A domain decomposition method for the Helmholtz equation in a multilayer domain

The two-dimensional Helmholtz equation for problems where the physical domain consists of layers with different material properties is studied. An efficient preconditioner for iterative solution of the problem is constructed. The problem is discretized with fourth-order accurate finite difference operators. For the construction of the radiation boundary conditions a fourth-order finite element method also is used. The large, sparse, complex, indefinite, and ill-conditioned system of equations that arises is solved with preconditioned restarted GMRES. A domain decomposition method is used, in which the preconditioning is based on the Schur complement algorithm with "fast Poisson-type" preconditioners for the subdomains. The memory requirements for the preconditioner are nearly linear in the number of unknowns. The arithmetic complexity for each iteration is low, whereas the construction of the preconditioner is a bit more expensive. Electromagnetic wave propagation in a three-layered waveguide is used as a model problem. Numerical experiments show that convergence is achieved in a few iterations. Compared with banded Gaussian elimination, which is a standard solution method, the iterative method shows significant gain in both memory requirements and arithmetic complexity. Furthermore, the relative gain grows when the problem size increases.