Domain decomposition method for harmonic wave propagation: A general presentation
A general presentation of nonoverlapping domain decomposition methods for harmonic wave propagation models is given. The key points to overcome the main difficulties in nonoverlapping iterative methods (communications through the interfaces) are the choice of good transmission conditions and the choice of the iterative scheme. The model problem which is considered in the major part of the article is the Helmholtz equation. The authors reformulate the transmission problem in terms of a new one whose unknowns are quantities defined at the interfaces between the defined subdomains. They identify an abstract general form of this problem via the introduction of appropriate linear operators. Various iterative domain decomposition algorithms and their variants using local transmission problems and global transmission operators are discussed. Some convergence results for two relaxed algorithms (Jacobi and Gauss-Seidel) are established. In the particular case of two subdomains, the authors analyze in a more quantitive way the speed of convergence of the presented algorithms. Various extensions within the abstract formalism are presented too. The theoretical conclusions are illustrated by results for 2D numerical simulations on simple test examples where the space domain is a rectangle and the space discretization is obtained by the mixed finite element method.
scientific article; zbMATH DE number 1488462
scientific article