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Additive Schwarz methods with nonreflecting boundary conditions for the parallel computation of Helmholtz problems. (English) Zbl 0910.65086

Mandel, Jan (ed.) et al., Domain decomposition methods 10. The 10th international conference, Boulder, CO, USA, August 10–14, 1997. Providence, RI: AMS, American Mathematical Society. Contemp. Math. 218, 325-333 (1998).
Recent advances in discretizations and preconditioners for solving the exterior Helmholtz problem are combined in a single code and their benefits evaluated on a parameterized model. Motivated by large-scale simulations, we consider iterative parallel domain decomposition algorithms of additive Schwarz-type. The preconditioning action in such algorithms can be built out of nonoverlapping or overlapping subdomain solutions with homogeneous Sommerfeld-type transmission conditions on the artificially introduced subdomain interfaces. Generalizing the usual Dirichlet Schwarz interface conditions, such Sommerfeld-type conditions avoid the possibility of resonant modes and thereby assure the uniqueness of the solution in each subdomain.
Although the discretized Helmholtz linear system matrix is sparse, for a large number of equations direct methods are inadequate. Moreover, the Helmholtz operator tends to be indefinite for practical values of the wavenumber and the mesh parameter, leading to ill-conditioning. As a result conventional iterative methods do not converge for all values of the wavenumber, or may converge very slowly. For example, resonances can occur when conventional Schwarz-based preconditioners are assembled from Dirichlet subdomain problems. In view of these difficulties, this work focuses on developing a family of parallel Krylov-Schwarz algorithms for Helmholtz problems based on subdomain problems with approximate local transmission boundary conditions.
For the entire collection see [Zbl 0895.00045].

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65Y05 Parallel numerical computation
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
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