Mixed finite element methods for general quadrilateral grids

We study a new mixed finite element of lowest order for general quadrilateral grids which gives optimal order error in the H(div)-norm. This new element is designed so that the H(div)-projection Ph satisfiesr Ph = Phdiv. A rigorous optimal order error estimate is carried out by proving a modified version of the Bramble–Hilbert lemma for vector variables. We show that a local H(div)-projection reproducing certain polynomials suffices to yield an optimal L-error estimate for the velocity and hence our approach also provides an improved error estimate for original Raviart–Thomas element of lowest order. Numerical experiments are presented to verify our theory. 2011 Elsevier Inc. All rights reserved.