz problems in piecewise smooth media

M. Bernkopf, T. Chaumont-Frelet, J. M. Melenk

Abstract

We present a wavenumber-explicit convergence analysis of the hp finite element method applied to a class of heterogeneous Helmholtz problems with piecewise analytic coefficients at large wavenumber $k$. Our analysis covers the heterogeneous Helmholtz equation with Robin, exact Dirichlet-to-Neumann, and second order absorbing boundary conditions, as well as perfectly matched layers.

Wavenumber-explicit stability and convergence analysis of hp finite element discretizations of Helmholtz problems in piecewise smooth media

Abstract

We present a wavenumber-explicit convergence analysis of the hp finite element method applied to a class of heterogeneous Helmholtz problems with piecewise analytic coefficients at large wavenumber . Our analysis covers the heterogeneous Helmholtz equation with Robin, exact Dirichlet-to-Neumann, and second order absorbing boundary conditions, as well as perfectly matched layers.
Paper Structure (23 sections, 26 theorems, 243 equations, 1 table)

This paper contains 23 sections, 26 theorems, 243 equations, 1 table.

Table of Contents

  1. Introduction
  2. Assumptions and Problem Specific Notation
  3. Wavenumber
  4. Generic constants
  5. Geometry
  6. Function spaces
  7. Informal explanation of the proof of the main result and the assumptions
  8. Time-harmonic wave propagation problem
  9. Auxiliary positive problem
  10. Abstract Contraction Argument
  11. Filtering operators
  12. A contraction argument
  13. Regularity splitting of time-harmonic solutions
  14. Stability and Convergence of Abstract Galerkin Discretizations
  15. Abstract Galerkin discretizations
  16. ...and 8 more sections

Key Result

Lemma 2.2

Let $\omega_1$, $\omega_2 \subset {\mathbb R}^d$ be bounded open and $g: \omega_1 \rightarrow \omega_2$ be a bijection and analytic on the closed set $\overline{\omega}_1$. Let $f_1$ be analytic on the closed set $\overline{\omega}_2$ and $f_2 \in \mathfrak{A}(M_f,\gamma_f,\omega_2)$. Then there are

Theorems & Definitions (60)

  • Remark 2.1
  • Lemma 2.2: melenk-sauter21
  • Remark 2.3
  • Remark 2.4: On the stability assumption \ref{['WP2']}
  • Proposition 3.1: Surface filters
  • proof
  • Proposition 3.2: Volume filters
  • proof
  • Remark 3.3: Other constructions of high and low pass filters
  • Lemma 3.4: Unified contraction argument
  • ...and 50 more