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

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 singulariti es. Based on suitable norms involving discrete solutions at specific points , superconvergence rates on solution derivatives are exploited by using fiv e combinations, e.g., the nonconforming combination, the penalty combinatio n, Combinations I and II, and symmetric combination. For quasi-uniform rect angular grids, the superconvergence rates, O(h(2-delta)), of solution deriv atives by all five combinations can be achieved, where h is the maximal mes h length of difference grids used in the finite difference method, and delt a(> 0) is an arbitrarily small number. Superconvergence analysis in this paper lies in estimates on error bounds c aused by the coupling techniques and their incorporation with finite differ ence methods. Therefore, a similar analysis and conclusions may be extended to linear finite element methods using triangulation by referring to exist ing references. Moreover, the five combinations having O(h(2-delta)) of sol ution derivatives are well suited to solving engineering problems with mult iple singularities and multiple interfaces. (C) 2001 Elsevier Science Ltd. All rights reserved.