Superconvergence Of Least-Squares Mixed Finite Elements For Elliptic Problems

Lately, there has been much interest in the study of the least-squares mixed finite element method. The main task of this paper is to investigate the least-squares mixed method for the elliptic boudary-value problems. By introducing the projections and duality problems, we do some analysis for the approximations and finally obtain the general superconvergence results.This paper consists of two chapters. The purpose of the first part is to in-vestigate superconvergence phenomena for the second order elliptic boundary-value problems over triangulations, by using least-squares mixed finite ele-ment method. First, we present the equivalent variatial formulations of the least-squares mixed method and prove the existence and uniques for the weak problems. On the basis of L2-projections and Raviart-Thomas projections, we obtain the superconvergence of the least-squares mixed finite element approx-imations on uniform triangulations, where triangular mixed finite elements of the lowest order Raviart-Thomas spaces are used to approximate the flux p.In the second chapter, we briefly recall the standard and mixed finite methods for second order elliptic problems, and introduce a modified least-squares mixed method. In view of the model problem, we do some analysis for the approximations respectively; We compare uh with the standard finite element approximation of u in Vh, and ph with the usual (non-least-squares) mixed finite element approximation of p, provided of course that the same approximating spaces are used. It turns out that under weak conditions, they are "almost" equal, i.e., higher order perturbations of each other. Apart from improved a priori bounds, the result also gives us the possibility to extend superconvergence results from the standard and mixed method to the least-squares mixed method. Also, it might open ways to the reverse.Finally, we sum up the main ideas of this paper. Combining these results, we obtain general conclusions for the model problems and also make some comments on the prospect of the superconvergence analysis.