Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems
Assyr Abdulle, Yun Chao Bai, Gilles Vilmart · Discrete and Continuous Dynamical Systems - S · 2014
The reduced basis finite element heterogeneous multiscale method (RB-FE-HMM) fora class of nonlinear homogenization elliptic problems of nonmonotone type is introduced.In this approach, the solutions of the micro problems needed to estimate the macroscopicdata 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 nonlinearnumerical 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.