MATHEMATICS OF COMPUTATION Volume 66, Number 220, October 1997, Pages 1389–1406 S 0025-5718(97)00886-7 ANALYSIS OF A CELL–VERTEX FINITE VOLUME METHOD FOR CONVECTION–DIFFUSION PROBLEMS K. W. MORTON, MARTIN STYNES, AND ENDRE S ¨ ULI Abstract. A cell-vertex finite volume approximation of elliptic convection- dominated diffusion equations is considered in two dimensions. The scheme is shown to be stable and second-order convergent in a mesh-dependent L 2 -norm. 1. Introduction A finite volume formulation is the preferred technique for discretising systems of partial differential equations where conservation is the most important property to be modelled, compressible gas dynamics being the prime example–see Jameson [3] and a large subsequent literature. Of the various formulations that are possible, the cell-vertex scheme is often advocated for its compactness and its accuracy for first-order equations on distorted meshes (see Morton and Paisley [8] and S¨ uli [17]); moreover, Morton et al. [6] and Crumpton et al. [2] have also demonstrated the effectiveness of the cell-vertex scheme for the compressible Navier-Stokes equations (see also Mackenzie and Morton [7]). Numerous practical computations have, in- deed, shown this discretisation to be of very general utility, with recent extensions to unstructured three-dimensional meshes on general domains, and applicable to the very high aspect ratio meshes encountered with high Reynolds number, turbulent flows. However, the resulting system of discrete equations is difficult to solve and its accuracy is hard to analyse. Some of these issues can be studied with simple model problems on rectangular meshes. In the earlier form of the method, for purely hyperbolic problems, when it was referred to as the finite difference box scheme of Thomas [18], Preissmann [13], Wendroff [19], Keller [4] and others, the equations were always solved by marching in a special coordinate direction. This is not possible with the equations for steady inviscid transonic flow and various alternatives have been developed, based on the work of Ni [11]; marching techniques are even less appropriate when second-order viscous terms are present, but Ni’s techniques are still effective (see [2] and [7]). The present paper is one in a series devoted to the analysis of the resulting cell-vertex finite volume schemes. Scalar convection-dominated diffusion problems, with general convective velocity fields, show both the remarkable approximation properties of cell-vertex methods Received by the editor November 22, 1994 and, in revised form, January 26, 1996 and June 12, 1996. 1991 Mathematics Subject Classification. Primary 65N99, 65L10; Secondary 76M25. Key words and phrases. Finite volume methods, stability, error analysis. The authors are grateful to the British Council and Forbairt for the generous financial support of this project. c 1997 American Mathematical Society 1389 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use