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

Published inDiscrete and continuous dynamical systems. Series S, vol. 8, no. 1, p. 91-118
Publication date2015
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.
Affiliation Not a UNIGE publication
Citation (ISO format)
ABDULLE, Assyr, BAI, Yun, VILMART, Gilles. Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems. In: Discrete and continuous dynamical systems. Series S, 2015, vol. 8, n° 1, p. 91–118. doi: 10.3934/dcdss.2015.8.91
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ISSN of the journal1937-1179
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