Krause, Rolf Conen, Lea 2015-01-07 In this thesis, we present a two-level domain decomposition method for the iterative solution of the heterogeneous Helmholtz equation. The Helmholtz equation governs wave propagation and scattering phenomena arising in a wide range of engineering applications. Its discretization with piecewise linear finite elements results in typically large, ill-conditioned, indefinite, and non- Hermitian linear systems of equations, for which standard iterative and direct methods encounter convergence problems. Therefore, especially designed methods are needed. The inherently parallel domain decomposition methods constitute a promising class of preconditioners, as they subdivide the large problems into smaller subproblems and are hence able to cope with many degrees of freedom. An essential element of these methods is a good coarse space. Here, the Helmholtz equation presents a particular challenge, as even slight deviations from the optimal choice can be fatal. We develop a coarse space that is based on local eigenproblems involving the Dirichlet-to-Neumann operator. Our construction is completely automatic, ensuring good convergence rates without the need for parameter tuning. Moreover, it naturally respects local variations in the wave number and is hence suited also for heterogeneous Helmholtz problems. Apart from the question of how to design the coarse space, we also investigate the question of how to incorporate the coarse space into the method. Also here the fact that the stiffness matrix is non-Hermitian and indefinite constitutes a major challenge. The resulting method is parallel by design and its efficiency is investigated for two- and three-dimensional homogeneous and heterogeneous numerical examples. application/pdf https://susi.usi.ch/global/documents/318572 https://localhost:5000/ark:/12658/srd1318572 https://susi.usi.ch/documents/318572/files/2015INFO001.pdf eng info:eu-repo/semantics/altIdentifier/urn/urn:nbn:ch:rero-006-113732 info:eu-repo/semantics/altIdentifier/ark/12658/srd1318572 info:eu-repo/semantics/openAccess License undefined Helmholtz equation Overlapping domain decomposition Iterative solver Parallel method Coarse space Dirichlet-to-Neumann operator Finite element method info:eu-repo/classification/udc/004 Domain decomposition preconditioning for the Helmholtz equation : a coarse space based on local Dirichlet-to-Neumann maps http://purl.org/coar/resource_type/c_db06