The method of fundamental solutions for the numerical solution of the biharmonic equation

Abstract

The method of fundamental solutions (MFS) is a relatively new technique for the numerical solution of certain elliptic boundary value problems. It falls in the class of methods generally called boundary methods, and, like the well-known boundary integral equation method, is applicable when a fundamental solution of the differential equation is known. In the MFS, the approximate solution is a linear combination of fundamental solutions with singularities placed outside the domain of the problem. The locations of the singularities are either preassigned or determined along with the coefficients of the fundamental solutions so that the approximate solution satisfies the boundary conditions as well as possible. In many applications, these quantities are determined by a least squares fit of the boundary conditions, a nonlinear problem, which is solved using standard software. In this paper, the MFS is formulated for biharmonic problems and is applied to a variety of standard test problems as well as to problems arising in elasticity and fluid flow. © 1987.