Finite volume methods and their analysis. (English) Zbl 0729.65087

Morton, K. W.; Süli, E.

IMA J. Numer. Anal. 11, No. 2, 241-260 (1991).

This paper is a very nice first attempt to analyze mathematically the convergence properties of the finite volume methods which are really successful, in particular for fluid dynamics problems. In this way, the authors use model problems in one and two dimensions:
i) in one dimension, for the one-dimensional Poisson equation \(\phi ''=f\) and for the convection-diffusion equation \(\epsilon w''-(aw)'=f,\) they show that finite volume methods give accurate flux values at volume boundaries and that they can be more accurate than finite element methods;
ii) in two dimensions, for the pure advection problem, they do the error analysis of the cell-vertex scheme. They prove the insensitivity of the method to mesh stretching in the coordinate directions so that the second-order accuracy is maintained on any mesh.

Reviewer: M.Bernadou (Le Chesnay)

MSC:

65N35	Spectral, collocation and related methods for boundary value problems involving PDEs
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
35J25	Boundary value problems for second-order elliptic equations
76R10	Free convection
76R50	Diffusion

Keywords:

error estimates; convergence; finite volume methods; Poisson equation; convection-diffusion equation; finite element methods; advection problem; error analysis; cell-vertex scheme