ABSTRACT
The purpose of this paper is to present results of an attempt at a numerical treatment of the initial value problem of the Laplace equation in two spatial dimensions. A numerical solution is sought for the problem in which the Dirichlet and Neumann data are simultaneously imposed on a part of the boundary of the domain. The 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 treatment is based on the method of least squares, and a functional is minimized by the method of the steepest descent. After numerical computations by using the finite elements, it is concluded that our scheme is stable and that the numerical solutions are convergent.
The second purpose of this paper is to present a variational approach to numerical identification of boundary conditions for resolution of the Cauchy problem governed by the Navier equations in plane elastostatics. The Cauchy problem is described by a local Dirichlet Neumann map, which may contain some noises. The problem is re-formulated as a minimization problem of a functional with constraints, then the minimization problem is recast into successive primary and dual boundary value problems with no constraints in the corresponding plane elasticity. Displacement approach and traction approach are outlined in this paper. For approximate solution of a simple elastostatic problem, the conventional direct boundary element method is scrutinized.
