An offline–online homogenization strategy to solve quasilinear two‐scale problems at the cost of one‐scale problems

Abstract

SUMMARYInspired by recent analyses of the finite element heterogeneous multiscale method and the reduced basis technique for nonlinear problems, we present a simple and concise finite element algorithm for the reliable and efficient resolution of elliptic or parabolic multiscale problems of nonmonotone type. Solutions of appropriate cell problems on sampling domains are selected by a greedy algorithm in an offline stage and assembled in a reduced basis (RB). This RB is then used in an online stage to solve two-scale problems at a computational cost comparable to the single-scale case. Both the offline and the online cost 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.KEY WORDS: nonlinear nonmonotone problems, numerical homogenization, reduced basis method, a posteriori error estimator, finite element method

A time dependent approach for removing the cell boundary error in elliptic homogenization problems

“…For other approaches to decrease the computational burden in linear and quasi-linear elliptic multiscale PDEs see e.g. [6][7][8].…”

“…where û is given in (8). For an analysis of the FEM version of the HMM for elliptic problems we refer the reader to [9], see also [10] for a fully discrete analysis.…”

Numerical Homogenization Methods for Parabolic Monotone Problems

“…Following [5] we propose a linearized scheme. The idea is to decouple the micro-solutions in (99) and to consider…”

“…We next mention classical estimates for FEM with numerical quadrature that are needed in the analysis below [23,Thms 4,5]. Assuming (Q1) and appropriate regularity of the homogenised solution u 0 we have for all v H , w H ∈ S 0 (Ω , T H ) (where µ = 0 or 1),…”

Numerical Homogenization

“…As mentioned in §5a, for this type of nonlinear problem, we have cell problems defined in ( with κ = (x, s). The offline settings and outputs can be seen in table 5, where the parameter range of s is estimated by an offline procedure proposed in [62]. Online stage.…”