A new kind of combinations between the Ritz-Galerkin and finite element methods for singularity problems

The coupling techniques of simplified hybrid plus penalty functions are first presented for matching the Ritz-Galerkin method and thek(k>-1)-order Lagrange finite element methods to solve complicated problems of elliptic equations, homogeneous or nonhomogeneous, in particular with singularities or unbounded domains. Optimal convergence rates of numerical solutions have been proved in the Sobolev norms. Moreover, the theoretical results obtained in this paper have been verified by numerical experiments for the singular Motz problem.