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Numerical analysis of a non‐singular boundary integral method: Part II: The general case
Author(s) -
Dreyfuss P.,
Rappaz J.
Publication year - 2002
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.302
Subject(s) - mathematics , mathematical analysis , laplace operator , singular integral , galerkin method , singular perturbation , operator (biology) , finite element method , dirichlet problem , numerical analysis , piecewise , boundary value problem , integral equation , biochemistry , chemistry , physics , repressor , gene , transcription factor , thermodynamics
Abstract In order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we have presented in a previous paper a method which consists in inverting, on a finite element space, a non‐singular integral operator for circular domains. This operator was described as a geometrical perturbation of the Steklov operator, and we have precisely defined the relation between the geometrical perturbation and the dimension of the finite element space, in order to obtain a stable and convergent scheme in which there are non‐singular integrals. We have also presented another point of view under which the method can be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the weakly singular single‐layer potential. In the present paper, we extend the results given in the previous paper to more general cases for which the Laplace problem is set on any ∞ domains. We prove that the properties of stability and convergence remain valid. Copyright © 2002 John Wiley & Sons, Ltd.