Reduced basis finite element heterogeneous multiscale method for quasilinearelliptic homogenization problems

Assyr Abdulle; Yun Bai; Gilles Vilmart

doi:10.3934/dcdss.2015.8.91

Article Contents

2015, Volume 8, Issue 1: 91-118. Doi: 10.3934/dcdss.2015.8.91

This issue Previous Article Multi-scales H-measures Next Article Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems

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

1.
ANMC, Section de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne
2.
École Normale Supérieure de Rennes, INRIA Rennes, IRMAR, CNRS, UEB, Campus de Ker Lann, 35170 Bruz

Received: March 31, 2013

Revised: October 31, 2013

Published: January 2015

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Abstract

The reduced basis finite 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 offline stage. It is shown that the use of reduced basis (RB) for nonlinear numerical homogenization reduces considerably the computational cost of the finite 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 efficiency of the proposed method.

Keywords:

Nonlinear nonmonotone elliptic problems,
numerical homogenization,
reduced basis method,
a posteriori error estimator,
finite element method.

Mathematics Subject Classification: 65N30, 65M60, 74D10, 74Q05.

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