Coupling strategy for matching different methods in solving singularity problems

For solving elliptic boundary value problems with singularities, we have proposed the combined methods consisting of the Ritz-Galerkin method using singular (or analytic) basic functions for one part,S ₂, of the solution domainS, where there exist singular points, and the finite element method for the remaining partS ₁ ofS, where the solution is smooth enough. In this paper, general approaches using additional integrals are presented to match different numerical methods along their common boundary Г₀. Errors and stability analyses are provided for such a general coupling strategy. These analyses are important because they form a theoretical basis for a number of combinations between the Ritz-Galerkin and finite element methods addressed in [7], and because they can lead to new combinations of other methods, such as the combined methods of the Ritz-Galerkin and finite difference methods. Moreover, the analyses in this paper can be applied or extended to solve general elliptic boundary value problems with angular singularities, interface singularity or unbounded domain.