Abstract
We consider weakly singular integral equations of the first kind on open surface pieces Γ in ℝ3. To obtain approximate solutions we use theh-version Galerkin boundary element method. Furthermore we introduce two-level additive Schwarz operators for non-overlapping domain decompositions of Γ 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.
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Supported by the German Research Foundation (DFG) under grant Ste 238/25-9.
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Mund, P., Stephan, E.P. & Weiße, J. Two-level methods for the single layer potential in ℝ3 . Computing 60, 243–266 (1998). https://doi.org/10.1007/BF02684335
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DOI: https://doi.org/10.1007/BF02684335