Superconvergence For Least-Squares Mixed Finite Elements Over Quadrilaterals

Lately, more and more attention has been paid to the least-squares mixed finite element method. We treat two kinds of elliptic problems (symmetric and unsymmetric) in this paper. The existence and uniquess for their discrete problems are presented. We compare the least-squares mixed finite elment solutions with the corresponding projections, and the superconvergence results are obtained for both the scalar functions and the flux functions. We summarize our work in the below.It contains five sections. We first review the advantages and disadvantages about the mixed finite element method, and show why we choose the least-squares mixed finite element method for the boundary-value problems. Some preliminary knowledge and notations are also included.In the following two sections, by using least-squares mixed finite elment method over quadrilaterals, we investigate superconvergence phenomena separately for those two kinds of boundary-value problems of symmetric or unsymmetric elliptic equations. If the finite element partition is h2-uniform, we obtain the superconvergence result of least-squares mixed finite element solutions on the basis of the L2-projection and some mixed finite element projections and the integral identities technique developed by Q.Lin and his collaborates. This error estimate indicates an accuracy of 0(hr-2} if Raviart-Thomas elements of order r are employed with optimal error estimate of O(hr+1).Finally, numerical results are presented for the model problems in the forth section, which confirm our theoretical estimates.In the last section, we sum up the main ideas of this paper, and also make some comments on the prospect of the superconvergence analysis.