Two-Scale FEM for Homogenization Problems

Sparse Two-Scale FEM for Homogenization Problems

We analyze two-scale Finite Element Methods for the numerical solution of elliptic homogenization problems with coefficients oscillating at a small length scale ε 1. Based on a refined two-scale regularity on the solutions, two-scale tensor product FE spaces are introduced and error estimates which are robust (i.e. independent of ε) are given. We show that under additional two-scale regularity ...

Homogenization for contact problems with periodically rough surfaces

We consider the contact of two elastic bodies with rough surfaces at the interface. The size of the micro-peaks and -valleys is very small compared with the macrosize of the bodies’ domains. This makes the direct application of the FEM for the calculation of the contact problem prohibitively costly. A method is developed that allows deriving a macrocontact condition on the interface. The method...

High-Dimensional Finite Elements for Elliptic Problems with Multiple Scales

Elliptic homogenization problems in a domain Ω R d with n + 1 separate scales are reduced to elliptic problems in dimension (n + 1)d. These one-scale problems are discretized by a sparse tensor product finite element method (FEM). We prove that this sparse FEM has accuracy, work, and memory requirements comparable to those in a standard FEM for singlescale problems in Ω, while it gives numerica...

Fully discrete analysis of the heterogeneous multiscale method for elliptic problems with multiple scales

A fully discrete analysis of the finite element heterogeneous multiscale method (FEHMM) for elliptic problems with N + 1 well separated scales is discussed. The FEHMM is a numerical homogenization method that relies on macroscopic scheme (macro FEM) for the approximation of the effective solution corresponding to the multiscale problem. The effective data are recovered from micro scale computat...

Homogenization via p-FEM for Problems