On the Numerical Solution of the Laplace Equation with Complete and Incomplete Cauchy Data Using Integral Equations

We consider the numerical solution of the Laplace equations in planar bounded domains with corners for two types of boundary conditions. The first one is the mixed boundary value problem (Dirichlet-Neumann), which is reduced, via a single-layer potential ansatz, to a system of well-posed boundary in...

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Bibliographic Details
Published in:Computer modeling in engineering & sciences 2014, Vol.101 (5), p.299-317
Main Authors: Babenko, Christina, Chapko, Roman, B. Tomas Johansson
Format: Article
Language:English
Subjects:
Boundaries
Boundary conditions
Boundary integral method
Boundary value problems
Cauchy problems
Dirichlet problem
Domains
Ill posed problems
Integral equations
Laplace equation
Mathematical analysis
Mathematical models
Quadratures
Regularization
Well posed problems
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Summary:We consider the numerical solution of the Laplace equations in planar bounded domains with corners for two types of boundary conditions. The first one is the mixed boundary value problem (Dirichlet-Neumann), which is reduced, via a single-layer potential ansatz, to a system of well-posed boundary integral equations. The second one is the Cauchy problem having Dirichlet and Neumann data given on a part of the boundary of the solution domain. This problem is similarly transformed into a system of ill-posed boundary integral equations. For both systems, to numerically solve them, a mesh grading transformation is employed together with trigonometric quadrature methods. In the case of the Cauchy problem the Tikhonov regularization is used for the discretized system. Numerical examples are included both for the well-posed and ill-posed cases showing that accurate numerical solutions can be obtained with small computational effort.
ISSN:1526-1492
1526-1506
1526-1506
DOI:10.3970/cmes.2014.101.299