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An integral equation method for the numerical solution of a Dirichlet problem for second-order elliptic equations with variable coefficients

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Abstract

We develop a numerical approximation involving boundary integral techniques for the solution of the Dirichlet problem for second-order elliptic equations with variable coefficients. Using the concept of a parametrix, the problem is reduced to a boundary-domain integral equation to be solved for two unknown densities. Via a change of variables based on shrinkage of the boundary curve of the solution domain a parameterised system of boundary-domain integrals is obtained. It is shown how to write the singularities in this system in an explicit form such that boundary integral techniques can be applied for analysis and discretisation. An effective discretisation involving the Nyström method is given, together with numerical experiments showing that the proposed approach can be turned into a practical working method.

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Authors and Affiliations

  1. Faculty of Applied Mathematics and Informatics, Ivan Franko National University of Lviv, Lviv, 79000, Ukraine

    Andriy Beshley & Roman Chapko

  2. School of Mathematics, Aston University, Birmingham, B4 7ET, UK

    B. Tomas Johansson

Authors
  1. Andriy Beshley
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  2. Roman Chapko
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  3. B. Tomas Johansson
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Correspondence to B. Tomas Johansson.

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Beshley, A., Chapko, R. & Johansson, B.T. An integral equation method for the numerical solution of a Dirichlet problem for second-order elliptic equations with variable coefficients. J Eng Math 112, 63–73 (2018). https://doi.org/10.1007/s10665-018-9965-7

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  • DOI: https://doi.org/10.1007/s10665-018-9965-7

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