Boundary-integral approach for the numerical solution of the Cauchy problem for the Laplace equation - Chapko, R.; Johansson, B. T. (Ukrains’kyi Matematychnyi Zhurnal, 2016)
We give a survey of a direct method of boundary integral equations for the numerical solution of the Cauchy problemfor the Laplace equation in doubly connected domains. The domain of solution is located between two closed boundarysurfaces (curves in the case of two-dimensional domains). This Cauchy problem is reduced to finding the values of aharmonic function and its normal derivative on one of the two closed parts of the boundary according to the informationabout these quantities on the other boundary surface. This is an ill-posed problem in which the presence of noise in theinput data may completely destroy the procedure of finding the approximate solution. We describe and present a resultsfor a procedure of regularization aimed at the stable determination of the required quantities based on the representationof the solution to the Cauchy problem in the form a single-layer potential. For the given data, this representation yieldsa system of boundary integral equations for two unknown densities. We establish the existence and uniqueness of thesedensities and propose a method for the numerical discretization in two- and three-dimensional domains. We also considerthe cases of simply connected domains of solution and unbounded domains. Numerical examples are presented both fortwo- and three-dimensional domains. These numerical results demonstrate that the proposed method gives good accuracywith relatively small amount of computations.