×

zbMATH — the first resource for mathematics

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

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
PDF BibTeX XML Cite
Full Text: DOI