Mixed covolume methods for quasi-linear second-order elliptic problems

Abstract

We consider covolume methods for the mixed formulations of quasi-linear second-order elliptic problems. Covolume methods for the mixed formulations of linear elliptic problem was rst considered by Russell [Rigorous Block-Centered Discretizations on Irregular Grids: Improved Simulation of Complex Reservoir Systems, Tech. report 3, Project Report, Reservoir Simulation Research Corporation, Tulsa, OK, 1995] and tested extensively in [ Cai et al., Comput. Geosci., 1( 1997), pp. 289-315], [Jones, A Mixed Finite Volume Element Method for Accurate Computation of Fluid Velocities in Porous Media, Ph. D. thesis, University of Colorado, Denver, 1995]. The analysis was carried out by Chou and Kwak [SIAM J. Numer. Anal., 37 (2000), pp. 758-771] for linear symmetric problems, where they showed optimal error estimates in L-2 norm for the pressure and in H ( div) norm for the velocity. In this paper we extend their results to quasi-linear problems by following Milner's argument [Math. Comp., 44 (1985), pp. 303-320] through an adaptation of the duality argument of Douglas and Roberts [Math. Comp., 44 (1985), pp. 39-52] for mixed covolume methods.