Analysis of the $hp$-version of a first order system least squares method for the Helmholtz equation

TL;DR

This paper provides an analysis of the $hp$-version least squares method for the Helmholtz equation, showing improved convergence rates under specific mesh and polynomial degree conditions, especially for analytic domains.

Contribution

It extends previous wavenumber-explicit analysis to demonstrate $L^2$-convergence with enhanced rates for the $hp$-method on analytic domains.

Findings

Improved convergence rates in mesh size $h$ and polynomial degree $p$.

Conditions on $hk/p$ and $p/ ext{log}k$ for optimal convergence.

Analysis applicable to domains with analytic boundaries.

Abstract

Extending the wavenumber-explicit analysis of [Chen & Qiu, J. Comput. Appl. Math. 309 (2017)], we analyze the $L^2$-convergence of a least squares method for the Helmholtz equation with wavenumber $k$. For domains with an analytic boundary, we obtain improved rates in the mesh size $h$ and the polynomial degree $p$ under the scale resolution condition that $hk/p$ is sufficiently small and $p/\log k$ is sufficiently large.