Research reports
Years: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991
Sparse tensor finite elements for elliptic multiple scale problems
by H. Harbrecht and Ch. Schwab
(Report number 2010-34)
Abstract
Locally periodic, elliptic multiscale problems in a bounded Lipschitz domain $D/subset/mathbb(R)^n$ with K > 2 separated scales are reduced to an elliptic system of K coupled, anisotropic elliptic one-scale problems in a cartesian product domain of total dimension Kn (e.g. 2; 3; 11; 26). In (23; 31), it has been shown how these coupled elliptic problems could be solved by sparse tensor wavelet Finite Element methods in log-linear complexity with respect to the number N of degrees of freedom required by multilevel solvers for elliptic one-scale problems in D with the same convergence rate. In the present paper, the high dimensional one-scale limiting problems are discretized by a sparse tensor product finite element method (FEM) with standard, one-scale FE basis functions as used in engineering FE codes. Sparse tensorization and multilevel preconditioning is achieved by a BPX multilevel iteration. We show that the resulting sparse tensor multilevel FEM resolves all physical length scales throughout the domain, with efficiency (i.e., accuracy versus work and memory) comparable to that of multigrid solvers for elliptic one-scale problems in the physical domain D In particular, our sparse tensor FEM gives numerical approximations of the correct homogenized limit as well as compressed numerical representations of all first order correctors, throughout the physical domain with performance independent of the physical problem's scale parameters. Numerical examples with standard FE shape functions and BPX multilevel preconditioners for elliptic problems with K=2 separated, physical scales in spatial dimension n=2 confirm the theoretical results. In particular, the present approach allows to avoid the construction of wavelet FE bases necessary in previous work (23; 31) while achieving resolution of all scales throughout the physical domain in log-linear complexity, with the logarithmic exponent behaving linearly in the number K of scales.
Keywords: Multiscale elliptic boundary value problem, reiterated homogenization, two-scale convergence, sparse tensor product FEM, BPX multilevel preconditioner, frames
BibTeX@Techreport{HS10_436, author = {H. Harbrecht and Ch. Schwab}, title = {Sparse tensor finite elements for elliptic multiple scale problems }, institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich}, number = {2010-34}, address = {Switzerland}, url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2010/2010-34.pdf }, year = {2010} }
Disclaimer
© Copyright for documents on this server remains with the authors.
Copies of these documents made by electronic or mechanical means including
information storage and retrieval systems, may only be employed for
personal use. The administrators respectfully request that authors
inform them when any paper is published to avoid copyright infringement.
Note that unauthorised copying of copyright material is illegal and may
lead to prosecution. Neither the administrators nor the Seminar for
Applied Mathematics (SAM) accept any liability in this respect.
The most recent version of a SAM report may differ in formatting and style
from published journal version. Do reference the published version if
possible (see SAM
Publications).