Abstract
In this paper we generalize and improve a recently developed domain decomposition preconditioner for the iterative solution of discretized Helmholtz equations. We introduce an improved method for transmission at the internal boundaries using perfectly matched layers. Simultaneous forward and backward sweeps are introduced, thereby improving the possibilities for parallellization. Finally, the method is combined with an outer two-grid iteration. The method is studied theoretically and with numerical examples. It is shown that the modifications lead to substantial decreases in computation time and memory use, so that computation times become comparable to that of the fastests methods currently in the literature for problems with up to 108 degrees of freedom.
Highlights
The linear systems resulting from discretizing the high-frequency Helmholtz equation have been a challenge for mathematicians for a long time [6, 7]
We propose to combine the domain decomposition with a two-grid method, in such a way that the exact inverse at the coarse level of a two-grid preconditioner is replaced by an approximate inverse given by a domain decomposition preconditioner
We show that the method produces an exact solution on the strip with constant k, similar to the domain decomposition method of [15]
Summary
The linear systems resulting from discretizing the high-frequency Helmholtz equation have been a challenge for mathematicians for a long time [6, 7]. The second is the use of simultaneous forward and backward sweeps as opposed to consecutive ones This idea has been previously tried with other types of domain decomposition in [20]. The numerical examples below show that the good convergence properties carry over to the TGSP method, i.e. when the exact coarse level solver is replaced by a domain decomposition preconditioner. A theoretical result concerning the domain decomposition method with new transmission and simultaneous forward and backward sweeps is presented. In 2-D we study problems with up to 7 · 106 degrees of freedom, and in 3-D with up to 108 degrees of freedom In both cases it is possible to use quite thin PML layers for the domain decomposition preconditioner, e.g. wpml = 3 or 4 grid cells thick. In an Appendix we discuss the discretization of the operators when PML layers and multigrid are combined using modified mesh coarsening in the PML layers
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