Higher order multipoint flux mixed finite element methods on quadrilaterals and hexahedra

We develop higher order multipoint flux mixed finite element (MFMFE) methods for solving elliptic problems on quadrilateral and hexahedral grids that reduce to cell-based pressure systems. The methods are based on a new family of mixed finite elements, which are enhanced Raviart-Thomas spaces with bubbles that are curls of specially chosen polynomials. The velocity degrees of freedom of the new spaces can be associated with the points of tensor-product Gauss-Lobatto quadrature rules, which allows for local velocity elimination and leads to a symmetric and positive definite cell-based system for the pressures. We prove optimal $k$-th order convergence for the velocity and pressure in their natural norms, as well as $(k+1)$-st order superconvergence for the pressure at the Gauss points. Moreover, local postprocessing gives a pressure that is superconvergent of order $(k+1)$ in the full $L^2$-norm. Numerical results illustrating the validity of our theoretical results are included.