Superconvergence of coupling techniques in combined methods for elliptic equations with singularities

Abstract

Several coupling techniques, such as the nonconforming constraints, penalty, and hybrid integrals, of the Ritz-Galerkin and finite difference methods are presented for solving elliptic boundary value problems with singularities. Based on suitable norms involving discrete solutions at specific points, superconvergence rates on solution derivatives are exploited by using five combinations, e.g., the nonconforming combination, the penalty combination, Combinations I and II, and symmetric combination. For quasi-uniform rectangular grids, the superconvergence rates, O( h 2− δ ), of solution derivatives by all five combinations can be achieved, where h is the maximal mesh length of difference grids used in the finite difference method, and δ(> 0) is an arbitrarily small number. Superconvergence analysis in this paper lies in estimates on error bounds caused by the coupling techniques and their incorporation with finite difference methods. Therefore, a similar analysis and conclusions may be extended to linear finite element methods using triangulation by referring to existing references. Moreover, the five combinations having O( h 2− δ ) of solution derivatives are well suited to solving engineering problems with multiple singularities and multiple interfaces.