A finite element method for large domains

Abstract

A combined analytical and numerical method is devised to solve elliptic boundary value problems in large or infinite domains. First the domain is divided by an artificial boundary B into a small computational domain Ω and a large or infinite residual domain D. In D the problem is solved analytically, and from the solution an exact nonlocal relation between the solution and its derivatives on B is deduced. This relation is used as a boundary condition to complete the formulation of a problem in Ω which has exactly the same solution there as the original problem. Then a finite element formulation of this new problem in Ω is presented. The exact nonlocal boundary condition is given explicitly for Laplace's equation, for the equations of plane stress and plane strain in linear elastostatics, and for certain equations governing beams and axisymmetric cylindrical shells. The artificial boundary is chosen to be a sphere or a circle. The properties and computational implications of the boundary condition are discussed. Some numerical examples are presented, and the results are compared with those obtained by the standard finite element method using different approximate local boundary conditions on the artificial boundary.