Iterative and multigrid methods for wave problems with complex-valued boundaries

The ultimate goal of many a research is to develop an efficient solver for wave and scattering problems that are described by the Helmholtz equation defined on infinite domains. With this thesis we contribute our ideas and insights to this rich but still incomplete branch of applied mathematics.;Exterior complex scaling is used to enforce outgoing boundary conditions on the truncated numerical domain. In this light, we have developed an innovative multigrid inverted preconditioner for the numerical solution of indefinite Helmholtz problems with Krylov subspace methods, that is based on complex stretched grids. The multigrid method is further stabilized with non-standard smoothing components for the inversion of competitive preconditioners of which the wave number dependency is examined. The ideas are founded on new theoretical results on the spectrum of the Helmholtz operator with a constant wave number that is discretized with complex-valued mesh widths. Numerical results, on the one hand, confirm the analysis and, on the other hand, verify its usefulness for more general heterogeneous two-and three-dimensional Helmholtz models that originate from both acoustics and quantum mechanical break-up problems.