Adaptive FEM for Helmholtz Equation with Large Wave Number

TL;DR

This paper develops an adaptive finite element method for the Helmholtz equation with large wave numbers, providing rigorous error bounds, an adaptive algorithm, and numerical validation to improve solution accuracy in challenging regimes.

Contribution

It introduces a new adaptive FEM algorithm with proven convergence and quasi-optimality for high wave number Helmholtz problems, addressing pollution errors.

Findings

Standard residual error estimators underestimate true error in preasymptotic regime.

Adaptive CIP-FEM reduces pollution error significantly.

Numerical tests confirm theoretical error bounds and estimator reliability.

Abstract

A posteriori upper and lower bounds are derived for the linear finite element method (FEM) for the Helmholtz equation with large wave number. It is proved rigorously that the standard residual type error estimator seriously underestimates the true error of the FE solution for the mesh size $h$ in the preasymptotic regime, which is first observed by Babu\v{s}ka, et al. for an one dimensional problem. By establishing an equivalence relationship between the error estimators for the FE solution and the corresponding elliptic projection of the exact solution, an adaptive algorithm is proposed and its convergence and quasi-optimality are proved under condition that $k^3h_0^{1+\al}$ is sufficiently small, where $h_0$ is the initial mesh size and $\frac12<\al\le 1$ is a regularity constant depending on the maximum reentrant angle of the domain. Numerical tests are given to verify the theoretical findings and to show that the adaptive continuous interior penalty finite element method (CIP-FEM) with appropriately selected penalty parameters can greatly reduce the pollution error and hence the residual type error estimator for this CIP-FEM is reliable and efficient even in the preasymptotic regime.