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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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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


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:
  • Anthony T. Patera
  • Affiliation: Mechanical Engineering Department, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, Massachusetts 02139-4307
  • Email:
  • 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:
  • MathSciNet review: 3194123