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A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations

Patera, Adolf, 1836-1912, Rovas Dimitrios, C. Prud’homme, Veroy, Karen

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URIhttp://purl.tuc.gr/dl/dias/E7230C93-2C9B-4B6F-B93D-5C660B6DA4D8-
Identifierhttp://augustine.mit.edu/methodology/papers/atpAIAA2003.pdf-
Languageen-
Extent18 pagesen
TitleA posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations en
CreatorPatera, Adolf, 1836-1912en
CreatorRovas Dimitriosen
CreatorΡοβας Δημητριοςel
CreatorC. Prud’hommeen
CreatorVeroy, Karenen
PublisherAmerican Institute of Aeronautics and Astronauticsen
Content SummaryWe present a technique for the rapid and reliable prediction of linear–functional out- puts of elliptic partial differential equations with affine parameter dependence. The essential components are (i) rapidly convergent global reduced–basis approximations — (Galerkin) projection onto a space WN spanned by solutions of the governing partial dif- ferential equation at N selected points in parameter space; (ii) a posteriori error estimation — relaxations of the error-residual equation that provide inexpensive yet sharp bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures — methods which decouple the generation and projection stages of the approximation process. The operation count for the on–line stage — in which, given a new parameter value, we calculate the output of interest and associated error bound — depends only on N (typically very small) and the parametric complexity of the problem. In this paper we develop new a posteriori error estimation procedures for noncoercive linear, and certain nonlinear, problems that yield rigorous and sharp error statements for all N. We consider three particular examples: the Helmholtz (reduced-wave) equation; a cubically nonlinear Poisson equation; and Burgers equation — a model for incompressible Navier-Stokes. The Helmholtz (and Burgers) example introduce our new lower bound constructions for the requisite inf-sup (singular value) stability factor; the cubic nonlin- earity exercises symmetry factorization procedures necessary for treatment of high-order Galerkin summations in the (say) residual dual-norm calculation; and the Burgers equa- tion illustrates our accommodation of potentially multiple solution branches in our a posteriori error statement. Numerical results are presented that demonstrate the rigor, sharpness, and efficiency of our proposed error bounds, and the application of these bounds to adaptive (optimal) approximation.en
Type of ItemΑφίσα σε Συνέδριοel
Type of ItemConference Posteren
Licensehttp://creativecommons.org/licenses/by/4.0/en
Date of Item2015-10-17-
Date of Publication2003-
SubjectGreek mathematicsen
Subjectmathematics greeken
Subjectgreek mathematicsen
SubjectHelmholtz equationen
Bibliographic CitationK. Veroy, C. Prud’homme, D.V. Rovas, A.T. Patera.(2003).A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations.Presented at f the 16th AIAA computational fluid dynamics conference .[online].Available:http://augustine.mit.edu/methodology/papers/atpAIAA2003.pdfen

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