ABSTRACT
The aim of this paper is to develop a technique for solving discrete boundary value problems arising in finite element and finite difference analysis analogous to the boundary element method [1]. The approach being developed is based on the elimination of the unknowns corresponding to the interior part of the domain. Such elimination results in a reduced system of equations with respect to the boundary unknowns. The basic idea of the method is not new and is used in many computational techniques. Thus, the superelement method extensively used in the finite element analysis of large and complex structural systems is based on the elimination of the interior degrees of freedom of substructures by means of static condensation [2]. A method closely related to the approach under investigation was introduced by Ryabenkii and is referred to as the difference potential method [3]. In this method a system of discrete boundary equations is formulated by using boundary projectors; this system can be considered as a discrete analogue of the boundary operator equations introduced by Calderon [4] and then investigated by Seeley [5]. In contrast to Ryabenkii’s method, in this paper we investigate constructive formulations of the discrete boundary equations related to the direct and indirect formulations of the boundary integral equations in the boundary element method. In the context of the developed approach we assume that the inverse of the discrete operator with constant coefficients L−1 can be computed as a convolution with a discrete fundamental solution. The algorithms for calculation of a discrete fundamental solution are discussed in [6]. The self-adjoint formulations of the discrete boundary equations presented in this paper are similar to the hypersingular boundary integral equations in the continuous case [7]. But in the discrete case we do not face problems of regularisation and interpretation of hypersingular boundary integral operators. The obtained symmetric formulations of the discrete boundary equations are very suitable for the development of effective algorithms and for using discrete boundary equations in combination with the finite element method.
