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

Andriy Beshley, Roman Chapko, B. Tomas Johansson

Research output: Contribution to journalArticle

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.
Original languageEnglish
Pages (from-to)63-73
Number of pages11
JournalJournal of Engineering Mathematics
Volume112
Issue number1
Early online date11 Jun 2018
DOIs
Publication statusPublished - Oct 2018

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Integral Equation Method
Second Order Elliptic Equations
Variable Coefficients
Dirichlet Problem
Integral equations
Boundary Integral
Numerical Solution
Discretization
Change of Variables
Integral domain
Shrinkage
Numerical Approximation
Integral Equations
Numerical Experiment
Singularity
Unknown
Curve
Experiments

Keywords

  • elliptic equation
  • Nyström method
  • parametrix

Cite this

Beshley, Andriy ; Chapko, Roman ; Johansson, B. Tomas. / An integral equation method for the numerical solution of a Dirichlet problem for second-order elliptic equations with variable coefficients. In: Journal of Engineering Mathematics. 2018 ; Vol. 112, No. 1. pp. 63-73.
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An integral equation method for the numerical solution of a Dirichlet problem for second-order elliptic equations with variable coefficients. / Beshley, Andriy; Chapko, Roman; Johansson, B. Tomas.

In: Journal of Engineering Mathematics, Vol. 112, No. 1, 10.2018, p. 63-73.

Research output: Contribution to journalArticle

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