A parallel domain decomposition method for the Helmholtz equation

A parallel solver for the Helmholtz equation in a domain consisting of layers with different material properties is presented. A fourth-order accurate finite difference discretization is used. The arising system of equations is solved with a preconditioned Krylov subspace method. A domain decomposition framework is employed, where fast transform subdomain preconditioners are used. Three ways of treating the Schur complement of the preconditioner are investigated, and the corresponding preconditioned iterative methods are compared with a standard direct method. It is noted that the convergence rate of the iterative methods is closely related to how the Schur complement system for the preconditioner is formed, and how accurately it is solved. However, in almost all cases, the gain in both memory requirements and arithmetic complexity is large compared with the direct method. Furthermore, the gain increases with problem size, allowing problems with many unknowns to be solved efficiently. The efficiency is further improved by parallelization using message-passing, enabling us to solve even larger Helmholtz problems in less time