Global superconvergence in combinations of Ritz-Galerkin-FEM for singularity problems

This paper combines the piecewise bilinear elements with the singular functions to seek the corner singular solution of elliptic boundary value problems. The global superconvergence rates O(h2−δ) can be achieved by means of the techniques of Lin and Yan (The Construction and Analysis of High Efficient FEM, Hobei University Publishing, Hobei, 1996) for different coupling strategies, such as the nonconforming constraints, the penalty integrals, and the penalty plus hybrid integrals, where δ(>0) is an arbitrarily small number, and h is the maximal boundary length of quasiuniform rectangles −qij used. A little effort in computation is paid to conduct a posteriori interpolation of the numerical solutions, uh, only on the subregion used in finite element methods. This paper also explores an equivalence of superconvergence between this paper and Z.C. Li, Internat. J. Numer. Methods Eng. 39 (1996) 1839–1857 and J. Comput. Appl. Math. 81 (1997) 1–17.