An offline–online homogenization strategy to solve quasilinear two‐scale problems at the cost of one‐scale problems
Assyr Abdulle, Yu-Shan Bai, Gilles Vilmart · International Journal for Numerical Methods in Engineering · 2014
SUMMARY A simple and concise finite element algorithm for the reliable and efficient resolution of nonlinear elliptic or parabolic multiscale problems of nonmonotone type is presented. Our method, based on a macroscopic and a microscopic discretization, combines reduced‐order modeling techniques with numerical homogenization. By precomputing a low dimensional reduced basis for the solution of the cell problems, one drastically reduces the cost of the iteration scheme at the macroscopic level, which become similar to the cost of solving a single scale nonlinear problem. A crucial step of our algorithm is in the selection of the reduced basis performed in an offline stage. This selection procedure relies on a new a posteriori nonlinear estimator. Both the offline and online costs are independent of the smallest scale in the physical problem. The performance and accuracy of the algorithm are illustrated on 2D and 3D stationary and evolutionary nonlinear multiscale problems.Copyright © 2014 John Wiley & Sons, Ltd.