Part I Boundary Methods for Solving Elliptic Problems with Singularities

In this part, boundary approximation techniques are described for solving homogeneous self-adjoint elliptic equations (see also Li (1986a, 1988), Li and Mathon (1990a, b) and Li et al. (1987)). Piecewise expansions of particular solutions on subdomains are used which approximate both the boundary and interface conditions in a least squares sense. Convergence of such approximations is provided and error estimates are derived in a natural norm. Stability analysis is also given for choosing better divisions of solution domains. Besides, numerical experiments are reported for the Motz problem, an infinite domain problem, and an interface problem, which yield extremely accurate solutions and very small condition numbers with only modest computational effort.