Finite volumes and mixed Petrov-Galerkin finite elements : the unidimensional problem

François Dubois (1, 2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 LMSSC - Laboratoire de Mécanique des Structures et des Systèmes Couplés

For Laplace operator in one space dimension, we propose to formulate the heuristic finite volume method with the help of mixed Petrov-Galerkin finite elements. Weighting functions for gradient discretization are parameterized by some universal function. We propose for this function a compatibility interpolation condition and we prove that such a condition is equivalent to the inf-sup property when studying stability of the numerical scheme. In the case of stable scheme and under two distinct hypotheses concerning the regularity of the solution, we demonstrate convergence of the finite volume method in appropriate Hilbert spaces and with optimal order of accuracy.

Mots clés: Poisson equation; finite volumes; mixed finite elements; Petrov-Galerkin variational formulation; inf-sup condition; Poisson equation.

Domaines: Analyse numérique [math.NA]

François Dubois. Finite volumes and mixed Petrov-Galerkin finite elements : the unidimensional problem. Numerical Methods for Partial Differential Equations, 2000, 16 (3), p. 335-360. ⟨10.48550/arXiv.1401.0824⟩. ⟨hal-00923745⟩