A Priori Error Estimate of a Multiscale Finite Element Method for Transport Modeling

Franck Ouaki (1) , Grégoire Allaire (2, 3) , Sylvain Desroziers (4) , Guillaume Enchéry (4)

1 CEA - Commissariat à l'énergie atomique et aux énergies alternatives
2 DeFI - Shape reconstruction and identification
3 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
4 IFPEN - IFP Energies nouvelles

This work proposes an \textit{a priori} error estimate of a multiscale finite element method to solve convection-diffusion problems where both velocity and diffusion coefficient exhibit strong variations at a scale which is much smaller than the domain of resolution. In that case, classical discretization methods, used at the scale of the heterogeneities, turn out to be too costly. Our method, introduced in~\cite{ECCOMAS}, aims at solving this kind of problems on coarser grids with respect to the size of the heterogeneities by means of particular basis functions. These basis functions are defined using cell problems and are designed to reproduce the variations of the solution on an underlying fine grid. Since all cell problems are independent from each other, these problems can be solved in parallel, which makes the method very efficient when used on parallel architectures. This article focuses on the proof of an \textit{a priori} error estimate of this method.

Mots clés: convection-diffusion; periodic homogenization; multiscale finite element method

Domaines: Analyse numérique [math.NA]

Franck Ouaki, Grégoire Allaire, Sylvain Desroziers, Guillaume Enchéry. A Priori Error Estimate of a Multiscale Finite Element Method for Transport Modeling. SeMA Journal: Boletin de la Sociedad Española de Matemática Aplicada, 2015, 67 (1), pp.1-37. ⟨hal-01071637⟩