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

TL;DR

This paper provides a detailed analysis of the stability and convergence of hp finite element methods for solving Helmholtz problems with heterogeneous media at high wavenumbers, including various boundary conditions.

Contribution

It offers the first wavenumber-explicit convergence analysis for hp finite element discretizations of complex Helmholtz problems with piecewise analytic coefficients.

Findings

Establishes stability bounds explicit in wavenumber k

Derives convergence rates for hp finite element methods at large k

Includes analysis for multiple boundary conditions and PMLs

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.