On the Numerical Solution of a Cauchy Problem for the Laplace Equation Via a Direct Integral Equation Approach

Research output: Contribution to journalArticle

Authors

Colleges, School and Institutes

Abstract

We investigate the problem of determining the stationary temperature field on an inclusion from given Cauchy data on an accessible exterior boundary. On this accessible part the temperature (or the heat flux) is known, and, additionally, on a portion of this exterior boundary the heat flux (or temperature) is also given. We propose a direct boundary integral approach in combination with Tikhonov regularization for the stable determination of the temperature and flux on the inclusion. To determine these quantities on the inclusion, boundary integral equations are derived using Green's functions, and properties of these equations are shown in an L-2-setting. An effective way of discretizing these boundary integral equations based on the Nystrom method and trigonometric approximations, is outlined. Numerical examples are included, both with exact and noisy data, showing that accurate approximations can be obtained with small computational effort, and the accuracy is increasing with the length of the portion of the boundary where the additionally data is given.

Details

Original languageEnglish
Pages (from-to)25-38
Number of pages14
JournalInverse Problems and Imaging
Volume6
Issue number1
Publication statusPublished - 1 Feb 2012

Keywords

  • Green's function, Tikhonov regularization, Logarithmic- and hypersingularities, Integral equation of the first kind, Laplace equation, Cauchy problem, single- and double layer potentials