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

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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