A first order system least squares method for the Helmholtz equation

TL;DR

This paper introduces a first order system least squares (FOSLS) method for high-frequency Helmholtz problems, providing explicit error estimates and demonstrating quasi-optimal accuracy under certain conditions.

Contribution

It develops a novel FOSLS approach for the Helmholtz equation that ensures positive definiteness and offers explicit error bounds dependent on mesh size, polynomial degree, and wave number.

Findings

The FOSLS method yields a Hermitian positive definite system.

Error estimates depend explicitly on h, p, and k.

Numerical results confirm theoretical error bounds.

Abstract

We present a first order system least squares (FOSLS) method for the Helmholtz equation at high wave number k, which always deduces Hermitian positive definite algebraic system. By utilizing a non-trivial solution decomposition to the dual FOSLS problem which is quite different from that of standard finite element method, we give error analysis to the hp-version of the FOSLS method where the dependence on the mesh size h, the approximation order p, and the wave number k is given explicitly. In particular, under some assumption of the boundary of the domain, the L2 norm error estimate of the scalar solution from the FOSLS method is shown to be quasi optimal under the condition that kh/p is sufficiently small and the polynomial degree p is at least O(\log k). Numerical experiments are given to verify the theoretical results.