Mathematics of Computation

Author(s): C. Schwab; W. L. Wendland.
Journal: Math. Comp. 68 (1999), 91-122.
MSC (1991): Primary 45F15, 15N38, 45K05; Secondary 47G30, 58G15, 35J25
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Abstract: In this paper we develop and analyze a bootstrapping algorithm for the extraction of potentials and arbitrary derivatives of the Cauchy data of regular three-dimensional second order elliptic boundary value problems in connection with corresponding boundary integral equations. The method rests on the derivatives of the generalized Green's representation formula, which are expressed in terms of singular boundary integrals as Hadamard's finite parts. Their regularization, together with asymptotic pseudohomogeneous kernel expansions, yields a constructive method for obtaining generalized jump relations. These expansions are obtained via composition of Taylor expansions of the local surface representation, the density functions, differential operators and the fundamental solution of the original problem, together with the use of local polar coordinates in the parameter domain. For boundary integral equations obtained by the direct method, this method allows the recursive numerical extraction of potentials and their derivatives near and up to the boundary surface.

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C. Schwab
Affiliation: Seminar für Angewandte Mathematik, ETH Zürich, CH--8092 Zürich, Switzerland
Email: schwab@sam.math.ethz.ch

W. L. Wendland
Affiliation: Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, D--70569 Stuttgart, Germany
Email: wendland@mathematik.uni-stuttgart.de

DOI: 10.1090/S0025-5718-99-01044-3
PII: S 0025-5718(99)01044-3
Keywords: Boundary integral equation methods, derivatives of the Cauchy data, regularization of hypersingular potentials
Received by editor(s): April 29, 1997
Dedicated: This work is dedicated to Professor Dr. G. C. Hsiao on the occasion of his 60$^{th}$ birthday
Copyright of article: Copyright 1999, American Mathematical Society

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