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# Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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1569-3945
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# Numerical solution of an elliptic 3-dimensional Cauchy problem by the alternating method and boundary integral equations

Ihor Borachok
/ Roman Chapko
/ B. Tomas Johansson
Published Online: 2016-01-28 | DOI: https://doi.org/10.1515/jiip-2015-0053

## 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.

MSC 2010: 35R25; 35J05; 65R20

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Revised: 2015-09-12

Accepted: 2015-12-23

Published Online: 2016-01-28

Published in Print: 2016-12-01

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 24, Issue 6, Pages 711–725, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219,

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