## 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
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