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

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 ? I _h u),? _h v _h ) _h may be estimated as order O(h ²) when u ∈ H ³(Ω), where I _h u denotes the bilinear interpolation of u, v _h 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(h ²)/O(h ³) in broken H ¹-norm, which is one/two order higher than its interpolation error when u ∈ H ³(Ω)/H ⁴(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(h ³), two order higher than traditional error estimate, is obtained by constructing a new suitable extrapolation scheme.