Generalized Lagrangian Coordinates for Transport and Two-Phase Flows in Heterogeneous Anisotropic Porous Media

We show how Lagrangian coordinates provide an effective representation of how difficult non-linear, hyperbolic transport problems in porous media can be dealt with. Recalling Lagrangian description first, we then derive some basic but remarkable properties useful for the numerical com- putation of projected transport operators. We furthermore introduce new generalized Lagrangian coordinates with their application to the Darcy–Muskat two-phase flow models. We show how these generalized Lagrangian coordinates can be constructed from the global mass conservation, and that they are related to the existence of a global pressure previously defined in the literature about the subject. The whole representation is developed in two or three dimensions for numerical purposes, for isotropic or anisotropic heterogeneous porous media

Recursive POD expansion for the advection-diffusion-reaction equation

This paper deals with the approximation of advection-diffusion-reaction equation solution by reduced order methods. We use the Recursive POD approximation for multivariate functions introduced in [M. AZAÏEZ, F. BEN BELGACEM, T. CHACÓN REBOLLO, Recursive POD expansion for reactiondiffusion equation, Adv.Model. and Simul. in Eng. Sci. (2016) 3:3. DOI 10.1186/s40323-016-0060-1] and applied to the low tensor representation of the solution of the reaction-diffusion partial differential equation. In this contribution we extend the Recursive POD approximation for multivariate functions with an arbitrary number of parameters, for which we prove general error estimates. The method is used to approximate the solutions of the advection-diffusion-reaction equation. We prove spectral error estimates, in which the spectral convergence rate depends only on the diffusion interval, while the error estimates are affected by a factor that grows exponentially with the advection velocity, and are independent of the reaction rate if this lives in a bounded set. These error estimates are based upon the analyticity of the solution of these equations as a function of the parameters (advection velocity, diffusion, reaction rate). We present several numerical tests, strongly consistent with the theoretical error estimates.Ministerio de Economía y CompetitividadAgence nationale de la rechercheGruppo Nazionale per il Calcolo ScientificoUE ERA-PLANE

Recursive POD expansion for the advection-diffusion-reaction equation

This paper deals with the approximation of advection-diffusion-reaction equation solution by reduced order methods. We use the Recursive POD approximation for multivariate functions introduced in [M. AZAÏEZ, F. BEN BELGACEM, T. CHACÓN REBOLLO, Recursive POD expansion for reactiondiffusion equation, Adv.Model. and Simul. in Eng. Sci. (2016) 3:3. DOI 10.1186/s40323-016-0060-1] and applied to the low tensor representation of the solution of the reaction-diffusion partial differential equation. In this contribution we extend the Recursive POD approximation for multivariate functions with an arbitrary number of parameters, for which we prove general error estimates. The method is used to approximate the solutions of the advection-diffusion-reaction equation. We prove spectral error estimates, in which the spectral convergence rate depends only on the diffusion interval, while the error estimates are affected by a factor that grows exponentially with the advection velocity, and are independent of the reaction rate if this lives in a bounded set. These error estimates are based upon the analyticity of the solution of these equations as a function of the parameters (advection velocity, diffusion, reaction rate). We present several numerical tests, strongly consistent with the theoretical error estimates.Ministerio de Economía y CompetitividadAgence nationale de la rechercheGruppo Nazionale per il Calcolo ScientificoUE ERA-PLANE

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

Un nouveau préconditionneur pour les problèmes elliptiques à coefficients variables

On présente dans cette Note un nouveau préconditionneur pour l’inversion du système algébrique issu de la discrétisation par méthode spectrale d’un problème elliptique du second ordre à coefficients variables et non séparables. Ce préconditionneur est construit en discrétisant un problème similaire à l’original et obtenu par moyenne des coefficients. L’inversion du préconditionneur utilise une méthode directe connue sous le nom de diagonalisation successive

A cure for instabilities due to advection-dominance in POD solution to advection-diffusion-reaction equations

In this paper, we propose to improve the stabilized POD-ROM introduced by S. Rubino in [37] to deal with the numerical simulation of advection-dominated advection-diffusion-reaction equations. In particular, we introduce a stabilizing post-processing strategy that will be very useful when considering very low diffusion coefficients, i.e. in the strongly advection-dominated regime. This strategy is applied both for the offline phase, to produce the snapshots, and the reduced order method to simulate the new solutions. The new process of a posteriori stabilization is detailed in a general framework and applied to advection-diffusion-reaction problems. Numerical studies are performed to discuss the accuracy and performance of the new method in handling strongly advection-dominated cases

Error bounds for POD expansions of parameterized transient temperatures

We focus on the convergence analysis of the POD expansion for the parameterized solutionof transient heat equations. The parameter of interest is the conductivity coefficient. We provethat this expansion converges with exponential accuracy, uniformly if the conductivity coeffi-cient remains within a compact set of positive numbers. This convergence result is independentof the regularity of the temperature with respect to the space and time variables. We presentsome numerical experiments to show that a reduced number of modes allows to represent withhigh accuracy the family of solutions corresponding to parameters that lie in the compact setunder study.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo Regiona

Recursive POD expansion for reaction-diffusion equation

This paper focuses on the low-dimensional representation of multivariate functions. We study a recursive POD representation, based upon the use of the power iterate algorithm to recursively expand the modes retained in the previous step. We obtain general error estimates for the truncated expansion, and prove that the recursive POD representation provides a quasi-optimal approximation in L2 norm. We also prove an exponential rate of convergence, when applied to the solution of the reaction-diffusion partial differential equation. Some relevant numerical experiments show that the recursive POD is computationally more accurate than the Proper Generalized Decomposition for multivariate functions. We also recover the theoretical exponential convergence rate for the solution of the reaction-diffusion equation

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