On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach

Roman Chapko, B. Tomas Johansson

    Research output: Contribution to journalArticlepeer-review

    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 L2-setting. An effective way of
    discretizing these boundary integral equations based on the Nystr¨om 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.
    Original languageEnglish
    Pages (from-to)25–38
    Number of pages13
    JournalInverse Problems and Imaging
    Volume6
    Issue number1
    DOIs
    Publication statusPublished - Feb 2012

    Keywords

    • cauchy problem
    • integral equation of the first kind
    • logarithmic- and hypersingularities
    • single- and double layer potentials
    • Green’s function
    • Laplace equation
    • Tikhonov regularization

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