An intrinsic Proper Generalized Decomposition for parametric symmetric elliptic problems

We introduce in this paper a technique for the reduced order approximation of parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optimal subspace of at most that dimension which realizes the best approximation in mean of the error with respect to the parameter in the quadratic norm associated to the elliptic operator, between the exact solution and the Galerkin solution calculated on the subspace. This is analogous to the best approximation property of the Proper Orthogonal Decomposition (POD) subspaces, excepting that in our case the norm is parameter-depending, and then the POD optimal sub-spaces cannot be characterized by means of a spectral problem. We apply a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, directly in the on-line step. We prove that the partial sums converge to the continuous solutions, in mean quadratic elliptic norm.Comment: 18 page

Condition Inf-Sup vue par les méthodes spectrales

Il est bien connu que l'approximation des équations aux dérivées partielles sous contraintes nécessite la prise en compte d'une condition Inf-Sup. C'est le moyen mathématique, introduit dans [6, 7], pour assurer la compatibilité entre l'EDP et la contrainte. Quand celle-ci est assurée par l'Introduction d'un multiplicateur de Lagrange alors la condition Inf-Sup assure l'unicité de ce dernier. Le choix de la méthode d'approximation in ue de manière significative sur celui des espaces d'approximation compatibles ainsi que sur le comportement de la condtion Inf-Sup discrète. Dans le cadre de cette contribution, nous ferons le point sur cette question dans le cas d'une approximation par méthodes spectrales. Comme exemples d'EDP, nous allons considérer deux cas :i) les équations de Darcy et ii) les équations de Stoke

ITERATED QUASI-REVERSIBILITY METHOD APPLIED TO ELLIPTIC AND PARABOLIC DATA COMPLETION PROBLEMS

International audienceWe study the iterated quasi-reversibility method to regularize ill-posed elliptic and parabolic problems: data completion problems for Poisson's and heat equations. We define an abstract setting to treat both equations at once. We demonstrate the convergence of the regularized solution to the exact one, and propose a strategy to deal with noise on the data. We present numerical experiments for both problems: a two-dimensional corrosion detection problem and the one-dimensional heat equation with lateral data. In both cases, the method prove to be efficient even with highly corrupted data

Improvements on open and traction boundary conditions for Navier-Stokes time-splitting methods

no abstrac