A note on the relation between the boundary- and finite-element method with application to Laplace's equation in two dimensions

The relation between the boundary- and finite-element method for boundary-value problems governed by linear, homogeneous, and elliptic differential equations is discussed. By design, the boundary-element formulation results in a system of linear equations involving boundary values alone, referred to as the boundary-value system, whereas the finite-element formulation results in a linear system involving boundary as well as interior values. To compare the two approaches, the finite-element system is reduced to a boundary-value system using incomplete Gauss-Jordan elimination, and the coefficient matrices are used to construct an effective matrix determining the difference between the boundary- and the finite-element solutions. In the case of Laplace's equation in a two-dimensional disk-like domain with linear boundary elements and associated three-node linear finite elements, the eigenvalues of this matrix are shown to cluster around zero as the discretization level is raised, which means that the corresponding eigenvectors representing boundary waves are rapidly removed up to the working spatial resolution. (c) 2005 Elsevier Ltd. All rights reserved.