A first order system least squares method for the Helmholtz equation

Abstract

We present a first order system least squares (FOSLS) method for the Helmholtz equation at high wave number k, which always leads to a Hermitian positive definite algebraic system. By utilizing a non-trivial solution decomposition to the dual FOSLS problem which is quite different from that of the standard finite element methods, we give an 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(logk). Numerical experiments are given to verify the theoretical results.