Computer Methods in Applied Mechanics and Engineering
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Discrete Dispersion Relation for hp-Version Finite Element Approximation at High Wave Number
SIAM Journal on Numerical Analysis
Journal of Scientific Computing
A Refined Galerkin Error and Stability Analysis for Highly Indefinite Variational Problems
SIAM Journal on Numerical Analysis
The Trefftz method for the Helmholtz equation with degeneracy
Applied Numerical Mathematics
Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number
SIAM Journal on Numerical Analysis
Dispersive and Dissipative Behavior of the Spectral Element Method
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
Plane Wave Discontinuous Galerkin Methods for the 2D Helmholtz Equation: Analysis of the $p$-Version
SIAM Journal on Numerical Analysis
Convergence analysis of a multigrid algorithm for the acoustic single layer equation
Applied Numerical Mathematics
Wavenumber-Explicit $hp$-BEM for High Frequency Scattering
SIAM Journal on Numerical Analysis
General DG-Methods for Highly Indefinite Helmholtz Problems
Journal of Scientific Computing
An analysis of discretizations of the Helmholtz equation in L2 and in negative norms
Computers & Mathematics with Applications
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We develop a stability and convergence theory for a class of highly indefinite elliptic boundary value problems (bvps) by considering the Helmholtz equation at high wavenumber $k$ as our model problem. The key element in this theory is a novel $k$-explicit regularity theory for Helmholtz bvps that is based on decomposing the solution into two parts: the first part has the Sobolev regularity properties expected of second order elliptic PDEs but features $k$-independent regularity constants; the second part is an analytic function for which $k$-explicit bounds for all derivatives are given. This decomposition is worked out in detail for several types of bvps, namely, the Helmholtz equation in bounded smooth domains or convex polygonal domains with Robin boundary conditions and in exterior domains with Dirichlet boundary conditions. We present an error analysis for the classical $hp$-version of the finite element method ($hp$-FEM) where the dependence on the mesh width $h$, the approximation order $p$, and the wavenumber $k$ is given explicitly. In particular, under the assumption that the solution operator for Helmholtz problems is polynomially bounded in $k$, it is shown that quasi optimality is obtained under the conditions that $kh/p$ is sufficiently small and the polynomial degree $p$ is at least O(log $k$).