An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions
The authors extend the alternating method to the Cauchy problem for the Laplace equation in semi-infinite regions, that is the region is the intersection between a half-plane and the exterior of a bounded closed curved contained in the half-plane. The Cauchy data are given on the unbounded part of the boundary of the region and the aim of the authors is to construct the solution on the boundary of the inclusion. The authors describe the well-posedness of the corresponding mixed problems using a boundary integral equations method and present convergence of the alternating procedure. Moreover, they show how to solve the mixed problems numerically and how to discretize the suitable boundary integrals. The approach of the authors involves Green's function techniques and numerical schemes for the calculations of integrals on the real line with various singularities, and can easily be adjusted to the three-dimensional case. Finally, the reconstruction of the solution on the inclusion boundary is accurately found in the case of exact data. In the case of noisy data, the authors employ a discrepancy principle to stop the iteration.
trigonometric- and sinc- quadrature rules
scientific article; zbMATH DE number 5362899