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An improved error bound for reduced basis approximation of linear parabolic problems

Authors: Karsten Urban and Anthony T. Patera
Journal: Math. Comp. 83 (2014), 1599-1615
MSC (2010): Primary 35K15, 65M15, 65M60
Published electronically: October 23, 2013
MathSciNet review: 3194123
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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

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

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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.