New mixed finite element for elliptic problem on quadrilateral grid

Abstract

In this study, we introduce a new family of mixed finite element spaces of higher order(k≥1) on general quadrilateral grids and consider a control volume(covolume) method for second order elliptic PDEs with the rotated-Q_1 nonconforming finite element on rectangular grids. A typical element has two fewer degrees of freedom than the well-known Raviart-Thomas finite element RT_[k] [18], yet enjoys an optimal order approximation for the velocity in L^2-norm. The order of approximation in H(div;Ω)-norm is one less than the velocity, as is common to all other known elements, except a recent element introduced by Arnold et al[4]. However, we introduce a local post-processing technique to obtain an optimal order in H(div;Ω)-norm. This technique can be used to enhance the result of RT_[k] element also, hence can be easily incorporated into existing codes. In pressure the new element has one lower order of approximation than the RT_[k] element. However, the pressure also can be locally post-processed to produce an optimal order approximation. The greatest advantage of our finite element lies in that it has the fewest degrees of freedom among all the known quadrilateral mixed finite elements and thus together with the post-processing techniques provides a very efficient way of computing flow variables in mixed formulation. For the covolume method, various diagonal tensor coefficients including discontinuous ones for second order elliptic PDEs are considered. We prove the first order convergence in H^1 norm and second order convergence in L^2 norm for the pressure variable. Unlike the usual nonconforming finite element method(FEM), the L^2-error estimate cannot be derived by a simple application of Aubin-Nitsche trick. The reason lies in that our covolume scheme is at most formulated in Petrov-Galerkin sense. Numerical examples are in quite good agreement with the theory even for the case of almost degenerate quadrilateral grids and the new covolume scheme shows ...