A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations
Résumé
This paper develops the first rigorous error bounds for noncoercive problems (the Helmholtz problem — see also Y Maday, et al., A Blackbox Reduced-Basis Output Bound Method for Noncoercive Linear Problems, in Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar Volume XIV (eds D Cioranescu and J-L Lions), Elsevier Science B.V., pp. 533–569, 2002, and Rovas thesis) and for nonlinear problems (a monotonic cubic nonlinearity); additionally, we discuss a first quasi-rigorous treatment for the Burgers equation.
The proposals in this paper for the inf-sup lower bound (required for our error estimators for noncoercive equations), the treatment of greater-than-quadratic nonlinearities, and the error bounds for the Burgers equation, have all been subsequently improved: see Papers 12 and 18, Paper 13, and Papers 5 and 7 and 9, respectively.
This paper also introduces an adaptive sampling procedure which exploits our error bounds to construct very efficient RB samples $S_N$ and hence spaces $W_N$ — essentially a POD-like selection approach but in which only the retained snapshots (amongst a very large number of candidates) are actually computed on the FEM truth approximation space. This sampling approach is further articulated in Paper 7, Paper 8, and Paper 9.
Origine : Fichiers produits par l'(les) auteur(s)
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