UMA Home page for Patrick Ciarlet
Research activities
Propagation of fields at the interface between metals or meta-materials and classical dielectric media. In physics, these models allow one to take into account the onset and propagation of surface plasmons. With the help of the T-coercivity theory, one can solve those models mathematically and numerically when they are well-posed. One can also build configurations for which the problem is ill-posed. Current research focuses on the redefinition of the model in this case and on the design of adequate numerical methods.
Computation of electromagnetic fields in a cavity, or near an obstacle, with a non-smooth boundary. When the computational domain is non-convex, the presence of reentrant edges and/or vertices, called geometrical singularities, produces intense fields. Numerically, it can be interesting to use an ad hoc discretization method, such as the singular complement method (SCM), the Fourier singular complement method (FSCM), or the weighted regularization method (WRM). I also study "classical" numerical schemes, based on edge finite elements, and more recently on Staggered Discontinuous Galerkin techniques.
External collaborations: domain decomposition methods to solve the neutron diffusion equation in realistic configurations (Coll. CEA/Saclay) ; discretization techniques for linear elasticity problems, based on a direct approximation of the stress tensor (Coll. CityU Hong Kong and Univ. Zürich) ; design of efficient numerical methods for visco-elastic wave propagation in high-contrast media (Coll. Shell Corp.).
A link to Google Scholar
Computation of electromagnetic fields in a cavity, or near an obstacle, with a non-smooth boundary. When the computational domain is non-convex, the presence of reentrant edges and/or vertices, called geometrical singularities, produces intense fields. Numerically, it can be interesting to use an ad hoc discretization method, such as the singular complement method (SCM), the Fourier singular complement method (FSCM), or the weighted regularization method (WRM). I also study "classical" numerical schemes, based on edge finite elements, and more recently on Staggered Discontinuous Galerkin techniques.
External collaborations: domain decomposition methods to solve the neutron diffusion equation in realistic configurations (Coll. CEA/Saclay) ; discretization techniques for linear elasticity problems, based on a direct approximation of the stress tensor (Coll. CityU Hong Kong and Univ. Zürich) ; design of efficient numerical methods for visco-elastic wave propagation in high-contrast media (Coll. Shell Corp.).
A link to Google Scholar
Teaching activities
Associate Professor, Ecole Polytechnique, 1999-2000 and 2005-2007.
Guest Professor (Graduate Studies), University of Houston, USA, March 2002.
Guest Professor (Graduate Studies), University of Strasbourg, October 2002.
Visiting Professor at the Chinese University of Hong Kong, China, January-July 2010.
Guest Professor (Graduate Studies), Universidade de Santiago de Compostela, Spain, March 2011.
Guest Professor (Graduate Studies), University of Houston, USA, March 2002.
Guest Professor (Graduate Studies), University of Strasbourg, October 2002.
Visiting Professor at the Chinese University of Hong Kong, China, January-July 2010.
Guest Professor (Graduate Studies), Universidade de Santiago de Compostela, Spain, March 2011.
Other activities
Member of the Editorial Boards of "Differential Equations and Applications" (2008-2015), "Computers and Mathematics with Applications" (2012-...) and "ESAIM: Mathematical Modelling and Numerical Analysis" (2013-...).
Publications
Papers in peer-reviewed journals
To appear
- Domain decomposition methods for the diffusion equation with low-regularity solution
, and
Computers & Mathematics with Applications, to appear
2018
- Mesh requirements for the finite element approximation of problems with sign-changing coefficients
, and
Numerische Mathematik, vol. 138, pp. 801--838, 2018 [
]
2017
- Eigenvalue problems with sign-changing coefficients
, and
C. R. Acad. Sci. Paris, Ser. I, vol. 355, pp. 671-675, 2017 - On the Approximation of Electromagnetic Fields by Edge Finite Elements. Part 2: A Heterogeneous Multiscale Method for Maxwell's equations
, and
Computers & Mathematics with Applications, vol. 73(9), pp. 1900-1919, may, 2017
- Theory and implementation of H-matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels
, and
Journal of Computational Physics, vol. 351, pp. 165–186, dec, 2017
2016
- Intrinsic finite element methods for the computation of fluxes for Poisson's equation
, , and
Numer. Math., vol. 132, pp. 433-462, 2016 - On the approximation of electromagnetic fields by edge finite elements. Part 1: Sharp interpolation results for low-regularity fields
Computers Math. Applic., vol. 71, pp. 85-104, 2016 - On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients
, , and
J. Comput. Phys., vol. 322, pp. 224-247, 2016
2014
- Edge element methods for Maxwell's equations with strong convergence for Gauss' laws
, and
SIAM J. Numer. Anal., vol. 52, pp. 779-807, 2014 - Finite Element Heterogeneous Multiscale Method for the Classical Helmholtz Equation
and
C. R. Acad. Sci. Paris, Ser. I, vol. 352, pp. 755-760, 2014 [ ] 
- T-coercivity for the Maxwell problem with sign-changing coefficients
, and
Communications in Partial Differential Equations, vol. 39, pp. 1007-1031, 2014 - Two-dimensional Maxwell's equations with sign-changing coefficients
, and
Applied Numerical Mathematics, vol. 79, pp. 29-41, 2014 [ ]
2013
- Analysis of the Scott-Zhang interpolation in the fractional order Sobolev spaces
J. Numer. Math., vol. 21, pp. 173-180, 2013 - Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwells equations on Cartesian grids
, and
J. Comput. Phys., vol. 235, pp. 14-31, 2013 - Fast non-overlapping Schwarz domain decomposition methods for solving the neutron diffusion equation
and
J. Comput. Phys., vol. 241, pp. 445-463, 2013 - Scalar transmission problems between dielectrics and metamaterials: T-coercivity for the Discontinuous Galerkin approach.
and
J. Comput. Appl. Math., vol. 239, pp. 189-207, 2013 - T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients
and
Numer. Math., vol. 124, pp. 1-19, 2013 [ ]
2012
- T-coercivity for scalar interface problems between dielectrics and metamaterials
, and
Math. Mod. Num. Anal., vol. 46, pp. 1363-1387, 2012 [ ] - T-coercivity: application to the discretization of Helmholtz-like problems
Computers Math. Applic., vol. 64, pp. 2234, 2012
2011
- Compact imbeddings in electromagnetism with interfaces between classical materials and meta-materials
and
SIAM J. Math. Anal., vol. 43(5), pp. 2150-2169, 2011 [ ] - Lagrange multipliers in intrinsic elasticity
, , , and
Math. Models Meth. App. Sci., vol. 21, pp. 651-666, 2011 - Numerical solution of Maxwells equations in axisymmetric domains with the Fourier Singular Complement Method
and
Differential Equations and Applications, vol. 3, pp. 113-155, 2011
2010
- A Lagrangian approach to intrinsic linearized elasticity
, , , and
C. R. Acad. Sci. Paris, Ser. I, vol. 348, pp. 587-592, 2010 - Electrowetting of a 3D drop: numerical modelling with electrostatic vector fields
and
ESAIM: Mathematical Modelling and Numerical Analysis, vol. 44 (4), pp. 647-670, 2010 - Time harmonic wave diffraction problems in materials with sign-shifting coefficients
, and
Journal of Computational and Applied Mathematics, vol. 234(6), pp. 1912-1919, 2010 - Weak vector and scalar potentials. Applications to Poincaré’s theorem and Korn’s inequality in Sobolev spaces with negative exponents.
, and
Analysis and Applications, vol. 8, pp. 1-17, 2010 - Weighted regularization for composite materials in electromagnetism
, , and
ESAIM: Mathematical Modelling and Numerical Analysis, vol. 44, pp. 75-108, 2010
2009
- Direct computation of stresses in planar linearized elasticity
and
Math. Models Meth. App. Sci., vol. 19 (7), pp. 1043-1064, 2009 - Numerical analysis of the generalized Maxwell equations (with an elliptic correction) for charged particle simulations
and
Math. Models Meth. App. Sci., vol. 19 (11), pp. 1959-1994, 2009 - Solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements
, and
Numerische Mathematik., vol. 113 (4), pp. 497-518, 2009
2008
- A characterization of singular electromagnetic fields by an inductive approach
, and
Numerical Analysis and Modeling, vol. 5, pp. 491-515, 2008 - A new approach for approximating linear elasticity problems
and
C. R. Acad. Sci. Paris, Ser. I, vol. 346 (5-6), pp. 351-356, 2008 - A new compactness result for electromagnetic waves. Application to the transmission problem between dielectrics and metamaterials
, and
Math. Models Meth. App. Sci., vol. 18 (9), pp. 1605-1631, 2008 - Computing electromagnetic eigenmodes with continuous Galerkin approximations
and
Comput. Methods Appl. Mech. Engrg., vol. 198 (2), pp. 358-365, 2008
2007
- Characterization of the kernel of the operator CURL CURL
, , and
Comptes Rendus Mathématique, vol. 344 (I), pp. 305-308, 2007 - Continuous Galerkin methods for solving the time-dependent Maxwell equations in 3D geometries
and
J. Comput. Phys., vol. 226 (1), pp. 1122-1135, 2007 - Generalized formulations of Maxwell's equations for numerical Vlasov-Maxwell simulations
, and
Math. Models Meth. App. Sci., vol. 17 (5), pp. 657-680, 2007 - Multiscaled asymptotic expansions for the electric potential: surface charge densities and electric fields at rounded corners
and
Math. Models Meth. App. Sci., vol. 17 (6), pp. 845-876, 2007 - Two-field and three-field formulations for wave transmission between media with opposite sign dielectric constants
, and
Journal of Computational and Applied Mathematics, vol. 204 (2 SPEC), pp. 408-417, 2007 - Vector and scalar potentials, Poincaré's theorem and Korn's inequality
, and
C. R. Acad. Sci. Paris, Ser. I, vol. 345, pp. 603-608, dec, 2007 - Well-posedness of the Drude-Born-Fedorov model for chiral media
and
Math. Models Meth. App. Sci., vol. 17(3), pp. 461-484, 2007
2006
- A justification of Peek's empirical law in electrostatics
et
Comptes Rendus Mathématique, vol. 343 (10), pp. 671-674, 2006 - The Fourier Singular Complement Method for the Poisson problem. Part II: axisymmetric domains
, , , and
Numerische Mathematik, vol. 102, pp. 583-610, 2006 - Time-dependent Maxwell's equations with charges in singular geometries
, , and
Computer Methods in Applied Mechanics and Engineering, vol. 196(1-3), pp. 665-681, 2006
2005
- Another approach to linearized elasticity and a new proof of Korn's inequality
and
Mathematical Models and Methods in Applied Sciences, vol. 15(02), pp. 259-271, 2005 - Augmented formulations for solving Maxwell equations
Computer Methods in Applied Mechanics and Engineering, vol. 194(2-5), pp. 559-586, feb, 2005 - Singular electromagnetic fields: inductive approach
, and
C. R. Acad. Sci. Paris, Ser. I, vol. 341 (10), pp. 605-610, 2005 - The Fourier Singular Complement Method for the Poisson problem. Part I: prismatic domains
, , , and
Numerische Mathematik, vol. 101, pp. 423-450, 2005
2004
- Another approach to linearized elasticity and Korn's inequality
and
Comptes Rendus Mathematique, vol. 339(4), pp. 307-312, 2004 - Solving Maxwell equations in 3D prismatic domains
, and
C. R. Acad. Sci. Paris, Ser. I, vol. 339 (10), pp. 721-726, 2004
2003
- La Méthode du Complément Singulier pour des problèmes scalaires 2d
et
Comptes Rendus Mathematique, vol. 336(4), pp. 353-358, fév 2003 - Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: The Singular Complement Method
, , and
Journal of Computational Physics, vol. 191(1), pp. 147-176, oct, 2003 - Solution of axisymmetric Maxwell equations
, and
Mathematical Methods in the Applied Sciences, vol. 26(10), pp. 861-896, 2003 - Some observations on generalized saddle-point problems
, and
SIAM Journal on Matrix Analysis and Applications, vol. 25(1), pp. 224-236, 2003 - Système de Stokes avec flux de vitesse et pression imposés
Comptes Rendus Mathematique, vol. 337(2), pp. 119-124, 2003
2002
- Condition inf-sup pour l'élément fini de Taylor-Hood P2-iso-P1, 3D; application aux équations de Maxwell
et
Comptes Rendus Mathematique, vol. 335(10), pp. 827-832, 2002 - Theoretical tools to solve the axisymmetric Maxwell equations
, and
Mathematical Methods in the Applied Sciences, vol. 25 (1), pp. 49-78, 2002
2001
- On traces for functional spaces related to Maxwell's equations. Part I: an integration by parts formula in Lipschitz polyhedra
and
Mathematical Methods in the Applied Sciences, vol. 24 (1), pp. 9-30, 2001 - On traces for functional spaces related to Maxwell's equations. Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications
and
Mathematical Methods in the Applied Sciences, vol. 24 (1), pp. 31-48, 2001
2000
- Numerical solution to the time-dependent Maxwell equations in two-dimensional singular domains: the Singular Complement Method
, and
Journal of Computational Physics, vol. 161(1), pp. 218-249, jun, 2000 - Résolution des équations de Maxwell instationnaires avec charges dans un domaine singulier bidimensionnel
, et
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, vol. 330, pp. 391-396, mar 2000 - Un résultat de fermeture pour les équations de Maxwell en géométrie axisymétrique
, et
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, vol. 331(4), pp. 293-298, aoû 2000
1999
- A characterization of the singular part of the solution to Maxwell's equations in a polyhedral domain
, , and
Mathematical Methods in the Applied Sciences, vol. 22, pp. 485-499, 1999 - Caractérisation de la partie singulière et résolution des équations de Maxwell en géométrie singulière axisymétrique
, and
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, vol. 328(9), pp. 767-772, may, 1999 - Fully discrete finite element approaches for time-dependent Maxwell's equations
and
Numerische Mathematik, vol. 82(2), pp. 193-219, apr, 1999
1998
- Les équations de Maxwell dans un polyèdre : un résultat de densité
, et
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, vol. 326(11), pp. 1305-1310, 1998 - Quelques résultats sur la régularité en temps des équations de Maxwell instationnaires
et
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, vol. 327(8), pp. 719-724, oct 1998 - Resolution of the Maxwell equations in a domain with reentrant corners
, and
Modélisation Mathématique et Analyse Numérique, vol. 32(3), pp. 359-389, 1998
1997
- A decomposition of the electric field. Application to the Darwin model
and
Mathematical Models and Methods in Applied Sciences, vol. 7(8), pp. 1085-1120, dec, 1997 - Finite element convergence for the Darwin model to Maxwell's equations
and
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, vol. 31(2), pp. 213-250, 1997 - On the optimality of the median cut spectral bisection graph partitioning method
, and
SIAM Journal on Scientific Computing, vol. 18(3), pp. 943-948, 1997 - Une caractérisation de l'orthogonal de $\Delta (H^2(\Omega)\cap H^1_0(\Omega))$ dans $L^2(\Omega)$
et
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, vol. 325(6), pp. 605-610, 1997
1996
- Does contraction preserve triangular meshes?
and
Numerical Algorithms, vol. 13, pp. 201-223, 1996 - On the validity of a front-oriented approach to partitioning large sparse graphs with a connectivity constraint
and
Numerical Algorithms, vol. 12, pp. 193-214, 1996 - Résolution des équations de Maxwell dans un domaine avec un coin rentrant
, et
C. R. Acad. Sci. Paris, Ser. I, vol. 323, pp. 203-208, 1996
1994
- Implementation of a domain decomposition method well-suited for parallel architectures
High Speed Computing, vol. 6, pp. 157-182, 1994 - Repeated Red-Black ordering: a new approach
Numerical Algorithms, vol. 7, pp. 295-324, 1994
1993
- A decomposition of L2(Ω)3 and an application to magnetostatic equations
Mathematical Models and Methods in Applied Sciences, vol. 3, pp. 289-301, 1993
Books
2010
- La Méthode des Eléments Finis. De la Théorie à la Pratique. II. Compléments
, , et
Coll. Les Cours, Les Presses de l'ENSTA, 288 pages, nov 2010
2009
- La Méthode des Eléments Finis. De la Théorie à la Pratique. I. Concepts Généraux
et
Coll. Les Cours, Les Presses de l'ENSTA, 200 pages, mar 2009
International conferences with proceedings
2017
- Criticality computation with finite element method on non-conforming meshes
, and
Mathematics & Computational Methods Applied to Nuclear Science & Engineering, Jeju, Korea, 2017 - Nonlocal models for interface problems between dielectrics and metamaterials
and
Metamaterials’2017, Marseille, France, 2017
2014
- Domain decomposition for the neutron SPN equations
, , and
DDM'21, Lecture Notes in Computational Science and Engineering, 98, pp. 677-685, Springer, 2014
2013
- Negative materials and corners in electromagnetism
, , , and
Report No.3/2013 of Mathematisches Forschungsinstitut Oberwolfach "Computational Electromagnetism and Acoustics", Oberwolfach., Allemagne, 2013 [ ] - Plasmonic cavity modes: black-hole phenomena captured by Perfectly Matched Layers
, , and
PIERS, Stockholm, aug, 2013 [ ]
2008
- A multiscale approach for solving Maxwell's equations in waveguides with conical inclusions
and
ICCS'08, Lecture Notes in Computer Science, vol. 5102, pp. 331-340, Springer, 2008
2006
- Vlasov-Maxwell simulations in singular geometries
and
ICCS'06, Lecture Notes in Computer Science, vol. 3994, pp. 623-630, Springer, 2006
2005
- Continuous Galerkin methods for solving Maxwell equations in 3D geometries
and
Numerical mathematics and advanced applications (ENUMATH 2005), Santiago de Compostela, Espagne, pp. 547-554, Springer, 2005
2002
- The singular complement method
, , and
13th Conf. on Domain Decomposition Methods, 2000, N. Debit et al, Eds, pp. 161-189, CIMNE, Barcelona, 2002
2000
- Numerical solution to Maxwell equations in singular domains: the singular complement method
, and
Mathematical and numerical aspects of wave propagation (WAVES 2000), Santiago de Compostela, Espagne, pp. 714-718, SIAM, Philadelphia, 2000
1995
- On the influence of the partitioning schemes on the efficiency of overlapping domain decomposition methods
, and
5th Symp. on the frontiers of massively parallel computation (FRONTIERS'95), pp. 375-384, IEEE, 1995
1994
- A comparison of three iterative algorithms based on domain decomposition methods
7th Conf. on Domain Decomposition Methods, 1993, D. E. Keyes and J. Xu, Eds, vol. 180, pp. 387-393, Contemporary Mathematics, 1994
Conferences without proceedings
2015
- Plasmonic waveguide : T-coercivity approach for Maxwell's equations
, and
WAVES, Karlsrhuhe, jul, 2015
2014
- Leaky modes in a non dissipative plasmonic waveguide with a bounded cross section
, , and
OWTNM, Nice, France, jun, 2014 - Revealing guides modes in a plasmonic waveguide using Perfectly Matched Layers at the corners
, , and
KOZWaves, Newcastle, Australia, feb, 2014 [ ]
2013
- Negative materials and corners in electromagnetism
, , , and
Conference on Computational Electromagnetism and Acoustics, Oberwolfach, Allemagne, 2013 [ ] - Plasmonic cavity modes with sign changing permittivity
, , , and
WAVES, Tunisia, jun, 2013 [ ]
2012
- Black holes in metamaterials
, , , and
MATHmONDES 2012, Reading, jul, 2012 - Mathematical analysis of time harmonic Maxwell equations with sign changing coefficients
, and
Journées de Metz 2012, 2012 - On the use of Perfectly Matched Layers to capture strongly confined plasmonic edge waves
, , , and
ETOPIM (Electrical, Transport and Optical Properties of Inhomogeneous Media), Marseille, sep, 2012 - Time harmonic Maxwell equations with sign changing coefficients
, , and
Workshop: around scattering by obstacles and billiards, Aveiro, 2012 [ ] - Transmission eigenvalue problems with sign-changing coefficients
, , and
Picof conference, Palaiseau, 2012 [ ] - Transmission problems with sign-changing coefficients
, and
Congrès de MÉthodes numériques et MOdélisation, Tunis, dec, 2012
2011
- Les problèmes mathématiques et numériques posés par la simulation des interfaces entre diélectriques et métamatériaux
, and
Colloque National Métamatériaux 2011, Orsay, 2011 [ ] - Problèmes de propagation d'ondes dans un milieu diélectrique métamatériau
, and
Congrès d'inauguration de la Fondation Mathématique Jacques Hadamard, Ecole Polytechnique, Palaiseau, 2011 [ ] - Radiation condition for a non-smooth interface between a dielectric and a metamaterial
, , and
Numerical Electromagnetics & Industrial Applications (NELIA 2011), Santiago de Compostela, 2011 [ ] - Scalar waves transmission problems with sign-changing coefficients
, and
ICIAM conference, Vancouver, 2011 [ ]
2010
- Electromagnetic wave propagation at classical/metamaterial interfaces - Mathematical aspects
, and
Workshop : Metamaterials: Applications, Analysis and Modeling, IPAM (Los Angeles), 2010 [ ]
2008
- Etude de formulations éléments finis en présence dune interface entre un diélectrique et un métamatériau
, et
NUMELEC, déc 2008
2007
- Time-harmonic wave transmission problems with sign-shifting material coefficients
, and
8th International Conference on Mathematical and Numerical Aspects of Waves (WAVES'07), Reading, jul, 2007
2005
- A variational approach for wave transmission between media with opposite sign dielectric constants
, and
7th International Conference on Mathematical and Numerical Aspects of Waves (WAVES'05), 2005 - A variational approach for wave transmission between media with opposite sign dielectric constant
, and
7th International Conference on Mathematical and Numerical Aspects of Waves (WAVES'05), Providence, jun, 2005
Habilitation these
1998
- Préconditionnements matriciels et théorie des graphes. Résolution des équations de Maxwell et de modèles approchés
Université Paris 6, 1998 [ ]
Doctoral these
1992
- Etude de préconditionnements parallèles pour la résolution d'EDP. Une décomposition de l'espace L2(Ω)3
Université Paris 6, 1992
Research reports
2015
2009
- Time-domain study of the DrudeBornFedorov model for a class of heterogeneous chiral materials
, and
Rapport de Recherche hal-00430196, 2009
2005
- The Fourier Singular Complement Method for the Poisson problem. Part III: implementation issues
, , , and
Rapport de Recherche SFB393/05-12, Technische Universität Chemnitz, 2005
2003
- The Fourier Singular Complement Method for the Poisson problem in prismatic domains
and
Rapport de Recherche CUHK-2003-04, Chinese University of Hong Kong, 2003
1998
- Caractérisation de la partie singulière de la solution des équations de Maxwell dans un domaine polyédrique
, , et
Rapport de Recherche CEA-N-2826, Bruyères-le-Châtel, 1998
1997
- Développements récents dans le code de résolution des équations de Vlasov-Maxwell en 2D structuré MANET
, et
Rapport de Recherche CEA/DRIF/DCSA/MLS/B3 DO/224/97, Bruyères-le-Châtel, 1997
1994
- Recursive partitioning methods and greedy partitioning methods: a comparison on finite element graphs
and
Rapport de Recherche CAM 94-09, UCLA, 1994
Lecture notes
2016
- Calcul scientifique parallèle. Rappels de calcul scientifique
et
Cours du M2 Analyse, Modélisation et Simulation, 165 pages, 2016
2011
- Notes de cours sur les équations de Maxwell
Cours de 3ème année, 82 pages, ENSTA et M2 Modélisation & Simulation, 2011
2010
- Lecture Notes on Computational and Applied Mathematics
and
Cours de 2ème année, 199 pages, Chinese University of Hong Kong, 2010
2007
- Optimisation quadratique
et
Cours de 1ère année, 109 pages, ENSTA, 2007
2005
- Introduction au calcul scientifique. Aspects algorithmiques
Cours de 2ème année, 120 pages, ENSTA, 2005
2002
- Modèles et méthodes pour les équations de Maxwell
et
Cours de 3ème année, 160 pages, ENSTA, 2002 - Optimisation et algèbre linéaire
et
Cours de 1ère année, 237 pages, ENSTA, 2002