An Efficient Online-Offline Method for Elliptic Homogenization Problems

A Nitsche Hybrid multiscale method with non-matching grids

“…If we use some other numerical upscaling methods, e.g., [18,32,19,29], then b ε h ∈ M(λ , Λ ; D) with certain constants λ and Λ , which depend on λ and Λ, but not exactly the same. To quantity the approximation error for the effective matrix, we define e(HMM): = max x∈ K2 (A − A h )(x) F , where…”

“…, where à is the effective matrix associated with ãε . Since there is no analytical formula for Ã, we use the least-squares based method in [32,29] to obtain a higher-order approximation to A so that e(HMM) is negligible, which is denoted by Ãh . The homogenized solution u 0 is computed by solving Problem (1.3) with…”

A Concurrent Global–Local Numerical Method for Multiscale PDEs

“…We let ε = 0.0063 for the sake of comparison with those in [2] and compute u ε over a uniform mesh with mesh size 3.33e − 4. By Corollary 2.2 and the identity (2.6), the effective matrix A = χ K0 a + (1 − χ K0 ) A and the approximating effective matrix A h = χ K0 a+(1−χ K0 ) A h , where A h is an approximation of the effective matrix associated with a ε through a fast solver based on the discrete least-squares reconstruction in the framework of HMM (see [28] and [25] for details of such fast algorithm). We reconstruct A h to high accuracy so that the reconstruction error is negligible.…”