SUPERCONVERGENCE AND POSTPROCESSING OF FLUXES FROM LOWEST-ORDER MIXEDMETHODS ON TRIANGLES AND TETRAHEDRA

Certain finite difference methods on rectangular grids for second-orde r elliptic equations are known to yield superconvergent flux approxima tions. A class of related finite difference methods recently have been defined for triangular meshes by applying special quadrature rules to an extended version of a mixed finite element method by Arbogast, Daw son, and Keenan [Mixed Finite Element Methods as Finite Difference Met hods for Solving Elliptic Equations on Triangular Elements, Tech. repo rt 93-53, Dept. of Computational and Applied Mathematics, Rice Univers ity, Houston, TX, 1993; in Computational Methods in Water Resources, K luwer Academic Publishers, Norwell, MA, 1994]; the usual hybrid mixed method can also be applied to meshes of triangular and tetrahedral ele ments. Unfortunately, the flux vectors from these methods are only fir st-order accurate. Empirical evidence indicates that a local postproce ssing technique described by Keenan in [An Efficient Postprocessor for Velocities from Mixed Methods on Triangular Elements, Dept. of Comput ational and Applied Mathematics, Tech. report 94-22, Rice University, Houston, TX, 1994] recovers second-order accurate velocities at specia l points. In this paper, a class of local postprocessing techniques ge neralizing the one in [An Efficient Postprocessor for Velocities from Mixed Methods on Triangular Elements, Tech. report 94-22, Rice Univers ity, Houston, TX, 1994] are presented and analyzed. These postprocesso rs are shown to recover second-order accurate velocity fields on three lines meshes. Numerical experiments illustrate these results and inve stigate more general situations, including meshes of tetrahedral eleme nts.