A mixed nonoverlapping covolume method on quadrilateral grids for elliptic problems

A covolume method is proposed for the mixed formulation of second-order elliptic problems. The solution domain is divided by a quadrilateral grid, corresponding to which a nonoverlapping dual grid is constructed. The velocity and pressure are approximated by the lowest-order Raviart–Thomas space on quadrilaterals. We prove its first order optimal rate of convergence for the approximate velocities in the H(div)H(div)-norm as well as for the approximate pressures in the L2L2-norm. A second order error estimate between a suitable projection of the exact velocity (or pressure) and the approximate velocity (or approximate pressure) is also presented. Numerical experiments are provided to illustrate the error behavior of the scheme.