Generalized p-FEM in homogenization

Abstract

A new nite element method for elliptic problems with locally periodic microstructure of length " > 0 is developed and analyzed. It is shown that the method converges, as " ! 0, to the solution of the homogenized problem with optimal order in " and exponentially in the number of degrees of freedom independent o f " > 0. The computational work of the method is bounded independently of ". Numerical experiments demonstrate the feasibility and con rm the theoretical results.

Numerical methods for multiscale elliptic problems

“…This idea has been used in the analytical work of Allaire [3], E [18] and Nguetseng [40], among others. The work of Schwab et al [36,44,45] develop this into a numerical tool. The following result on the accuracy of this method is proved in [37] Here U n (p,k) 6 Cp À(kÀn + 1) and U n (l,s) 6 Cl À(sÀn + 1) .…”

Two-scale FEM for homogenization problems

“…For any f ∈ L 2 (R n ), (1.1), (1.6) admits a unique solution u ε ∈ H 1 (Ω ∞ ε ). We will exploit that u ε admits the representation [6,7,10] u ε (x) = 1 (2π) n/2 t∈R nf (t)ψ (x, ε, t) dt, x ∈ Ω ∞ ε , (1.11) where the kernel ψ(x, ε, t) is the distributional solution of…”

“…These two-scale approximation results are quite general and applicable whenever the solution has the two-scale regularity; in particular, the representation (1.11) which is valid only in the linear setting is not necessary. In contrast, in [4][5][6][7]9] a different (in general smaller) space V ε N than (1.18) is proposed. In that approach the kernel φ(y, ε, t) in (1.17) is incorporated directly in the FE-space via shape functions φ(y, ε, t) sampled at suitable points t j in the Fourier space.…”

A posteriori error estimates in quantities of interest for the finite element heterogeneous multiscale method

“…Babuška and Osborn [1] developed the pioneering work for multiscale FEM for elliptic problems using multiscale basis functions. We further mention the multiscale finite element method (MsFEM) developed by Hou et al [2] (see also the book by Efendiev and Hou [3]), the two-scale FEM proposed by Matache, Babuška and Schwab [4], the variational multiscale method by Hughes et al [5], and the sparse FEM introduced by Hoang and Schwab [6]. In this work we use the framework of the heterogeneous multiscale method (HMM) proposed by E and Engquist [7,8].…”