$L^∞$-Error Estimates and Superconvergence in Maximum Norm of Mixed Finite Element Methods for NonFickian Flows in Porous Media

Abstract

On the basis of the estimates for the regularized Green's functions with memory terms, optimal order $L^∞$-error estimates are established for the nonFickian flow of fluid in porous media by means of a mixed Ritz-Volterra projection. Moreover, local $L^∞$-superconvergence estimates for the velocity along the Gauss lines and for the pressure at the Gauss points are derived for the mixed finite element method, and global $L^∞$-superconvergence estimates for the velocity and the pressure are also investigated by virtue of an interpolation post-processing technique. Meanwhile, some useful a-posteriori error estimators are presented for this mixed finite element method.