Influence of perturbations in transmission conditions on the convergence of the domain decomposition method for the Helmholtz equation

The domain decomposition is a widely used technique applied in parallel iterative solvers of the Helmholtz equation with the convergence rate controlled by the quality of transmission conditions. The optimal conditions are those based on the Dirichlet-to-Neumann map. However, if grid-based numerical methods are used to solve the Helmholtz equation, this map needs to be localized, which induces artificial grid-independent perturbations in the transmission conditions. In this paper it is proved that if the perturbations are nonsymmetric, i.e. different errors are induced in the adjoint subdomains, then the domain decomposition method converges to a solution of a problem differing from the original one. In other words, there exists an irreducible error.