Publications

2009

• Couplage des méthodes modale et éléments finis pour la diffraction des ondes élastiques guidées : Application au Contrôle Non Destructif
  
  • Baronian Vahan
  , 2009. En vue de simuler une expérience de contrôle non destructif par ondes ultrasonores guidées, on considère un guide élastique 2D (une plaque) ou 3D (une barre) qui comporte un défaut (fissure, hétérogénéité locale due à une soudure etc...). L'objectif est de résoudre numériquement le problème de la diffraction d'un mode du guide par le défaut. Nous nous sommes attachés à mettre au point une méthode couplant des éléments finis dans une portion (aussi petite que possible) du guide, contenant le défaut, avec des décompositions modales de part et d'autre du défaut. La difficulté consiste à écrire la bonne condition de raccord entre ces deux représentations. Le point important est d'avoir à sa disposition une relation d'orthogonalité permettant de projeter la solution éléments finis sur les modes. Ceci conduit à formuler le problème à l'aide de vecteurs hybrides déplacement/contrainte pour lesquels il existe une relation de bi-orthogonalité : la relation dite de Fraser. On peut alors écrire une condition exacte (ou transparente) à la troncature modale près, sur les frontières artificielles du domaine de calcul. Il faut enfin intégrer cette condition aux limites dans une approche variationnelle (en déplacements) en vue de développer une méthode d'éléments finis. Du fait du caractère hybride de la condition, on doit pour cela introduire comme inconnue supplémentaire la composante normale de la contrainte normale définie sur la frontière artificielle et écrire une formulation mixte. Nous avons traité numériquement les cas bidimensionnel et tridimensionnel d'un guide isotrope à bords libres. Les modes du guide sont calculés numériquement par une approche originale utilisant à nouveau les vecteurs hybrides déplacement/contrainte, qui permet de conserver au niveau discret la relation de biorthogonalité. Le code développé permet de calculer très rapidement la "matrice de scattering
• Schémas numériques pour la résolution de l’équation des ondes
  
  • Bécache Eliane
  , 2009. L’objectif de ce cours est d’appréhender les problèmes de propagation d’ondes, de les étudier sur le plan mathématique et de proposer et d’analyser des méthodes numériques pour les résoudre. Cette partie du cours est plus spécifiquement consacrée aux méthodes numériques et nous renvoyons au cours de P. Joly pour ce qui concerne l’étude mathématique des équations continues.
• Optimized higher order time discretization of second order hyperbolic problems: Construction and numerical study
  
  • Joly Patrick
  • Rodríguez Jerónimo
  Journal of Computational and Applied Mathematics, Elsevier, 2009, 234 (6), pp.1953-1961. We investigate explicit higher order time discretizations of linear second order hyperbolic problems. We study the even order (2m2m) schemes obtained by the modified equation method. We show that the corresponding CFL upper bound for the time step remains bounded when the order of the scheme increases. We propose variants of these schemes constructed to optimize the CFL condition. The corresponding optimization problem is analyzed in detail. The optimal schemes are validated through various numerical results. (10.1016/j.cam.2009.08.046)
  DOI : 10.1016/j.cam.2009.08.046
• Improved Successive Constraint Method Based A Posteriori Error Estimate for Reduced Basis Approximation of 2D Maxwells Problem
  
  • Chen Yanlai
  • Hesthaven Jan Sickmann
  • Maday Yvon
  • Rodríguez Jerónimo
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2009, 43 (6), pp.1099--1116. In a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations, the construction of lower bounds for the coercivity and inf-sup stability constants is essential. In [7], the authors presented an efficient method, compatible with an off-line/on-line strategy, where the on-line computation is reduced to minimizing a linear functional under a few linear constraints. These constraints depend on nested sets of parameters obtained iteratively using a greedy algorithm. We improve here this method so that it becomes more efficient due to a nice property, namely, that the computed lower bound is monotonically increasing with respect to the size of the nested sets. This improved evaluation of the inf-sup constant is then used to consider a reduced basis approximation of a parameter dependent electromagnetic cavity problem both for the greedy construction of the elements of the basis and the subsequent validation of the reduced basis approximation. The problem we consider has resonance features for some choices of the parameters that are well captured by the methodology.
• On the theoretical justification of Pocklington's equation
  
  • Claeys Xavier
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2009, 19 (8), pp.1325-1355. Pocklington's model consists in a one-dimensional integral equation relating the current at the surface of a straight finite wire to the tangential trace of an incident electromagnetic field. It is a simplification of the more usual single layer potential equation posed on a two-dimensional surface. We are interested in estimating the error between the solution of the exact integral equation and the solution of Pocklington's model. We address this problem for the model case of acoustics in a smooth geometry using results of asymptotic analysis. (10.1142/S0218202509003802)
  DOI : 10.1142/S0218202509003802
• Energy preserving scheme for non linear systems of wave equations. Application to piano strings.
  
  • Chabassier Juliette
  • Joly Patrick
  , 2009, pp.00. The linear wave equation does not describe the com- plexity of the piano strings vibration enough for physics based sound synthesis. The nonlinear cou- pling between transversal and longitudinal modes has to be taken into account, as does the "geometrically exact" model. This system of equations can be clas- sified among a general energy preserving class of sys- tems. We present an implicit, centered, second order accurate, numerical scheme that preserves a discrete energy, leading to unconditional stability of the nu- merical scheme. The complete model takes into ac- count the bridge coupling the strings, and the ham- mer non linear attack on the strings.
• Non Spurious Mixed Spectral Element Methods for Maxwell's Equations
  
  • Cohen Gary
  • Sinding Alexandre
  , 2009. In this paper, we describe a new continuous approximation of Maxwell’s equations well-suited to mass-lumping and which ensures low storage. Then, we introduce a dissipative jump derived from Discontinuous Galerkin Methods (DGM) to get rid of spurious waves for both edge and continuous elements. This new approach leads to efficient spectral elements for Maxwell’s equations which are cheaper than DGM. On the other hand, this approach provides a good approximation of singularities generated by reentrant corners.
• Analyse mathématique et numérique de problèmes de propagation des ondes dans des milieux périodiques infinis localement perturbés
  
  • Fliss Sonia
  , 2009. Les milieux périodiques présentent des propriétés intéressantes dans un grand nombre d'applications (les cristaux photoniques en optique, les matériaux composites en mécanique,...). Dans ces applications, on rencontre souvent ces milieux présentant des défauts localisés, c'est-à-dire des milieux qui diffèrent de milieux périodiques dans des régions bornées. Il nous semble intéressant de proposer des méthodes mathématiques et numériques nouvelles spécifiques au traitement des structures périodiques de grande taille, pouvant présenter des défauts localisés. Les caractéristiques du problème rendant très souvent les méthodes d'homogénéisation inapplicables, l'idée est d'exploiter la structure particulière des milieux périodiques pour restreindre les calculs au voisinage du défaut. Nous avons donc approfondi la question de trouver des conditions aux bords parfaitement transparentes. C'est pourquoi nous avons cherché à généraliser les techniques de conditions transparentes non locales, de type Neumann-to-Dirichlet, bien établies pour les milieux homogènes à l'extérieur de la perturbation. La difficulté est que lorsque le milieu extérieur est homogène, on ne dispose plus d'une représentation explicite de la solution. Nous traitons successivement trois situations de difficulté croissante : le cas mono-dimensionnel qui est un cas classique mais dont l'étude a des vertus pédagogiques, le problème du guide périodique localement perturbé et le problème plus complexe du milieu périodique dans les deux dimensions. Pour chaque situation, la démarche est la même : elle consiste tout d'abord à résoudre le problème pour un milieu absorbant puis pour un milieu non absorbant par absorption limite. Nous pouvons alors montrer que les opérateurs DtN peuvent être caractérisés en utilisant la solution de problèmes de cellule locaux, l'utilisation d'outils mathématiques tels que la Transformée de Floquet-Bloch et la solution d'équations quadratiques et linéaires à valeurs et inconnus opérateurs.
• Computation of the band structure of two-dimensional photonic crystals with hp finite elements
  
  • Schmidt Kersten
  • Kauf Peter
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2009, 198, pp.1249-1259. The band structure of 2D photonic crystals -- a periodic material with discontinuous dielectrical properties -- and their eigenmodes can be efficiently computed with the finite element method (FEM). For second order elliptic boundary value problems with piecewise analytic coefficients it is known that the solution converges extremly fast, i.e. exponentially, when using {\em p}-FEM for smooth and {\em hp}-FEM for polygonal interfaces and boundaries. In this article we discretise the variational eigenvalue problems for photonic crystals with smooth and polygonal interfaces in scalar variables with quasi-periodic boundary conditions by means of {\em p}- and {\em hp}-FEM -- this for the transverse electric (TE) and transverse magnetic (TM) modes. Our computations show exponential convergence of the numerical eigenvalues for smooth and polygonal lines of discontinuity of dielectric material properties.
• Computations of lossy bloch waves in two-dimensional photonic crystals
  
  • Engström Christian
  • Hafner Christian
  • Schmidt Kersten
  Journal of computational and theoretical nanoscience, American Scientific Publishers, 2009, 6, pp.775-783. In this article we compute lossy Bloch waves in two-dimensional photonic crystals with dispersion and material loss. For given frequencies these waves are determined from non-linear eigenvalue problems in the wave vector. We applied two numerical methods to a demanding test case, a photonic crystal with embedded quantum dots that exhibits very strong and anamolous dispersion. The first method is based on the formulation with periodic boundary conditions leading to a quadratic eigenvalue problem. We discretize this problem by the finite element method (FEM), first of quadratic order and, second, of higher orders using curved cells (p-FEM). Second, we use the multiple-multipole method (MMP) with artificial sources and compute extrema in the field response determining the eigenvalues. Both MMP and FEM provide robust solutions for the investigated dispersive and lossy photonic crystal, and can approximate the Bloch waves to a high accuracy. Moreover, the MMP method and p-FEM show low computational effort for very accurate solutions.
• Advances on 3D geoelectric forward solver techniques
  
  • Blome Mark
  • Maurer Hansruedi
  • Schmidt Kersten
  Geophysical Journal International, Oxford University Press (OUP), 2009, 176, pp.740-752. Modern geoelectrical data acquisition systems allow large amounts of data to be collected in a short time. Inversions of such data sets require powerful forward solvers for predicting the electrical potentials. State-of-the-art solvers are typically based on finite elements. Recent developments in numerical mathematics led to direct matrix solvers that allow the equation systems arising from such finite element problems to be solved very efficiently. They are particularly useful for 3D geoelectrical problems, where many electrodes are involved. Although modern direct matrix solvers include optimized memory saving strategies, their application to realistic, large-scale 3D problems is still somewhat limited. Therefore, we present two novel techniques that allow the number of grid points to be reduced considerably, while maintaining a high solution accuracy. In the areas surrounding an electrode array we attach infinite elements that continue the electrical potentials to infinity. This does not only reduces the number of grid points, but also avoid the artificial Dirichlet or mixed boundary conditions that are well known to be the cause of numerical inaccuracies. Our second development concerns the singularity removal in the presence of significant surface topography. We employ a fast multipole boundary element method for computing the singular potentials. This renders unnecessary mesh refinements near the electrodes, which results in substantial savings of grid points of up to more than 50%. By means of extensive numerical tests we demonstrate that combined application of infinite elements and singularity removal allows the number of grid points to be reduced by a factor of $\approx$ 6 -- 10 compared with traditional finite element methods. This will be key for applying finite elements and direct matrix solver techniques to realistic 3D inversion problems.
• A distribution framework for the generalized Fourier transform associated with a Sturm--Liouville operator
  
  • Hazard Christophe
  , 2009, pp.18. The generalized Fourier transform associated with a selfadjoint Sturm--Liouville operator is a unitary transformation which converts the action of this operator into a simple product by a spectral variable. For a particular operator defined on the half-line and which involves a step function, we show how to extend such a transformation to generalized functions, or distributions, with a suitable definition of such distributions. This extension is based essentially on the fact that, as the usual Fourier transform, this transformation has the property to exchange regularity and decay between the physical and spectral variables.
• Convergence results of the fictitious domain method for a mixed formulation of the wave equation with a Neumann boundary condition
  
  • Bécache Eliane
  • Rodríguez Jerónimo
  • Tsogka Chrysoula
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2009, 43 (2), pp.377-398. The problem of modeling acoustic waves scattered by an object with Neumann boundary condition is considered. The boundary condition is taken into account by means of the fictitious domain method, yielding a first order in time mixed variational formulation for the problem. The resulting system is discretized with two families of mixed finite elements that are compatible with mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always correctly taken into account when the first family of mixed finite elements is used. We, therefore, introduce the second family of mixed finite elements for which a theoretical convergence analysis is presented and error estimates are obtained. A numerical study of the convergence is also considered for a particular object geometry which shows that our theoretical error estimates are optimal. © 2009 EDP Sciences SMAI. (10.1051/m2an:2008047)
  DOI : 10.1051/m2an:2008047
• La méthode des éléments finis : de la théorie à la pratique. Tome 1 : Concepts généraux
  
  • Ciarlet Patrick
  • Lunéville Éric
  , 2009, pp.194. La méthode des éléments finis, apparue dans les années 50 pour traiter des problèmes de mécanique des structures, a connu depuis lors un développement continu et est utilisée, aujourd’hui, dans tous les domaines d’applications : mécanique, physique, chimie, économie, finance et biologie. Elle est maintenant utilisée dans la plupart des logiciels de calcul scientifique, et de nombreux ingénieurs y sont confrontés dans le cadre de leur activité de modélisation et de simulation numérique. Il est donc important d’en maîtriser les divers aspects. Cet ouvrage recouvre un cours enseigné à l'ENSTA depuis plusieurs années. On y présente tous les éléments essentiels de la méthode : les fondements théoriques (formulations variationnelles d’équations aux dérivées partielles, principes généraux et analyse numérique de la méthode), les considérations pratiques de mise en œuvre (structure creuse des matrices, principe d’assemblage), les algorithmes (en particulier ceux relatifs à la résolution des systèmes linéaires) et enfin des illustrations numériques.
• The fictitious domain method and applications in wave propagation
  
  • Bécache Eliane
  • Rodríguez Garcia Jerónimo
  • Tsogka Chrysoula
  , 2009. This paper deals with the convergence analysis of the fictitious domain method used for taking into account the Neumann boundary condition on the surface of a crack (or more generally an object) in the context of acoustic and elastic wave propagation. For both types of waves we consider the first order in time formulation of the problem known as mixed velocity-pressure formulation for acoustics and velocity-stress formulation for elastodynamics. The convergence analysis for the discrete problem depends on the mixed finite elements used. We consider here two families of mixed finite elements that are compatible with mass lumping. When using the first one which is less expensive and corresponds to the choice made in a previous paper, it is shown that the fictitious domain method does not always converge. For the second one a theoretical convergence analysis was carried out in [7] for the acoustic case. Here we present numerical results that illustrate the convergence of the method both for acoustic and elastic waves.
• Reactive transport in porous media
  
  • Apoung-Kamga Jean-Baptiste
  • Have Pascal
  • Houot Jean
  • Kern Michel
  • Semin Adrien
  ESAIM: Proceedings, EDP Sciences, 2009, 28, pp.227 - 245. We present a numerical method for coupling transport with chemistry in porous media. Our method is based on a fixed-point algorithm that enables us to coupled different transport and chemistry modules. We present the methods for solving the sub-problems, detail the formulation for the coupled problem and show numerical examples to validate the method. Résumé. Nous présentons une méthode numérique pour le couplage du transport et de la chimie en milieu poreux. Notre méthode utilise un algorithme de point fixe, qui nous permet de coupler des modules de transport et de chimie différents. Nous présentons les méthodes numériques utilisées pour chacun des sous-problèmes, ainsi qu'une formulation pour le problème couplé, et nous validons la méthode sur quelques exemples numériques. (10.1051/proc/2009049)
  DOI : 10.1051/proc/2009049
• Diffraction by a defect in an open waveguide: A Mathematical analysis based on a modal radiation condition
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Dakhia Ghania
  • Hazard Christophe
  • Chorfi Lahcène
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2009, 70 (3), pp.677-693. We consider the scattering of a time-harmonic acoustic wave by a defect in a twodimensional open waveguide. The scattered wave satisfies the Helmholtz equation in a perturbed layered half-plane. We introduce a modal radiation condition based on a generalized Fourier transform which diagonalizes the transverse contribution of the Helmholtz operator. The uniqueness of the solution is proved by an original technique which combines a property of the energy flux with an argument of analyticity with respect to the generalized Fourier variable. The existence is then deduced classically from Fredholm's alternative by reformulating the scattering problem as a Lippmann-Schwinger equation by means of the Green's function for the layered half-plane. © 2009 Society for Industrial and Applied Mathematics. (10.1137/080740155)
  DOI : 10.1137/080740155
• Solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements
  
  • Buffa Annalisa
  • Ciarlet Patrick
  • Jamelot Erell
  Numerische Mathematik, Springer Verlag, 2009, 113 (4), pp.497-518. A few years ago, Costabel and Dauge proposed a variational setting, which allows one to solve numerically the time-harmonic Maxwell equations in 3D polyhedral geometries, with the help of a continuous approximation of the electromagnetic field. In order to remove spurious eigenmodes, their method required a parameterization of the variational formulation. In order to avoid this difficulty, we use a mixed variational setting instead of the parameterization, which allows us to handle the divergence-free constraint on the field in a straightforward manner. The numerical analysis of the method is carried out, and numerical examples are provided to show the efficiency of our approach. © Springer-Verlag 2009. (10.1007/s00211-009-0246-2)
  DOI : 10.1007/s00211-009-0246-2
• Direct computation of stresses in planar linearized elasticity
  
  • Ciarlet Philippe G.
  • Ciarlet Patrick
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2009, 19 (7), pp.1043-1064. Given a simply-connected domain Ω in ℝ2, consider a linearly elastic body with Ω as its reference configuration, and define the Hilbert space E(Ω)={e(eαβ) ∈ L2s (Ω) ∂11e22- 2∂12e12}+∂22e11 = 0 in H-2(Ω)}. Then we recently showed that the associated pure traction problem is equivalent to finding a 2 × 2 matrix field = (∈αβ) ∈E(Ω) that satisfies j(∈)= inf e∈E(Ω) j(e), where j(e) = 1/2 ∫Ω Aαβστ eστ eαβ dx - l(e), where (A αβστ ) is the elasticity tensor, and l is a continuous linear form over E(Ω) that takes into account the applied forces. Since the unknown stresses (σαβ) inside the elastic body are then given by σαβ = Aαβστ eστ, this minimization problem thus directly provides the stresses. We show here how the above Saint Venant compatibility condition ∂11 e22 - 2∂12e12 + ∂22e11 = 0 in H-2(Ω) can be exactly implemented in a finite element space h, which uses "edge" finite elements in the sense of J. C. Nédélec. We then establish that the unique solution h of the associated discrete problem, viz., find ∈h ∈ Eh such that j(∈h)=inf eh∈Eh j(eh) converges to in the space L2 s(Ω). We emphasize that, by contrast with a mixed method, only the approximate stresses are computed in this approach. © 2009 World Scientific Publishing Company. (10.1142/s0218202509003711)
  DOI : 10.1142/s0218202509003711
• Application of Discontinuous Galerkin spectral method on hexahedral elements for aeroacoustic
  
  • Castel Nicolas
  • Cohen Gary
  • Duruflé Marc
  Journal of Computational Acoustics, World Scientific Publishing, 2009, 17 (2), pp.175-196. A discontinuous Galerkin method is developed for linear hyperbolic systems on general hexahedral meshes. The use of hexahedral elements and tensorized quadrature formulas to evaluate the integrals leads to an efficient matrix-vector product. It is shown for high order approximations, the reduction in computational time can be very important, compared to tetrahedral elements. Two choices of quadrature points are considered, the Gauss points or Gauss-Lobatto points. The method is applied to the aeroacoustic system (simplified Linearized Euler Equations). Some 3-D numericals experiments show the importance of penalization, and the advantage of using high order.
• Resonances of an elastic plate coupled with a compressible confined flow
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Mercier Jean-François
  Quarterly Journal of Mechanics and Applied Mathematics, Oxford University Press (OUP), 2009, 62 (2), pp.105-129. A theoretical study of the resonances of an elastic plate in a compressible flow in a two-dimensional duct is presented. Due to the fluid-structure coupling, a quadratic eigenvalue problem is involved, in which the resonance frequencies k solve the equations λ(k) = k2, where λ is the eigenvalue of a self-adjoint operator of the form A + kB. In a previous paper, we have proved that a linear eigenvalue problem is recovered if the plate is rigid or the fluid at rest. We focus here on the general problem for which elasticity and flow are jointly present and derive a lower bound for the number of resonances. The expression of this bound, based on the solution of two linear eigenvalue problems, points out that the coupling between elasticity and flow generally reduces the number of resonances. This estimate is validated numerically. © The author 2009. Published by Oxford University Press; all rights reserved. (10.1093/qjmam/hbp004)
  DOI : 10.1093/qjmam/hbp004
• Model reduction for a class of linear descriptor systems
  
  • Hechme Grace
  • Nechepurenko Yu.M.
  • Sadkane Miloud
  Journal of Computational and Applied Mathematics, Elsevier, 2009, 229 (1), pp.54-60. For linear descriptor systems of the form Bẋ=Ax+Cu, this paper constructs reduced order systems associated with a given part of the finite spectrum of the pencil P(λ)=A−λB. It is known that the reduction can be obtained by a block diagonalization of the generalized Schur decomposition of P(λ). In this paper we consider the special case when B = [(H, 0; 0, 0)]and A = [(J, G; - F*, 0)]. This case is suited, in particular, for linearized hydrodynamic problems. We derive a sufficient condition under which the reduced system can approximate the initial one and show that it can be obtained in significantly cheap and efficient approaches. We consider first in detail the case when F = G and H is the identity matrix and then treat the general case. © 2008 Elsevier B.V. All rights reserved. (10.1016/j.cam.2008.10.001)
  DOI : 10.1016/j.cam.2008.10.001
• Fast and accurate computation of layer heat potentials
  
  • Li Jing-Rebecca
  • Greengard Leslie
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2009. We discuss the numerical evaluation of single and double layer heat potentials in two dimensions on stationary and moving boundaries. One of the principal difficulties in designing high order methods concerns the local behavior of the heat kernel, which is both weakly singular in time and rapidly decaying in space. We show that standard quadrature schemes suffer from a poorly recognized form of inaccuracy, which we refer to as geometrically-induced stiffness, but that rules based on product integration of the full heat kernel in time are robust. When combined with previously developed fast algorithms for the evolution of the history part of layer potentials, diffusion processes in complex, moving geometries can be computed accurately and in nearly optimal time.
• Influence of Gauss and Gauss-Lobatto quadrature rules on the accuracy of a quadrilateral finite element method in the time domain.
  
  • Duruflé Marc
  • Grob Pascal
  • Joly Patrick
  Numerical Methods for Partial Differential Equations, Wiley, 2009, 25 (3), pp.526-551. In this paper, we examine the infl uence of numerical integration on finite element methods using quadrilateral or hexahedral meshes in the time domain. We pay special attention to the use of Gauss-Lobatto points to perform mass lumping for any element order. We provide some theoretical results through several error estimates that are completed by various numerical experiments. (10.1002/num.20353)
  DOI : 10.1002/num.20353
• Space-time mesh refinement for discontunuous Galerkin methods for symmetric hyperbolic systems
  
  • Ezziani Abdelaâziz
  • Joly Patrick
  Journal of Computational and Applied Mathematics, Elsevier, 2009, 234 (6), pp.1886-1895. We present a new non-conforming space-time mesh refinement method for the symmetric first order hyperbolic system. This method is based on the one hand on the use of a conservative higher order discontinuous Galerkin approximation for space discretization and a finite difference scheme in time, on the other hand on appropriate discrete transmission conditions between the grids. We use a discrete energy technique to drive the construction of the matching procedure between the grids and guarantee the stability of the method. (10.1016/j.cam.2009.08.094)
  DOI : 10.1016/j.cam.2009.08.094