Two-level methods for the single layer potential in R3

Patrick Mund

doi:10.1007/BF02684335

Two-level methods for the single layer potential in R3

1998-09-01

41

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AI

DOI：10.1007/BF02684335

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We consider weakly singular integral equations of the first kind on open surface pieces Gamma in R-3. To obtain approximate solutions we use the h-version Galerkin boundary element method. Furthermore we introduce two-level additive Schwarz operators for non-overlapping domain decompositions of Gamma and we estimate the conditions numbers of these operators with respect to the mesh size. Based on these operators we derive an a posteriori error estimate for the difference between the exact solution and the Galerkin solution. The estimate also involves the error which comes from an approximate solution of the Galerkin equations. For uniform meshes and under the assumption of a saturation condition we show reliability and efficiency of our estimate. Based on this estimate we introduce an adaptive multilevel algorithm with easily computable local error indicators which allows direction control of the local refinements. The theoretical results are illustrated by numerical examples for plane and curved surfaces.

Keyword:

weakly singular integral equation

boundary element method

domain decomposition

hierarchical basis

a posteriori error estimate

adaptive algorithm

Two-level methods for the single layer potential in R3

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