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A posteriori error bounds for reduced-basis approximations ofparametrized parabolic partial differential equations

Published online by Cambridge University Press:  15 March 2005

Martin A. Grepl  and
Anthony T. Patera
Martin A. Grepl
Affiliation:
Massachusetts Institute of Technology, Room 3-264, Cambridge, MA, USA.
Anthony T. Patera
Affiliation:
Massachusetts Institute of Technology, Room 3-266, Cambridge, MA, USA. patera@mit.edu (corresponding author).
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Abstract

In this paper, we extend the reduced-basis methods and associated a posteriori error estimators developed earlier for elliptic partial differential equations to parabolic problems with affine parameter dependence. The essential new ingredient is the presence of time in the formulation and solution of the problem – we shall “simply” treat time as an additional, albeit special, parameter. First, we introduce the reduced-basis recipe – Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter-time space – and develop a new greedy adaptive procedure to “optimally” construct the parameter-time sample set. Second, we propose error estimation and adjoint procedures that provide rigorous and sharp bounds for the error in specific outputs of interest: the estimates serve a priori to construct our samples, and a posteriori to confirm fidelity. Third, based on the assumption of affine parameter dependence, we develop offline-online computational procedures: in the offline stage, we generate the reduced-basis space; in the online stage, given a new parameter value, we calculate the reduced-basis output and associated error bound. The operation count for the online stage depends only on N (typically small) and the parametric complexity of the problem; the method is thus ideally suited for repeated, rapid, reliable evaluation of input-output relationships in the many-query or real-time contexts.

Keywords

Parabolic partial differential equationsdiffusion equationparameter-dependent systemsreduced-basis methodsoutput boundsGalerkin approximationa posteriori error estimation.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 1 , January 2005 , pp. 157 - 181
Copyright
© EDP Sciences, SMAI, 2005

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