A discontinuous least squares finite element method for Helmholtz equations

TL;DR

This paper introduces a novel discontinuous least squares finite element method for Helmholtz equations, achieving optimal convergence and facilitating adaptive mesh refinement with practical implementation advantages.

Contribution

It develops a new discontinuous least squares approach with explicit wavenumber error estimates and natural a posteriori error estimation for Helmholtz problems.

Findings

Achieves optimal convergence in energy norm

Provides explicit wavenumber error estimates

Enables efficient adaptive mesh refinement

Abstract

We propose a discontinuous least squares finite element method for solving the Helmholtz equation. The method is based on the L2 norm least squares functional with the weak imposition of the continuity across the interior faces as well as the boundary conditions. We minimize the functional over the discontinuous polynomial spaces to seek numerical solutions. The wavenumber explicit error estimates to our method are established. The optimal convergence rate in the energy norm with respect to a fixed wavenumber is attained. The least squares functional can naturally serve as a posteriori estimator in the h-adaptive procedure. It is convenient to implement the code due to the usage of discontinuous elements. Numerical results in two and three dimensions are presented to verify the error estimates.