A mixed nonoverlapping covolume method on quadrilateral grids for elliptic problems

Abstract

a b s t r a c tA 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)-norm as well as for the approximate pressures in the L 2 -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.

Error Bounds for Discontinuous Finite Volume Discretisations of Brinkman Optimal Control Problems

“…Moreover, we also have the following discrete Poincaré–Friedrichs inequality (see [ 11 , pp. 457]) and we can use Cauchy–Schwarz inequality and the definition of to readily obtain Proceeding analogously to [ 56 , Lemma 6], we can establish the coercivity of the bilinear form , stated in the following result.…”