A first order system least squares method for the Helmholtz equation

We present a first order system least squares (FOSLS) method for the Helmholtz equation at high wave number kk, 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 hphp-version of the FOSLS method where the dependence on the mesh size hh, the approximation order pp, and the wave number kk is given explicitly. In particular, under some assumption of the boundary of the domain, the L2L2 norm error estimate of the scalar solution from the FOSLS method is shown to be quasi optimal under the condition that kh/pkh/p is sufficiently small and the polynomial degree pp is at least O(logk)O(logk). Numerical experiments are given to verify the theoretical results.