Matrix-Free Parallel Preconditioned Iterative Solvers for the 2D Helmholtz Equation Discretized with Finite Differences

Abstract

We present a matrix-free parallel iterative solver for the Helmholtz equation related to applications in seismic problems and study its parallel performance. We apply Krylov subspace methods, GMRES, Bi-CGSTAB and IDR(s), to solve the linear system obtained from a second-order finite difference discretization. The Complex Shifted Laplace Preconditioner (CSLP) is employed to improve the convergence of Krylov solvers. The preconditioner is approximately inverted by multigrid iterations. For parallel computing, the global domain is partitioned blockwise. The standard MPI library is employed for data communication. The matrix-vector multiplication and preconditioning operator are implemented in a matrix-free way instead of constructing large, memory-consuming coefficient matrices. These adjustments lead to direct improvements in terms of memory consumption. Numerical experiments of model problems show that the matrix-free parallel solution method has satisfactory parallel performance and weak scalability. It allows us to solve larger problems in parallel to obtain more accurate numerical solutions.