Compactness of discrete approximate solutions to parabolic PDEs - Application to a turbulence model

Thierry Gallouët (1) , J.-C Latché (2)

1 I2M - Institut de Mathématiques de Marseille
2 IRSN - Institut de Radioprotection et de Sûreté Nucléaire

In this paper, we prove an adaptation of the classical compactness Aubin-Simon lemma to sequences of functions obtained through a sequence of discretizations of a parabolic problem. The main difficulty tackled here is to generalize the classical proof to handle the dependency of the norms controlling each function u (n) of the sequence with respect to n. This compactness result is then used to prove the convergence of a numerical scheme combining finite volumes and finite elements for the solution of a reduced turbulence problem. 1. Introduction. Let us suppose given a sequence of approximations of a parabolic problem (P), which, for instance, may be though of as discretizations of (P) by a numerical scheme, or resulting from the combination of a Faedo-Galerkin technique with a time discretization. In both cases, we are in presence of a family of finite-dimensional systems, and their solution, let us say (u (n)) n∈N , may be considered as a sequence of functions of time, taking their values in a finite dimensional subspace, let us say B (n) of a Banach space (usually a Lebesgue L q space, q ≥ 1). To show the convergence of such a process, a common path is to follow the following steps: 1. first, for each approximate problem, prove the existence of a solution, and derive estimates satisfied by any solution, 2. then use compactness arguments to show (possibly up to the extraction of a subsequence) the existence of a limit, 3. and, finally, prove that this limit satisfies the initial problem (P). Let us now focus on item 2, which is the issue addressed in this paper. The problem here is to prove a compactness result for a sequence of functions of time taking their value in a sequence of discrete spaces and controlled (themselves and their discrete time-derivative) in discrete norms; in the general case, both B (n) and the space part of the norms depend on n. The compactness result given in this paper relies on this particular structure, and consists in a generalization of the classical Aubin-Simon lemma to this specific case.

Mots clés: Partial differential equations; discretisation; convergence

Domaines: Analyse classique [math.CA]

Thierry Gallouët, J.-C Latché. Compactness of discrete approximate solutions to parabolic PDEs - Application to a turbulence model. Communications on Pure and Applied Analysis, 2012, 11 (6), pp.21. ⟨10.3934/cpaa.2012.11.2371⟩. ⟨hal-01283559⟩