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

A FETI method with a mesh independent condition number for the iteration matrix

We introduce a framework for FETI methods using ideas from the decomposition via Lagrange multipliers of H1 0 (Ω) derived by Raviart-Thomas [22] P.-A. Raviart, J.-M. Thomas, Primal Hybrid Finite Element Metho and complemented with the detailed work on polygonal domains developed by Grisvard [17] P. Grisvard, Singularities in Boundary value problems. Recherches en Mathématiques Appliquées, 22. Masson, 1992.. We compute the action of the Lagrange multipliers using the natural H 1/2 00 scalar product, therefore no consistency error appears. As a byproduct, we obtain that the condition number for the iteration matrix is independent of the mesh size and there is no need for preconditioning. This result improves the standard asymptotic bound for this condition number shown by Mandel-Tezaur in [19] J. Mandel, R. Tezaur, Convergence of a substructuring method with Lagrange multipliers. Numer. Math., 73 (1996), 473–487. Numerical results that confirm our theoretical analysis are presented.Nous proposons une nouvelle approche des méthodes FETI: la décomposition de domaine fait appel aux multiplicateurs de Lagrange tels qu’introduits par Raviart-Thomas [22] P.-A. Raviart, J.-M. Thomas, Primal Hybrid Finite Element Methods for second order eliptic equations. Math. Comp., 31 (1977), 391-413 et au traitement des domaines polygonaux dù á Grisvard [17] P. Grisvard, Singularities in Boundary value problems. Recherches en Mathématiques Appliquées, 22. Masson, 1992. Ces multiplicateurs utilisent le produit scalaire de H 1/2 00, de sorte qu’aucune erreur de consistance n’apparaît. En outre, nous prouvons que le nombre de condition de la matrice liée à chaque itération est indépendant de la taille du maillage, ce qui améliore le résultat de Mandel-Tezaur [19] J. Mandel, R. Tezaur, Convergence of a substructuring method with Lagrange multipliers. Numer. Math., 73 (1996), 473–487; par suite, aucun préconditionnement n’est nécessaire. Nous présentons des expériences numériques qui confirment notre analyse.Ministerio de Educación y Cienci

the WAF method for non-homogeneous SWE with pollutant

This paper deals with the extension of the WAF method to discretize Shallow Water Equations with pollutants. We consider two different versions of the WAF method, by approximating the intermediate waves using the flux of HLL or the direct approach of HLLC solver. It is seen that both versions can be written under the same form with different definitions for the approximation of the velocity waves. We also propose an extension of the method to non-homogeneous systems. In the case of homogeneous systems it is seen that we can rewrite the third component of the numerical flux in terms of an intermediate wave speed approximation. We conclude that – in order to have the same relation for non-homogeneous systems – the approximation of the intermediate wave speed must be modified. The proposed extension of the WAF method preserves all stationary solutions, up to second order accuracy, and water at rest in an exact way, even with arbitrary pollutant concentration. Finally, we perform several numerical tests, by comparing it with HLLC solver, reference solutions and analytical solutions

The Effort of Increasing Reynolds Number in Projection-Based Reduced Order Methods: From Laminar to Turbulent Flows

We present in this double contribution two different reduced order strategies for incompressible parameterized Navier-Stokes equations characterized by varying Reynolds numbers. The first strategy deals with low Reynolds number (laminar flow) and is based on a stabilized finite element method during the offline stage followed by a Galerkin projection on reduced basis spaces generated by a greedy algorithm. The second methodology is based on a full order finite volume discretization. The latter methodology will be used for flows with moderate to high Reynolds number characterized by turbulent patterns. For the treatment of the mentioned turbulent flows at the reduced order level, a new POD-Galerkin approach is proposed. The new approach takes into consideration the contribution of the eddy viscosity also during the online stage and is based on the use of interpolation. The two methodologies are tested on classic benchmark test cases

Finite element pressure stabilizations for incompressible flow problems

Discretizations of incompressible flow problems with pairs of finite element spaces that do not satisfy a discrete inf-sup condition require a so-called pressure stabilization. This paper gives an overview and systematic assessment of stabilized methods, including the respective error analysis

Nurses' perceptions of aids and obstacles to the provision of optimal end of life care in ICU

Contains fulltext : 172380.pdf (publisher's version ) (Open Access

Numerical investigation of algebraic oceanic turbulent mixing-layer models

In this paper we investigate the finite-time and asymptotic behaviour of algebraic turbulent mixing-layer models by numerical simulation. We compare the performances given by three different settings of the eddy viscosity. We consider Richardson number-based vertical eddy viscosity models. Two of these are classical algebraic turbulence models usually used in numerical simulations of global oceanic circulation, i.e. the Pacanowski–Philander and the Gent models, while the other one is a more recent model (Bennis et al., 2010) proposed to prevent numerical instabilities generated by physically unstable configurations. The numerical schemes are based on the standard finite element method. We perform some numerical tests for relatively large deviations of realistic initial conditions provided by the Tropical Atmosphere Ocean (TAO) array. These initial conditions correspond to states close to mixing-layer profiles, measured on the Equatorial Pacific region called the West-Pacific Warm Pool. We conclude that mixing-layer profiles could be considered as kinds of "absorbing configurations" in finite time that asymptotically evolve to steady states under the application of negative surface energy fluxes