On the $L^2$ Stability of Finite Volumes for Stationary First Order Systems
Résumé
The aim of this paper is two-folds. Firstly we study first order stationary systems of PDEs of the form $\sum_k A_k\partial_k U + KU = 0$ with $K_\eta tr0$ on $R^d$ . We prove that the classical assumption $K > 0$ is not necessary for the well-posedness of the system and is violated in the particular case of the first order Poisson problem. Secondly we prove the $L^2$ stability of the finite volume discretisations provided the term KU is appropriately discretised on faces. Our result relies on a discrete Gagliardo-Nirenberg-Sobolev inequality to be submitted.