Pascal Azerad Home Page

Maître de conférences, HdR.
Institut de Modélisation Mathématique de Montpellier UMR CNRS 5149

Research field: numerical and mathematical analysis of PDE's in fluid dynamics

My PhD dealt with Navier-Stokes equations in thin domains, namely geophysical fluids where the hydrostatic approximation (pression linear along the vertical axis) is in current use, for which I have designed a new mixed finite element, which is stable (see 1). I also proved an existence and convergence theorem for the linearized time-dependent hydrostatic approximation of the Navier-Stokes equations (see 6).

7,8,10

I also studied the transport equation, and developed a new variational formulation, useful both theoretically and numerically. It is based on a space-time least squares formulation of the advection equation, which turns it into an anisotropic diffusion equation (see 2,4). To recover the coerciveness, one must work in suitable spaces, and prove a Poincaré-like inequality (see 3,21). This method leads naturally to a space-time finite element discretization which enjoys very good stability properties at any Courant number for pure advection problems (see 2,4).

In collaboration with Eberhard Baensch (Erlangen-Nuremberg), I studied the hydrodynamic features of specific devices used in hematology-haemostasis (see 11).

Currently, I am working on the numerical simulation of coastal erosion, in the framework of the ANR projects COPTER (Conception Optimisation et Prototypage d'ouvrages de lutte contre l'ERosion en domaine littoral 2006-2008), MATHOCEAN (2009-2012) and OILDEBEACH (2009-2011). With Bijan Mohammadi, I supervised the Ph. D. thesis of Damien Isebe. (see 12,14,16,17,18,20 ). Damien is now research engineer at HORIBA-ABX, Montpellier

One year POSTDOC POSITION offered in the COPTER project. Start January 2007. Position appointed to Dr Borys Alvarez-Samaniego, extended until july 2008. Later on Borys was research fellow at McMaster University, Ontario.

I supervised with B. Mohammadi the Ph. D. thesis of Afaf Bouharguane, start fall 2008, successfully defended on june 20th, 2011. After a post-doc in Lab. Jean Kuntzmann, University Joseph Fourier, Grenoble, Afaf has now a permanent position of Maitre de conferences in Institut de Mathematiques de Bordeaux, University Bordeaux 1.

A recent topic which I started to investigate is PDE with non local terms, i.e. defined by integrals or in Fourier space. I started to look at the Fowler equation for dune morphodynamics (see 19, 20, 24 ). In a joint work with Mohamed Mellouk, I also studied a stochastic pde (fractional heat equation) (see 13 ). We also applied fractional calculus to signal processing (see 23, 24 ).

List of publications

P.A., Analyse et approximation du problème de Stokes en bassin peu profond, C.R. Acad. Sci. Paris, T. 318, serie I, pp.53-58, 1994
P. Perrochet and P. A., Space-Time Integrated Least-Squares: solving a pure advection with a pure diffusion operator, J. Comput. Phys., vol. 117, no 2, pp 183-193, 1995
P.A. and J. Pousin, Inégalité de Poincaré courbe pour le traitement variationnel de l'équation de transport, C.R. Acad. Sci. Paris, T. 322, serie I, pp.721-727, 1996.
P.A., P. Perrochet and J. Pousin, Space-Time Integrated Least-Squares: a simple, stable and precise finite element scheme to solve advection equations as if they were elliptic, Progress in partial differential equations: the Metz surveys 4, Pitman Research Notes in Mathematics Series 345, M Chipot and I Shafrir Eds, pp 161-174, Longman, 1996.
P.A., O.Besson, Numerical simulation of fluid flow in a stratified shallow lake Proceedings of the IXth International Conference on Finite Elements in Fluids, M. Morandi Cecchi, K. Morgan, J.Périaux, B.A. Schrefler, O. C. Zienkiewicz (Eds), pp. 1477-1486, Venice, 1995.
P.A., Analyse des équations de Navier-Stokes en bassin peu profond et de l'équation de transport, Thèse de doctorat ès sciences, Neuchâtel, 1996.
PostScript zipped file, 340 k
PDF file 1M
P.A., O.Besson and F. Guillén, Fluid flow in shallow domains: mathematical analysis and numerical simulation, Proceedings of the IVth Catalan Days of Applied Mathematics, C. Garcia, C. Olivé , M. Sanroma Eds, pp. 9-16, Universitat Rovira i Virgili, Tarragona, 1998.
P.A., and F. Guillén, Equations de Navier Stokes en bassin peu profond : l'approximation hydrostatique C.R. Acad. Sci. Paris, T. 329, serie I, pp.961-966, 1999.
P.A, Mathematical Analysis and Finite Element stategy for 3D numerical simulation of Navier-Stokes equations in thin domains, Proceedings of ECCOMAS, 8 p, Barcelona , 2000.
P.A. and F. Guillén, Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics, SIAM J. Math. Anal., Vol. 33, No. 4, pp. 847-859, 2001
P.A. and E. Baensch, Quasi-stability of the primary flow in a cone and plate viscometer, J. Math. Fluid Mech., 6, pp. 253-271, 2004.
P.A., D. Isebe, B. Ivorra, B. Mohammadi and F. Bouchette, Optimal shape design of coastal structures minimizing coastal erosion, Proceedings of workshop on inverse problems, CIRM Marseille, 2005.
P.A. and M. Mellouk, On a stochastic partial differential equation with non-local diffusion, Potential Analysis, 2007, Vol. 27, pp. 183-197, 2007.
D. Isebe, P.A., B. Mohammadi and F. Bouchette, Design of Passive Defense Structures in Coastal Engineering , Int. Rev. Mech. Eng.,1, 2007.
P.A., Contributions a l'etude de quelques equations aux derivees partielles, en mecanique des fluides et en genie cotier, HdR, 2007.
D. Isebe, P.A, B. Mohammadi, F. Bouchette, Optimal shape design of coastal structures minimizing water waves impact, Coastal Engineering, 2008, Vol. 55, pp. 35-46.
D. Isebe, P.A, F. Bouchette, B. Ivorra, B. Mohammadi, Shape optimization of geotextiles tubes for sandy beach protection, International Journal for Numerical Methods in Engineering, 2008, Vol. 74, pp. 1262-1277.
D. Isebe, B. Ivorra, P. A., B. Mohammadi and F. Bouchette, Progress in Global Optimization and shape design,Modeling, Simulation and Optimization of Complex Processes, Proceedings of the Third International Conference on High Performance Scientific Computing, Hanoi, Vietnam, H.G. Bock, E. Kostina, H.X. Phu and R. Rannacher, Eds, pp. 303-312, Springer, 2008
B. Alvarez-Samaniego, P. A., Existence of travelling wave solutions and local well-posedness of the Fowler equation, Disc. Cont. Dyn. Syst., Ser. B, 12, pp. 671-692, 2009.
N. Alibaud, P. A, D. Isebe, A non-monotone conservation law for dune morphodynamics, Differential and Integral Equations, 23, pp. 155-188, 2010.
A. Bouharguane, P.A., F. Bouchette, F.Marche and B. Mohammadi, Low complexity shape optimization and a posteriori high fidelity validation, Disc. Cont. Dyn. Syst., Ser. B, 13, pp. 759-772, 2010.
P. A., S. Brull , Inegalites de Poincare cinetiques Comptes Rendus Mathematique Volume 349, Issues 13-14, pp. 759-763, 2011
P. A., A. Bouharguane and J.-F. Crouzet, Simultaneous denoising and enhancement of signals by a fractal conservation law Communications in Nonlinear Science and Numerical Simulation Volume 17, Issue 2, pp. 867-881, 2012
P.A., A. Bouharguane, Finite difference approximations for a fractional diffusion/anti-diffusion equation preprint, 2011.

Curriculum Vitae

Some talks

Journee calcul scientifique et Modelisation mathematique, Amiens, june 2012
Seminaire de Calcul scientifique, Besancon, jan. 2012
Recent trends in differential equations, Bielefeld, nov. 2010 Analysis and discretisations methods.
projet COPTER le film! pilote réalisé par Zebra 3 Prod. Warning :fichier .mov 1 Go.
projet COPTER le documentaire en HD! pilote réalisé par Zebra 3 Prod.
Le projet COPTER, P. A, F. Bouchette et B. Mohammadi, Colloque Bilan ANR Blanc 2009, Paris, Cite des Sciences, La Villette. Warning :fichier ppt, 26 Mo.
Optimisation de forme, Calcul scientifique industriel et Analyse pour la protection du littoral, P. A, Journee Bordeaux-Montpellier-Pau-Toulouse, 2008, Toulouse.

Teaching activity

site du cours Mathematiques du web

Lecons de Mecanique des fluides. 23 p, 350 Ko PDF
7 lecons sur les Martingales. 33 p, 350 Ko PostScript PDF
Quelques notions d'analyse numérique. 27 p, 324 Ko PDF
Compléments pour l'épreuve de modélisation à l'agrégation de mathématiques, option probabilités-statistiques. 59 p, 780 Ko PDF
Some matlab scripts for FEM course (from femlab-1d) femlab-1d.tar.gz file 8 Ko
website of the graduate course on scientific computing