Asymptotic expansions and Richardson extrapolation of approximate solutions for second order elliptic problems on rectangular domains by mixed finite element methods

In this paper asymptotic error expansions for mixed finite element approximations of general second order elliptic problems are derived under rectangular meshes, and the Richardson extrapolation is applied to improve the accuracy of the approximations by two different schemes with the help of an interpolation postprocessing technique. The results of this paper provide new asymptotic expansions and new approximate solutions which are one-order and a half-order higher in accuracy than those obtained in [J. Wang, Math Comp., 56 (1991), pp. 477-503] and (H. Chen, R. E. Ewing, and R. Lazarov, Asymptotic Error Expansion for the Lowest Order Raviart-Thomas Rectangular Mixed Finite Elements, Technical report ISC-97-01, 1997], respectively. As a by-product, we illustrate that all the approximations of higher accuracy can be used to form a class of a posteriori error estimators for the mixed finite element method.