Sharp Stability Wavenumber-explicit Bounds for 2D Helmholtz Equations

TL;DR

This paper derives new sharp stability bounds for the 2D Helmholtz equation with mixed boundary conditions, providing insights into its solution behavior at large wavenumbers, which is crucial for improving numerical methods.

Contribution

It introduces novel wavenumber-explicit stability bounds for the 2D Helmholtz equation with mixed boundary conditions, using Fourier analysis and Rellich's identity.

Findings

Derived sharp stability bounds that are optimal in wavenumber dependence

Established the importance of stability bounds for numerical scheme development

Provided examples demonstrating the bounds' optimality

Abstract

Numerically solving the 2D Helmholtz equation is widely known to be very difficult largely due to its highly oscillatory solution, which brings about the pollution effect. A very fine mesh size is necessary to deal with a large wavenumber leading to a severely ill-conditioned huge coefficient matrix. To understand and tackle such challenges, it is crucial to analyze how the solution of the 2D Helmholtz equation depends on (perturbed) boundary and source data for large wavenumbers. In fact, this stability analysis is critical in the error analysis and development of effective numerical schemes. Therefore, in this paper, we analyze and derive several new sharp wavenumber-explicit stability bounds for the 2D Helmholtz equation with inhomogeneous mixed boundary conditions: Dirichlet, Neumann, and impedance. We use Fourier techniques, the Rellich's identity, and a lifting strategy to establish these stability bounds. Some examples are given to show the optimality of our derived wavenumber-explicit stability bounds.