Superconvergence analysis and extrapolation of quasi-Wilson nonconforming finite element method for nonlinear Sobolev equations

Abstract

Quasi-Wilson nonconforming finite element approximation for a class of nonlinear Sobolev equations is discussed on rectangular meshes. We first prove that this element has two special characters by novel approaches. One is that (∇h(u − Ihu),∇hvh)h may be estimated as order O(h2) when u ∈ H3(Ω), where Ihu denotes the bilinear interpolation of u, vh is a polynomial belongs to quasi-Wilson finite element space and ∇h denotes the piecewise defined gradient operator, h is the mesh size tending to zero. The other is that the consistency error of this element is of order O(h2)/O(h3) in broken H1-norm, which is one/two order higher than its interpolation error when u ∈ H3(Ω)/H4(Ω). Then we derive the optimal order error estimate and superclose property via mean-value method and the known high accuracy result of bilinear element. Furthermore, we deduce the global superconvergence through interpolation post processing technique. At last, an extrapolation result of order O(h3), two order higher than traditional error estimate, is obtained by constructing a new suitable extrapolation scheme.