Inspired by the reduced basis approach and modern numerical multiscale
methods, we present a new framework for an efficient treatment of
heterogeneous multiscale problems. The new approach is based on the
idea of considering heterogeneous multiscale problems as parametrized
partial differential equations where the parameters are smooth functions.
We then construct, in an offline phase, a suitable localized reduced
basis that is used in an online phase to efficiently compute approximations
of the multiscale problem by means of a discontinuous Galerkin method
on a coarse grid. We present our approach for elliptic multiscale
problems and discuss an a posteriori error estimate that can be used
in the construction process of the localized reduced basis. Numerical
experiments are given to demonstrate the efficiency of the new approach.
%0 Journal Article
%1 kaulmann2011local
%A Kaulmann, Sven
%A Ohlberger, Mario
%A Haasdonk, Bernard
%D 2011
%J Comptes Rendus Mathematique
%K anm ians imported
%N 23-24
%P 1233--1238
%R 10.1016/j.crma.2011.10.024
%T A new local reduced basis discontinuous Galerkin approach for heterogeneous
multiscale problems
%U http://www.sciencedirect.com/science/article/pii/S1631073X11003074
%V 349
%X Inspired by the reduced basis approach and modern numerical multiscale
methods, we present a new framework for an efficient treatment of
heterogeneous multiscale problems. The new approach is based on the
idea of considering heterogeneous multiscale problems as parametrized
partial differential equations where the parameters are smooth functions.
We then construct, in an offline phase, a suitable localized reduced
basis that is used in an online phase to efficiently compute approximations
of the multiscale problem by means of a discontinuous Galerkin method
on a coarse grid. We present our approach for elliptic multiscale
problems and discuss an a posteriori error estimate that can be used
in the construction process of the localized reduced basis. Numerical
experiments are given to demonstrate the efficiency of the new approach.
@article{kaulmann2011local,
abstract = {Inspired by the reduced basis approach and modern numerical multiscale
methods, we present a new framework for an efficient treatment of
heterogeneous multiscale problems. The new approach is based on the
idea of considering heterogeneous multiscale problems as parametrized
partial differential equations where the parameters are smooth functions.
We then construct, in an offline phase, a suitable localized reduced
basis that is used in an online phase to efficiently compute approximations
of the multiscale problem by means of a discontinuous Galerkin method
on a coarse grid. We present our approach for elliptic multiscale
problems and discuss an a posteriori error estimate that can be used
in the construction process of the localized reduced basis. Numerical
experiments are given to demonstrate the efficiency of the new approach.},
added-at = {2019-06-17T14:25:24.000+0200},
author = {Kaulmann, Sven and Ohlberger, Mario and Haasdonk, Bernard},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/2ae6948d949da5049a7aea70d2e861eee/britsteiner},
date-added = {2012-10-24T14:53:21GMT},
date-modified = {2014-01-29T11:58:01GMT},
doi = {10.1016/j.crma.2011.10.024},
file = {:http\://www.mathematik.uni-stuttgart.de/fak8/ians/publications/files/Kaulmann2011_www_CRAS_New_Local_RBDG.pdf:PDF},
interhash = {6bc15bc903a2ba02d9f28b724639b90e},
intrahash = {ae6948d949da5049a7aea70d2e861eee},
journal = {Comptes Rendus Mathematique},
keywords = {anm ians imported},
month = dec,
number = {23-24},
owner = {kaulmann},
pages = {1233--1238},
rating = {0},
read = {Yes},
timestamp = {2021-09-29T12:35:04.000+0200},
title = {{A new local reduced basis discontinuous Galerkin approach for heterogeneous
multiscale problems}},
url = {http://www.sciencedirect.com/science/article/pii/S1631073X11003074},
volume = 349,
year = 2011
}