TY - GEN A1 - Malqvist, Axel A1 - Peterseim, Daniel T1 - Localization of Elliptic Multiscale Problems N2 - 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 log(1/H) are sufficient to preserve the convergence rates of the classical P1-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]. KW - finite element method KW - a priori error estimate KW - convergence KW - multiscale method Y1 - 2011 UR - https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/904 UR - https://nbn-resolving.org/urn:nbn:de:0296-matheon-9046 ER -