Scientific Computing
Org: Christine Bernardi (CNRSParis VI), Anne Bourlioux (Montréal) and Brian Wetton (UBC) [PDF]
 MOHAMMED AMARA, IPRALMA UMR5142, Université de Pau, 64013 Pau Cedex, France
Mixed finite element approximation of a coupled reservoirwellbore model with heat transfer
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In order to interpret recorded temperatures in wellbores as well as a flowrate history at the surface and thus to better characterize reservoirs, we are interested in the thermomechanical coupling of a petroleum reservoir with a vertical wellbore, both written in 2D axisymmetric form.
The reservoir model, assumed to be a monophasic multilayered porous medium, is described by a DarcyForchheimer equation together with a nonstandard energy balance. The wellbore model is based on the compressible NavierStokes equations and an energy equation. The coupling between these two models is achieved by adequate transmission conditions at the perforations.
We obtain, at each time step, a mixed formulation and the uniqueness of the solution is established by means of a generalization of the BabuskaBrezzi theorem.
Concerning the spatial discretization, we approximate the heat and mass fluxes by the lowestorder RaviartThomas mixed elements, the pressure and the temperature by piecewise constant elements, the fluid's velocity by Q_{1} continuous elements while the Lagrange multipliers on the interface are taken piecewise constant. The density is updated by means of a thermodynamic module and the convective terms are treated by appropriated upwind schemes. A technical analysis of the discrete mixed formulation is carried on and the wellposedness of the problem is proved.
Numerical tests including real cases will be presented.
 FAKER BEN BELGACEM, Université de Technologie de Compiègne, Centre de Recherches de Royallieu, BP 20529, 60205 Compiègne Cedex, France
On the Lavrentiev Regularization of the Illposed Data Completion Problem
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The Lavrentiev regularization method is naturally fitted to the data completion problem, put under the variational form proposed in Ben Belgacem and El Fekih (Inverse Problems, 2005). We address the important issue of selecting the regularizing parameter. We study an a priori choice and the a posteriori choice by means of the discrepancy principle written on the KohnVogelius function. In both cases, we prove that the Lavrentiev method, though its simplicity, is superconvergent as we state estimates similar to those expected for Tikhonov's method.
 ZAKARIA BELHACHMI, Université Paul VerlaineMetz
Computing NeumannLaplacian eigenvalues in nonsmooth domains and applications
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The study of shape optimization problems involving the eigenvalues of an elliptic operator (the LaplacianD for instance) has strong relations with several applications (stability of vibrating bodies, ...). When Dirichlet boundary conditions are imposed on the free boundary ¶W, there are several situations where solving the optimization problem is made easier, in both theoretical and computational aspects, by the existence of a relaxation process. The same problems for the Laplace operator with Neumann conditions on the free boundary is a difficult subject where up to now very few results are available. In fact, proving existence of solutions in this case is related to the (deep) understanding of the behavior of the Neumann spectrum on nonsmooth domains (highly oscillating boundaries, cracks, etc). This is a challenging question and it is largely open.
In this talk, we consider the NeumannLaplacian eigenvalue problem in domains with multiple cracks. We derive a mixed variational formulation which holds on the whole geometric domain (including the cracks) and implement efficient finite element discretizations for the computation of eigenvalues. Optimal error estimates are given and several numerical examples are presented, confirming the efficiency of the method. As applications, we numerically investigate the behavior of the low eigenvalues of the NeumannLaplacian in domains with a high number of cracks. For particular cases of highly oscillating boundaries, these computations allows us to identify formally the problems with the limit spectra.
 ADEL BLOUZA, Université de Rouen
Finite element approximations of Naghdi's shell model in cartesian coordinates
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We present a mixed formulation of Naghdi's model for linearly elastic shells with little regularity, and a conforming finite element approximation thereof. The a posteriori analysis of the discrete problem leads to the construction of error indicators which satisfy optimal estimates. We then describe a mesh adaptativity strategy based on these indicators. We also use them to optimize the choice of the penalty parameter of our discretization. Numerical tests are given that validate and illustrate our approach.
 ANDRÉ FORTIN, Université Laval
An adaptive remeshing method for free surface viscoelastic fluid flow problems
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In this work, we describe an adaptive remeshing method which can be applied to the computation of viscoelastic fluid flow problems involving free surfaces. We first introduce a logconformation formulation of the constitutive equation and a level set method for the computation of free surfaces.
The methodology is then applied to the deformation of droplets in shear flow and to put in evidence the importance of secondary flows in coextrusion problems.
 FRÉDÉRIC HECHT, UPMC Univ. Paris 06, UMR 7598, Laboratoire JacquesLouis Lions, F75005, Paris, France
Some mesh adaptation schemes for solving PDE with high order finite element method
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Aujourd'hui il existe des générateurs de maillage triangulaire
(d=2) ou tétraédrique (d=3) qui utilisent des métriques pour prescrire les tailles de maille. Ces métriques M sont données comme des champs de matrice symétrique définie positive en tout point de l'espace. Ces champs dans le cas d'une interpolation élément fini P_{1} Lagrange, ces métriques dérivent directement du Hessien de la solution à approcher.
L'idée pour généraliser cette construction est de définir une représentation infinitésimale de l'erreur d'une fonction u donnée. Pour une approximation d'ordre k+1 (polynôme de dégre k), nous utiliserons simplement le reste de la formule de Taylor à l'ordre k+1, c'est à dire la dérivée D^{k+1} u (une forme k+1 linéaire) à l'ordre k+1 en négligant les termes d'ordre supérieur.
Alors nous construirons une métrique telle que:

1
(k+1)!

D^{k+1} u (x,...,x) £ (x, Mx), "x Î R^{d}, x = 1. 

Nous proposerons des schémas numériques pour construire des métriques respectant cette inégalité. Puis nous expliquerons comment construire des champs de métrique quasioptimaux asymptotiquement, pour construire un maillage
 avec un nombre de maille minimal pour une erreur donnée,
 avec un nombre de maille donné qui minimise l'erreur;
où l'erreur est l'erreur d'interpolation en norme L^{p} ou W^{1,p}.
Pour finir, nous présenterons le logiciel FreeFem++ pour illustrer ces propos.
 NICHOLAS KEVLAHAN, McMaster University, Hamilton, ON L8S 4K1
Stochastic Coherent Adaptive Large Eddy Simulation
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Stochastic Coherent Adaptive Large Eddy Simulation (SCALES) is an extension of Large Eddy Simulation that uses a waveletbased dynamic grid adaptation strategy to solve for the most energetic coherent structures in a turbulent flow field, while modelling the effect of the less energetic ones. A localized dynamic subgrid scale model is needed to fully exploit the ability of the method to track coherent structures. In this paper, new local Lagrangian models based on a modified Germano dynamic procedure, redefined in terms of wavelet thresholding filters, are proposed. These models extend the original pathline formulation of Meneveau et al. (J. Fluid Mech. 319, 1996) in two ways: as Lagrangian pathline diffusive and Lagrangian pathtube averaging procedures. The proposed models are tested for freely decaying homogeneous turbulence with initial Re_{l} = 72. It is shown that the SCALES results, obtained with fewer than 0.4% of the total nonadaptive nodes required for a DNS with the same wavelet solver, closely match reference DNS data. In contrast to classical LES, this agreement holds not only for large scale global statistical quantities, but also for energy and, more importantly, enstrophy spectra up to the dissipative wavenumber range.
 BOUALEM KHOUIDER, University of Victoria
Convectively coupled waves in a simple multicloud model
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Organized convection in the tropics involves a hierarchy of spacial and temporal scales ranging from the individual clouds of a few kilometres and a few hours to the mesoscale cloud clusters and superclusters and their intraseasonal/planetary scale wave envelopes, known as the MaddenJulian oscillation (Nakazawa, 1988). Interactions between large scale dynamics (and thermodynamics) and small scale convective processes are believed to play a crucial role in the generation and maintenance of these organized features. The analysis of Wheeler and Kiladis (1999) and Heartel and Kiladis (2004) showed that the dynamical fields and the power spectral peaks of the synoptic scale superclustersidentified as the convectively coupled waves, are closely related to the first few equatoriallytrapped linear waves of Matsuno (1966), but with a significantly reduced equivalent depth (i.e., phase speed). Despite recent research efforts and significant progress in computing power, current general circulation models (GCM), used for longterm weather and climate predictions, often simulate poorly the dynamics and structure of convectively coupled waves, and particularly the MJO, due to an inadequate treatment of organized convection by the convective parametrizations currently used by the GCMs (Moncrieff, 2004). To study these interesting waves, from both a practical and a theoretical perspective, we designed and used an idealized multicloud model that captures many of the physical and dynamical features attributed to the large scale organized convective systems.
Joint work with Andrew J. Majda.
 EMMANUEL LORIN, University of Ontario I.T.
Numerical Modeling of Intense and High Frequency LaserGas Interactions
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In this communication we are interested in mathematical, numerical and modeling tools for ultrashort, high intensity and high frequency lasermatter interactions in dense gaseous media. These last 15 years, in this extreme regime, several new phenomena have been experimentally discovered (High Harmonic Generation, Filamentation, ATI), and will likely be very useful for practical applications in laser technology (virtual antennas, atmospheric applications, communications, dynamic imaging, etc). We have recently presented one of the first multiscale models and a corresponding numerical 3D scheme for lasergas interaction in this regime. The model involves the coupling of macroscopic nonhomogeneous Maxwell's equations with TimeDependent Schroedinger Equations (TDSE's). Indeed, in order to consider high order nonlinearities, the polarization, response of the gas to the electromagnetic field, is computed using quantumlevel lasermolecule TDSE's; in contrast, perturbative nonlinear expansions are used in classical nonlinear Schroedinger models that do not model precisely important small scale phenomena and ionization. The complexity of the model requires the accurate computation in parallel of Maxwell's equations and thousands of 3D TDSE's.
In this contribution after a discussion on the mathematical modeling, we will present some recent numerical and mathematical results that involve in particular artificial boundary conditions for lasermolecule TDSE's and domain decomposition techniques for these multiscale MaxwellSchroedinger equations. Coupling macroscopic Maxwell's equations with thousands of TDSE's necessitates the use of HighPerformance Computers such as mammouth (http://ccs.usherbrooke.ca). Typical realistic simulations that will be presented with the actual code require several days on hundreds (up to 512) of processors and hundreds of GBytes of RAM.
 BERTRAND MAURY, Laboratoire de Mathématiques, Université ParisSud, 91405 Orsay Cedex
Direct simulation of dense suspensions
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The modelling fluid particle flows calls for efficient solvers of Poissonlike problems on domains with (possibly many) holes. After a description of the native difficulties of the problem, we will give an overview of the several methods which have been proposed to address this challenge, and present some of them, paying a special attention to numerical efficiency, conditionning aspects, and accuracy.
Among the methods we plan to present, let us mention the direct approach, based on a boundary fitted (and therefore unstructured) mesh, and some fictitious domain methods (based on a global mesh which covers the whole region of interest), like the Penalty Method, the Fat Boundary method, and saddlepoint approaches.
 BIJAN MOHAMMADI, Univ. Montpellier II, I3M  CC51  34095 Montpellier, France
Involutive completion for constrained PDE systems
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We propose to use the involutive form of the system of PDEs in numerical computations.
We illustrate our approach by applying it to Stokes system. As in case of the solution of differential algebraic equations our approach takes explicitly into account the integrability conditions of the system which are only implicitly present in the original formulation. The extra calculation cost is negligible while the discrete form becomes much simpler to handle. One interesting consequence is that the discretization need not verify the classical LBB stability condition.
The approach is very general and can be useful for a wide variety of systems not as well known as fluid flow equations.
The application of the approach to various PDE systems under constraint will be presented together with numerical examples.
 SERGE NICAISE, Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, ISTV, F59313 Valenciennes Cedex 9, France
Adaptive finite element methods: abstract framework and applications
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We consider a general abstract framework of a continuous elliptic problem set on a Hilbert space V that is approximated by a family of (discrete) problems set on a finitedimensional space of finite dimension not necessarily included into V. We give a series of realistic conditions on an error estimator that allows to conclude that the marking strategy of bulk type leads to the geometric converge of the adaptive algorithm. These conditions are then verified for different concrete problems like convectionreactiondiffusion problems approximated by conforming P_{1} finite elements or by a discontinuous Galerkin method with an estimator of residual type or obtained by equilibrated fluxes. Numerical tests that confirm the geometric converge will be presented.
 NILIMA NIGAM, McGill University, 805 Sherbrooke West, Montreal
A multigrid preconditioner for an integral equation of the first kind for acoustics
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We present a multigrid strategy for an integral equation of the first kind arising in acoustics. Such integral equations arise naturally, for example, in the study of screen scattering problems. Unfortunately, the integral operator involved has a spectrum whose behaviour is not directly conducive to multigrid strategies. We present an extension of ideas to circumvent this problem in the positivedefinite case due to Bramble et al., and demonstrate how their use leads to a convergent method. We also present some numerical experiments
 STEVE RUUTH, Simon Fraser University, Burnaby, BC
A Simple Technique for Solving Partial Differential Equations on Surfaces
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Many applications require the solution of timedependent partial differential equations (PDEs) on surfaces or more general manifolds.
Methods for treating such problems include surface parameterization,
methods on triangulated surfaces and embedding techniques. This talk
describes an embedding approach based on the closest point representation of the surface and describes some of its advantages over other embedding methods. Noteworthy features of the method are its generality with respect to the underlying surface and its simplicity.
In particular, the method requires only minimal changes to the corresponding threedimensional codes to treat the evolution of PDEs on surfaces.
 DOMINIK SCHOETZAU, University of British Columbia
Exactly divergencefree discontinuous Galerkin methods for incompressible fluid flow
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We present a class of discontinuous Galerkin methods for the incompressible NavierStokes equations yielding exactly divergencefree velocity approximations. Exact incompressibility is achieved by using divergenceconforming velocity spaces for the approximation of the velocities. The resulting methods are also locally conservative and energystable. We discuss the numerical analysis of the methods and illustrate their practical performance in a series of numerical experiments.
 JOHN STOCKIE, Simon Fraser University, Department of Mathematics, 8888 University Drive, Burnaby, BC V5A 1S6
A new approach for simulating flow through porous immersed boundaries
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Porous, deformable boundaries are encountered in a wide range of applications including cell membranes, vesicles, porous wave makers, and parachutes. The immersed boundary method has already proven to be a versatile and robust approach for simulating the interaction of an impermeable, elastic structure with an incompressible fluid flow. We show how to extend the method to handle porous boundaries by incorporating an explicit porous slip velocity through use of Darcy's law.
We derive a simple, radiallysymmetric exact solution, which is then used to validate numerical simulations of porous membranes in two dimensions.
