Finite element solutions for three-dimensional elliptic boundary value problems on unbounded domains

Abstract

The finite element (FE) solutions of a general elliptic equation -div([a ij] · ∇u) + u = f in an exterior domain Ω, which is the complement of a bounded subset of R 3, is considered. The most common approach to deal with exterior domain problems is truncating an unbounded subdomain Ω ∞, so that the remaining part Ω = Ω\Ω̄ ∞) is bounded, and imposing an artificial boundary condition on the resulted artificial boundary Γ a = Ω̄ ∞ ⊂ Ω̄ B. In this article, instead of discarding an unbounded subdomain Ω ∞ and introducing an artificial boundary condition, the unbounded domain is mapped to a unit ball by an auxiliary mapping. Then, a similar technique to the method of auxiliary mapping, introduced by Babuška and Oh for handling the domain singularities, is applied to obtain an accurate FE solution of this problem at low cost. This method thus does have neither artificial boundary nor any restrictions on f.