An alternating potential based approach for a Cauchy problem for the Laplace equation in a planar domain with a cut

R. Chapko, B.T. Johansson

Research output: Contribution to journalArticle

Abstract

We consider a Cauchy problem for the Laplace equation in a bounded region containing a cut, where the region is formed by removing a sufficiently smooth arc (the cut) from a bounded simply connected domain D. The aim is to reconstruct the solution on the cut from the values of the solution and its normal derivative on the boundary of the domain D. We propose an alternating iterative method which involves solving direct mixed problems for the Laplace operator in the same region. These mixed problems have either a Dirichlet or a Neumann boundary condition imposed on the cut and are solved by a potential approach. Each of these mixed problems is reduced to a system of integral equations of the first kind with logarithmic and hypersingular kernels and at most a square root singularity in the densities at the endpoints of the cut. The full discretization of the direct problems is realized by a trigonometric quadrature method which has super-algebraic convergence. The numerical examples presented illustrate the feasibility of the proposed method.
Original languageEnglish
Pages (from-to)315-335
Number of pages21
JournalComputational Methods in Applied Mathematics
Volume8
Issue number4
DOIs
Publication statusPublished - Jan 2008

Bibliographical note

© Institute of Mathematics, NAS of Belarus. This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. CC-BY-NC-ND 4.0

Keywords

  • logarithmic- and hypersingularities
  • Laplace equation
  • Cauchy problem
  • alternating method
  • mixed boundary
  • value problems
  • single- and double layer potentials
  • integral equation of the first kind
  • cosine- transformation
  • trigonometrical quadrature method

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