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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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An improved error bound for reduced basis approximation of linear parabolic problems
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by Karsten Urban and Anthony T. Patera PDF
Math. Comp. 83 (2014), 1599-1615 Request permission

Abstract:

We consider a space-time variational formulation for linear parabolic partial differential equations. We introduce an associated Petrov-Galerkin truth finite element discretization with favorable discrete inf-sup constant $\beta _{\delta }$, the inverse of which enters into error estimates: $\beta _{\delta }$ is unity for the heat equation; $\beta _{\delta }$ decreases only linearly in time for non-coercive (but asymptotically stable) convection operators. The latter in turn permits effective long-time a posteriori error bounds for reduced basis approximations, in sharp contrast to classical (pessimistic) exponentially growing energy estimates. The paper contains a full analysis and various extensions for the formulation introduced briefly by Urban and Patera (2012) as well as numerical results for a model reaction-convection-diffusion equation.
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Additional Information
  • Karsten Urban
  • Affiliation: University of Ulm, Institute for Numerical Mathematics, Helmholtzstr. 20, 89081 Ulm, Germany
  • Email: karsten.urban@uni-ulm.de
  • Anthony T. Patera
  • Affiliation: Mechanical Engineering Department, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, Massachusetts 02139-4307
  • Email: patera@mit.edu
  • Received by editor(s): June 16, 2012
  • Received by editor(s) in revised form: December 20, 2012
  • Published electronically: October 23, 2013
  • Additional Notes: The first author was supported by the Deutsche Forschungsgemeinschaft (DFG) under Ur-63/9 and GrK1100. This paper was partly written while the first author was a visiting professor at M.I.T
    The second author was supported by OSD/AFOSR/MURI Grant FA9550-09-1-0613 and by ONR Grant N00014-11-1-0713
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 83 (2014), 1599-1615
  • MSC (2010): Primary 35K15, 65M15, 65M60
  • DOI: https://doi.org/10.1090/S0025-5718-2013-02782-2
  • MathSciNet review: 3194123