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A domain decomposition method for the Helmholtz equation and related optimal control problems. (English) Zbl 0884.65118

A domain decomposition method (splitting the domain into smaller sub-domains and solving a sequence of similar sub-problems on these sub-domains) is described for the exterior Helmholtz equation with the lowest order absorbing boundary condition and for its extension to optimal control problems governed by this equation. The transmission conditions (interface conditions) are reformulated as mixed (Robin) boundary conditions to assert the problem being well-posed.

A Helmholtz adaption of the Schwarz algorithm for elliptic problems [cf. P. L. Lions, in: T. F. Chan et el. (ed.), Third Int. Symp. Domain Decomp. Meth., SIAM, Philadelphia, 202-223 (1990; Zbl 0704.65090)] is defined to ensure the transmission conditions through an iterative technique. At each step of that iterative procedure the resolution of each sub-problem is explicit. A slight modification by introduction of a relaxation parameter improves the convergence. All these techniques are repeated and generalized to linear optimal control problems, especially to those with non-local cost functions. Detailed hints to further problems [e.g. wave guides, relations to the method of perfectly matched layers, cf. J.-P. Berenger, J. Comput. Phys. 114, No. 2, 185-200 (1994; Zbl 0814.65129)] are given.

The numerical example found in the paper is convincing: a plane wave, arriving an annular resonator (with an open section of angle $\pi /8$), showing the scattered field with multiple reflections inside of the hard resonator, and the optimal control solution where the reflection is killed inside of the resonator. In the example each finite element is taken to be a sub-domain in the decomposition, thus, the algorithm reduces to explicit formulae (and transmission of data). Massively parallel strategy has been applied [cf. the first author in D. H. Bailey et al. (ed.), Seventh SIAM Conf. Parallel Proc. San Francisco, 90-95 (1995; Zbl 0836.65083)].

##### MSC:
 65N55 Multigrid methods; domain decomposition (BVP of PDE) 65F10 Iterative methods for linear systems 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 65N12 Stability and convergence of numerical methods (BVP of PDE) 35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation