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 Πh satisfies ∇ · Πh = 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 L2-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.