A new error bound for reduced basis approximation of parabolic partial differential equations

Urban, Karsten; Patera, Anthony T.

doi:10.1016/j.crma.2012.01.026

Partial Differential Equations/Numerical Analysis

A new error bound for reduced basis approximation of parabolic partial differential equations
[Une nouvelle borne pour lʼerreur dans le cadre des approximations par bases réduites des équations aux dériées partielles paraboliques]

Présenté par : Philippe G. Ciarlet

Urban, Karsten ¹ ; Patera, Anthony T.²

¹ University of Ulm, Institute for Numerical Mathematics, Helmholtzstr. 18, 89081 Ulm, Germany
² Mechanical Engineering Department, Massachusetts Institute of Technology, 77, Massachusetts Avenue, Cambridge, MA 02139-4307, USA

Comptes Rendus. Mathématique, Tome 350 (2012) no. 3-4, pp. 203-207.

Résumé
Abstract

Nous considérons une formulation variationnelle espace–temps pour les équations différentielles paraboliques linéaires. Nous y associons une discrétisation par éléments finis de Petrov–Galerkin pour laquelle la constante de stabilité inf-sup $β_{δ}$ possède des propriétés agréables : $β_{δ}$ est unité pour lʼéquation de la chaleur ; $β_{δ}$ a une croissance seulement linéaire en temps pour des opérateurs de convection non-coercifs (mais asymptotiquement stables). Dans le cadre des approximations par bases réduites, cette dernière propriété permet dʼobtenir des bornes efficaces pour lʼerreur a posteriori en temps long, en net contraste avec les estimateurs dʼerreur en énergie classiques (pessimistes) qui présentent une croissance exponentielle.

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 $β_{δ}$ : $β_{δ}$ is unity for the heat equation; $β_{δ}$ grows 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.

Reçu le : 2011-10-31
Accepté le : 2012-01-30
Publié le : 2012-02-01

DOI : 10.1016/j.crma.2012.01.026

Affiliations des auteurs :

Urban, Karsten ¹ ; Patera, Anthony T. ²

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     title = {A new error bound for reduced basis approximation of parabolic partial differential equations},
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     pages = {203--207},
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Urban, Karsten; Patera, Anthony T. A new error bound for reduced basis approximation of parabolic partial differential equations. Comptes Rendus. Mathématique, Tome 350 (2012) no. 3-4, pp. 203-207. doi : 10.1016/j.crma.2012.01.026. http://www.numdam.org/articles/10.1016/j.crma.2012.01.026/

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