Unique continuation for the Helmholtz equation using stabilized finite element methods

TL;DR

This paper develops a stabilized finite element method for the Helmholtz equation's unique continuation problem, providing stability estimates and error bounds that depend explicitly on the wave number, with numerical validation.

Contribution

It introduces a novel stabilized finite element approach with explicit wave number dependence and stability analysis for the Helmholtz equation's unique continuation problem.

Findings

Conditional stability estimates with linear growth in wave number

Explicit error bounds for finite element approximation

Numerical illustrations confirming theoretical results

Abstract

In this work we consider the computational approximation of a unique continuation problem for the Helmholtz equation using a stabilized finite element method. First conditional stability estimates are derived for which, under a convexity assumption on the geometry, the constants grow at most linearly in the wave number. Then these estimates are used to obtain error bounds for the finite element method that are explicit with respect to the wave number. Some numerical illustrations are given.