Expanded Mixed Finite Element Methods for
                   Linear Second-Order Elliptic Problems, I

                                Zhangxin Chen

                      Department of Mathematics, Box 156
                        Southern Methodist University
                       Dallas, Texas 75275--0156, USA.

                                   Abstract

We develop a new mixed formulation for the numerical solution of second-order
elliptic problems.  This new formulation expands the standard mixed
formulation in the sense that three variables are explicitly treated:  the
scalar unknown, its gradient, and its flux (the coefficient times the
gradient).  Based on this formulation, mixed finite element approximations of
the second-order elliptic problems are considered.  Optimal order error
estimates in the <i>L<sup>p</sup></i>- and <i>H<sup>-s</sup></i>-norms are
obtained for the mixed approximations.  Various implementation techniques for
solving the systems of algebraic equations are discussed.  A postprocessing
method for improving the scalar variable is analyzed, and superconvergent
estimates in the <i>L<sup>p</sup></i>-norm are derived.  The mixed formulation
is suitable for the case where the coefficient of differential equations is a
small tensor and does not need to be inverted.


This paper will appear in RAIRO Mod\`el. Math. Anal. Num\'er.