An energy method of fundamental solutions for solving the inverse Cauchy problems of the Laplace equation

The accuracy of the method of fundamental solutions (MFS) is heavily influenced by the distribution of source points. One often locates the source points along an offset to the problem boundary or a circle with a fixed radius. In this paper we propose an energy regularization technique to choose the source points and weighting factors in the numerical solution of the inverse Cauchy problems for the Laplace equation in arbitrary domain. An inequality is derived, which is a criterion to pick up the source points and weighting factors. This new technique can improve the accuracy of the numerical solution than the MFS with the distribution of source points using a fixed offset. Some numerical tests confirm that the energy MFS (EMFS) has a good stability and accuracy, and the computational cost is cheap.