Global superconvergence of simplified hybrid combinations for elliptic equations with singularities, I. Basic theorem

To solve the elliptic boundary value problems with singularities, the simplified hybrid combinations of the Ritz-Galerkin and finite element methods (RGM-FEM) are explored to lead to the global superconvergence rates on the entire solution domain, based on an a posteriori interpolation techniques of Lin and Yan [12] that only cost a little more computation. Let the solution domain S = S-1 boolean OR S-2 boolean OR Gamma(0) and S-1 boolean AND S-2 = 0. Suppose that S-1 can be partitioned into quasiuniform rectangles: S-1 = Sigma(ij) square(ij), a singular point occurs at partial derivative S-2, and the singular functions are chosen in S-2. Then for bilinear elements, it is proven that the simplified hybrid combinations of RGM-FEM can provide the global superconvergence rate O(h(2)) for solution gradients over the entire subdomains S-1 and S-2, where h is the maximal boundary length of square(ij). The global superconvergence O(h(2)) is better, compared to O(h(2-delta)), 0 < delta << 1 given in [4, 9]. Note that numerical stability of the simplified hybrid combinations of RGM-FEM is also optimal [6]. This paper presents the important results for the general case of Poisson-problems on a polygonal domain S estimates for the Sobolev norm \\ . \\(1), given in a much more general sense than known before, cf. [1-4, 14-18].