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Stable solution of the Cauchy problem for the Laplace equation using surface potentials. (de superficie.) (Spanish. English summary) Zbl 1381.35031

Summary: In this work the Cauchy problem for the Laplace equation in an annular bidimensional region is studied. The solution is sought like sum of surface potentials with densities defined on the boundary of the annular region which ones are numerically looking for a collocation method. The advantages of this technique are: conceptually simple, easy to implement numerically, can be applied to different curves representing the boundary and can be extended to the tridimensional case. Due to ill posedness of the Cauchy problem, the matrix obtained by collocation method, is ill conditioned presenting for that numerical instability, which is handled by the Tikhonov regularization method. Synthetic examples are presented for a circular annular region. In this case the potential is calculated in exact form and compared to the numerical approximation. The results of this work show the feasibility of the technique for the stable solution of the problem.

MSC:

35J25 Boundary value problems for second-order elliptic equations
30C30 Schwarz-Christoffel-type mappings
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
65N20 Numerical methods for ill-posed problems for boundary value problems involving PDEs
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