WIAS Preprint No. 119, (1994)

The qualocation method for Symm's integral equation on a polygon



Authors

  • Elschner, Johannes
  • Prössdorf, Siegfried
  • Sloan, Ian H.

2010 Mathematics Subject Classification

  • 65R20 45B05

Keywords

  • First kind boundary integral equation, qualocation method, nonlinear parametrization, Mellin convolution operator, superconvergence

Abstract

This paper discusses the convergence of the qualocation method for Symm's integral equation on closed polygonal boundaries in ℝ2 . Qualocation is a Petrov-Galerkin method in which the outer integrals are performed numerically by special quadrature rules. Before discretisation a nonlinear parametrisation of the polygon is introduced which varies more slowly than arc-length near each corner and leads to a transformed integral equation with a regular solution. We prove that the qualocation method using smoothest splines of any order k on a uniform mesh (with respect to the new parameter) converges with optimal order O(hk ). Furthermore, the method is shown to produce superconvergent approximations to linear functionals, retaining the same high convergence rates as in the case of a smooth curve.

Appeared in

  • Math. Nachr., 177 (1996), pp. 81--108

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