Numerical solution of an elliptic 3-dimensional Cauchy problem by the alternating method and boundary integral equations

Ihor Borachok, Roman Chapko, B. Tomas Johansson*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the Cauchy problem for the Laplace equation in 3-dimensional doubly-connected domains, that is the reconstruction of a harmonic function from knowledge of the function values and normal derivative on the outer of two closed boundary surfaces. We employ the alternating iterative method, which is a regularizing procedure for the stable determination of the solution. In each iteration step, mixed boundary value problems are solved. The solution to each mixed problem is represented as a sum of two single-layer potentials giving two unknown densities (one for each of the two boundary surfaces) to determine; matching the given boundary data gives a system of boundary integral equations to be solved for the densities. For the discretisation, Weinert's method [24] is employed, which generates a Galerkin-type procedure for the numerical solution via rewriting the boundary integrals over the unit sphere and expanding the densities in terms of spherical harmonics. Numerical results are included as well.

Original languageEnglish
Pages (from-to)711-725
Number of pages15
JournalJournal of Inverse and Ill-Posed Problems
Volume24
Issue number6
Early online date28 Jan 2016
DOIs
Publication statusPublished - Dec 2016

Keywords

  • alternating method
  • Cauchy problem
  • integral equation
  • Laplace equation

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