Open Access

ARTICLE

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

Christina Babenko¹, Roman Chapko², B. Tomas Johansson³

1 Ukrainian Engineering Pedagogics Academy, 61003 Kharkiv, Ukraine.
2 Faculty of Applied Mathematics and Informatics, Ivan Franko National University of Lviv, 79000 Lviv, Ukraine
3 Department of Science and Technology, Campus Norrköping, Linköping University, Norrköping, Sweden.

Computer Modeling in Engineering & Sciences 2014, 101(5), 299-317. https://doi.org/10.3970/cmes.2014.101.299

Abstract

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.

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Keywords

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