On identifying Dirichlet condition for under-determined problem of the Laplace equation by BEM

Abstract

The purpose of this paper is to present a numerical technique for the solution of an 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, respectively, imposed on two disjoint parts of the boundary of the domain, so that the union of the two parts does not constitute the whole boundary. This under-determined boundary value problem can be regarded as a boundary inverse problem, in which the proper Dirichlet condition is to be identified for the rest of the boundary. The solution of this problem is not unique. The technique is based on the direct variational method, and a functional introduced is minimized by the method of the steepest descent. Since the functional is convex, the minimum is attained uniquely. The minimization problem is recast into successive primary and adjoint boundary value problems of the Laplace equation. The boundary element method is applied for numerical solution of the boundary value problems. Some empirical tricks are proposed for increasing the efficiency of the numerical method. A few simple examples show that the method yields a convergent solution which corresponds to the minimum of the functional.