A parallel domain decomposition method for the Helmholtz equation in layered media

Abstract

An efficient domain decomposition method and its parallel implementation for the solution of the Helmholtz equation in three-dimensional layered media are considered. A modified trilinear finite element discretization scheme is applied to the equation system leading to fourth-order phase accuracy and thereby reducing the pollution error considerably. The resulting linear system is solved with the GMRES method using a multiplicative nonoverlapping domain decomposition preconditioner with layers defining the subdomains. This right preconditioner is constructed by embedding each layer into a rectangular domain and by employing a fast direct solver. Due to the construction of the preconditioner the iterations can be reduced to a subspace corresponding to the interfaces between the layers. Numerical experiments with several test cases demonstrate the effectiveness and scalability of the proposed method and ability to solve large-scale problems with up to billions of unknowns.