WIAS Preprint No. 142, (1995)

The collocation method for mixed boundary value problems on domains with curved polygonal boundaries



Authors

  • Elschner, Johannes
  • Jeon, Youngmok
  • Sloan, Ian H.
  • Stephan, Ernst P.

2010 Mathematics Subject Classification

  • 65N38 35J05 65N12 65R20

Keywords

  • polygonal domains, boundary integral equation, mixed Dirichlet-Neumann boundary value problem, Laplace equation, collocation method, mesh grading transformation, convergence, stability, Mellin transform technique

DOI

10.20347/WIAS.PREPRINT.142

Abstract

We consider an indirect boundary integral equation formulation for the mixed Dirichlet Neumann boundary value problem for the Laplace equation on a plane domain with a polygonal boundary. The resulting system of integral equations is solved by a collocation method which uses a mesh grading transformation and a cosine approximating space. The mesh grading transformation method yields fast convergence of the collocation solution by smoothing the singularities of the exact solution. A complete stability and solvability analysis of the transformed integral equations is given by use of a Mellin transform technique, in a setting in which each arc of the polygon has associated with it a periodic Sobolev space.

Appeared in

  • Numer. Math., 76 (1997), pp. 355-381

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