Localization of Elliptic Multiscale Problems

Details Localization of Elliptic Multiscale Problems Localization of Elliptic Multiscale Problems

2011-10-04

arxiv.org/abs/1110.0692v3

Mathematics

Numerical Analysis

Scientific paper

This note constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying diffusion tensor. The basis functions are solutions of local problems on vertex patches. The error of the corresponding generalized finite element method decays exponentially w.r.t. the number of element layers in the patches. Hence, on a uniform mesh of size $H$, patches of diameter H\log(1/H) are sufficient to preserve the convergence rates of the classical P_1-FEM for the Poisson problem. The analysis does not rely on regularity of the solution or scale separation in the coefficient. The result justifies the use of the class of variational multiscale methods, introduced in [Comput. Methods Appl. Mech. Engrg., 196:2313-2324, 2007].

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