Reduced basis A posteriori error bounds for the Stokes equations in parametrized domains: A penalty approach

Anna Lena Gerner, Karen Veroy

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

Abstract

We present reduced basis approximations and associated rigorous a posteriori error bounds for the Stokes equations in parametrized domains. The method is built upon a penalty formulation for saddle point problems and provides error bounds admitting affine geometric variations with relative ease. Essential features are: i) dimension reduction; ii) stable, good approximation of the pressure; iii) optimal and numerically stable approximations identified by an efficient Greedy sampling procedure; iv) certainty, through rigorous a posteriori bounds for the error in the reduced basis approximation; and v) efficiency, through an offline-online computational strategy. The method is applied to a flow problem in a two-dimensional channel with a parametrized rectangular obstacle. Numerical results show that the reduced basis approximation converges rapidly, the effectivities associated with the (inexpensive) rigorous a posteriori error bounds remain good even for reasonably small values of the penalty parameter.

Original languageEnglish
Title of host publicationICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics 2010
EditorsTheodore E. Simos, George Psihoyios, Ch. Tsitouras
Pages2041-2044
Number of pages4
DOIs
Publication statusPublished - 2010
Externally publishedYes
EventInternational Conference on Numerical Analysis and Applied Mathematics 2010, ICNAAM-2010 - Rhodes, Greece
Duration: 19 Sep 201025 Sep 2010

Publication series

NameAIP Conference Proceedings
PublisherAmerican Institute of Physics
Volume1281
ISSN (Print)0094-243X

Conference

ConferenceInternational Conference on Numerical Analysis and Applied Mathematics 2010, ICNAAM-2010
CountryGreece
CityRhodes
Period19/09/1025/09/10

Bibliographical note

Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

Keywords

  • a posteriori error estimation
  • error bound
  • Greedy sampling
  • offline-online procedure
  • penalty
  • real-time computation
  • reduced basis
  • reduced order model
  • saddle point problem
  • Stokes equations
  • successive constraint method

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