**
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE **

Vol. LXXVI, 1 (2007)

p. 111 - 136

Analysis tools for finite volume schemes

R. Eymard, T. Gallouet, R. Herbin and J.-C. Latche

**Abstract**.
This paper is devoted to a review of the analysis tools which have been developed for the
the mathematical study of cell centred finite volume schemes in the past years.
We first recall the general principle of the method and give some simple examples.
We then explain how the analysis is performed for elliptic equations and relate it to
the analysis of the continuous problem; the lack of regularity of the approximate
solutions is overcome by an estimate on the translates, which allows the use of the
Kolmogorov theorem in order to get compactness.
The parabolic case is treated with the same technique. Next we introduce a co-located
scheme for the incompressible Navier-Stokes equations,
which requires the definition of some discrete derivatives.
Here again, we explain how the continuous estimates can guide us for the discrete estimates.
We then give the basic ideas of the convergence analysis for non linear hyperbolic
conservation laws, and conclude with an overview of the recent domains of application.

**Keywords**:
Finite volume methods, elliptic equations, parabolic equations,
Navier-Stokes equations, hyperbolic equations.

**AMS Subject classification.** Primary: 65M12, 65N12, 76D05, 76D07, 76M12.

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