Numerical Solution of an Under-determined Problem of the Laplace Equatin

Abstract

The purpose of this paper is to present an attempt at a numerical treatment of a kind of under-determined problem of the Laplace equation in two spatial dimensions. A resolution is sought for the problem in which the Dirichlet and Neumann data are arbitrarily imposed on each part of the boundary of the domain. This new problem can be regarded as a boundary inverse problem, in which the proper boundary conditions are to be identified for the rest of the boundary. The solution of this problem is not unique. The treatment is based on the direct variational method, and a functional is minimized by the method of the steepest descent. The minimization problem is recast into successive primary and dual boundary value problems of the Laplace equation. After numerical computations by using the boundary elements, it is concluded that our scheme is stable, but the numerical solutions converge to the nearest local minimum.