Interior superconvergence error estimates for mixed finite element methods for second order elliptic problem

Abstract

The aim of this paper is to provide a local superconvergence analysis for mixed finite element methods of Poission equation. We shall prove that ifp is smooth enough in a local region$\Omega _o \subset \subset \Omega _1 \subset \subset \Omega$ and rectangular mesh is imposed on Ω{in1}, then local superconvergence for ∥π{inh}{itu}−{itu}{inh}∥0,2,Ω{in0}, and ∥P{inh}p−p{inh}∥0,2,Ω{in0}, are expected. Thus, by post-processing operators$\tilde P$ and$\tilde \pi$, we have obtained the following local superconvergence error estimate:$||p - \tilde P_{ph} ||o_1 \Omega _0 || + ||u - \tilde \pi u_h ||o_1 \Omega _o \leqslant c\left[ {h^{k + 2.5} ||p||k + 4_, \Omega _1 + h^{k + 1 + r - e} ||p||2 + r_, \Omega } \right]_,$ where 0≤r≤2 andk≥1.