Symmetrized method with optimized second-order conditions for the Helmholtz equation

A schwarz type domain decomposition method for the Helmholtz equation is considered. The interface conditions involve second order tangential derivatives which are optimized (OO2, Optimized Order 2) for a fast convergence. The substructured form of the algorithm is symmetrized so that the symmetric-QMR algorithm can be used as an accelerator of the convergence. Numerical results are shown. We consider the following type of problem: Find u such that L(u) = f in Ω (1) C(u) = g on ∂Ω (2) where L and C are partial differential operators. We consider Schwarz-type methods for the solving of this problem. The original Schwarz algorithm is based on a decomposition of the domain Ω into overlapping subdomains and the solving of Dirichlet boundary value problems in the subdomains. It has been proposed in [15] to use of more general boundary conditions for the subproblems in order to use a nonoverlapping decomposition of the domain. The convergence speed is also increased dramatically. More precisely, the computational domain Ω is decomposed into N nonoverlapping subdomains: