Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems

Assyr Abdulle (1) , Yun Bai (1) , Gilles Vilmart (2, 3)

1 EPFL - Ecole Polytechnique Fédérale de Lausanne
2 IPSO - Invariant Preserving SOlvers
3 IRMAR - Institut de Recherche Mathématique de Rennes

A reduced basis nite element heterogeneous multiscale method (RB-FE-HMM) for a class of nonlinear homogenization elliptic problems of nonmonotone type is introduced. In this approach, the solutions of the micro problems needed to estimate the macroscopic data of the homogenized problem are selected by a Greedy algorithm and computed in an o ine stage. It is shown that the use of reduced basis (RB) for nonlinear numerical homogenization reduces considerably the computational cost of the nite element heterogeneous multiscale method (FE-HMM). As the precomputed microscopic functions depend nonlinearly on the macroscopic solution, we introduce a new a posteriori error estimator for the Greedy algorithm that guarantees the convergence of the online Newton method. A priori error estimates and uniqueness of the numerical solution are also established. Numerical experiments illustrate the e ciency of the proposed method.

Mots clés: nonlinear nonmonotone elliptic problems; numerical homogenization; reduced basis method; a posteriori error estimator; nite element method

Domaines: Analyse numérique [math.NA]

Assyr Abdulle, Yun Bai, Gilles Vilmart. Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems. Discrete and Continuous Dynamical Systems - Series S, 2015, 8 (1), pp.91-118. ⟨10.3934/dcdss.2015.8.91⟩. ⟨hal-00811490v2⟩