TY  - EJOU
AU  - Babenko, Christina  
AU  - Chapko, Roman  
AU  - Johansson, B. Tomas  

TI  - On the Numerical Solution of the Laplace Equation with Complete and Incomplete Cauchy Data Using Integral Equations
T2  - Computer Modeling in Engineering \& Sciences

PY  - 2014
VL  - 101
IS  - 5
SN  - 1526-1506

AB  - 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.
KW  - Laplace equation
KW  -  Cauchy problem
KW  -  Corner domain
KW  -  Mixed problem
KW  -  Mesh grading transform
KW  -  Single-layer potential
KW  -  Tikhonov regularization

DO  - 10.3970/cmes.2014.101.299