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

Urban, Karsten; Patera, Anthony

doi:10.1090/S0025-5718-2013-02782-2

An improved error bound for reduced basis approximation of linear parabolic problems
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by Karsten Urban and Anthony T. Patera;

Math. Comp. 83 (2014), 1599-1615

DOI: https://doi.org/10.1090/S0025-5718-2013-02782-2

Published electronically: October 23, 2013

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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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