Matrix-Free Iterative Methods for Parallel Finite Element Computations in Acoustics

In this paper we consider the iterative solution of non-Hermitian, indefinite and complex-valued matrix problems arising from finite element discretization of time-harmonic acoustics problems in exterior domains. The computational model is based on using Galerkin least squares finite element methods for the acoustic fluid, combined with the non-local Dirichlet-to-Neumann map as the radiation boundary condition. The emphasis here is to develop efficient computational procedures that are suitable for parallel iterative solution of large-scale problems. In this context, we develop a low-storage implementation of the non-local DtN map which allows the use of this exact boundary condition without any storage penalties related to its non-local nature. In order to accelerate iterative convergence, we consider a multilevel preconditioning approach based on the h-version of the hierarchical finite element method. Finite element formulations that employ hierarchical shape functions yield better conditioned matrices than formulations based on the usual Lagrange functions. This improved conditioning translates into a faster rate of convergence if projections between nodal and hierarchical basis functions are used to construct the preconditioning operator. We present numerical results for the solution of two-dimensional scattering problems to examine convergence rates that are realized on practical discretizations.