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Technical University of Crete
Technical University of Crete
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| URI: | http://purl.tuc.gr/dl/dias/E7230C93-2C9B-4B6F-B93D-5C660B6DA4D8 | ||
| Year | 2003 | ||
| Type of Item | Conference Poster | ||
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| Bibliographic Citation | K. 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.pdf | ||
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We 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.
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