Some two-dimensional problems of elastostatics are governed by Laplace's equation. Using the terminology of elastostatics, if the face loads and body loads are not self-equilibrating, even when the displacement at infinity is restricted to zero, displacements in the near field will be infinite. However, the stress field within the domain is well behaved, and is of practical interest. In this paper the semi-analytical scaled boundary finite-element method is extended to permit the analysis of such problems. The solutions in the primary variable so obtained include an infinite component, but the difference in value between any two points in the domain can be computed accurately. The method is also extended to solve the non-homogeneous form of Laplace's equation. (C) 2003 Elsevier Science Ltd. All rights reserved.