An under-determined problem of the Laplace equation in two spatial dimension is considered. For this problem the Dirichlet and Neumann data are, respectively, prescribed on two disjoint parts of the boundary of the domain, so that the union of the two parts doesn't constitute the whole boundary. This under-determined boundary value problem can be regarded as a boundary inverse problem for determining the proper Dirichlet condition to the rest of the boundary. The main difficulty is that the solution of such problem is not unique. The aim of this paper is to present a numerical technique for the solution, which is based on the direct variational method. The functional is minimized by the method of steepest descent. It is proved that the functional is convex, so the minimum is attained uniquely. Three algoritms for finding the minimum are presented, all are based on the recast into succesive primary and adjoint boundary value problems of the Laplace equation, which are computed using the boundary element method. Several examples are included to show that the method yields a convergent solution corresponding to the minimum of the solution.