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# A staggered discontinuous Galerkin method for wave propagation in media with dielectrics and meta-materials

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*in*Journal of Computational and Applied Mathematics 239:189–207 · February 2013

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Abstract

Some electromagnetic materials exhibit, in a given frequency range, effective dielectric permittivity and/or magnetic permeability which are negative. In the literature, they are called negative index materials, left-handed materials or meta-materials. We propose in this paper a numerical method to solve a wave transmission between a classical dielectric material and a meta-material. The method we investigate can be considered as an alternative method compared to the method presented by the second author and co-workers. In particular, we shall use the abstract framework they developed to prove well-posedness of the exact problem. We recast this problem to fit later discretization by the staggered discontinuous Galerkin method developed by the first author and co-worker, a method which relies on introducing an auxiliary unknown. Convergence of the numerical method is proven, with the help of explicit inf–sup operators, and numerical examples are provided to show the efficiency of the method.

- ... Motivated by the study of metamaterials, many authors have investigated the questions of wellposedness and numerical approximation of boundary value problems involving sign changing coefficients. For scalar transmission problems, let us mention the works of Bonnet-Ben Dhia et al. [7, 5], Chesnel and Ciarlet Jr. [14], Chung and Ciarlet Jr. [15], Nicaise and Venel [25]. Time harmonic Maxwell's equations with sign changing coefficients have been investigated by Bonnet-Ben Dhia, Chesnel and Ciarlet Jr. [6], Fernandes and Raffetto [20], Oliveri and Raffetto [26]. ...... Consequently, a numerical study could be helpful to enlighten whether σ H is positive/negative or indefinite (see for instance Rohan et al. [28, 4, 29] for such numerical investigations in phononics). As the numerical approximation of sign changing coefficients problems requires specific tools [25, 14, 15], this task is beyond the scope of this paper and will be the investigated in a forthcoming work. However, more can be said about σ H in the particular case –but rather significant for applications– where σ(y) is invariant by the rotation Θ π/2 of angle π/2 centered in the middle of the reference cell Y (seeFigure 3 for simple illustrations). ...Article
- Jun 2015
- COMMUN MATH SCI

We investigate a periodic homogenization problem involving two isotropic materials with conductivities of different signs: a classical material and a metamaterial (or negative material). Combining the T−coercivity approach and the unfolding method for homogenization, we prove well-posedness results for the initial and the homogenized problems and we obtain a convergence result. These results are obtained under the condition that the contrast between the two conductivities is large enough in modulus. The homogenized matrix, is generally anisotropic and indefinite, but it is shown to be isotropic and (positive or negative) definite for particular geometries having symmetries. - ... The DG method has been widely used for fluid flow and other related problems with great success, see for example [6,18,20212226,30,33,35,31,34]. On the other hand, many works in literature show that the use of staggered meshes is an important technique in computational fluid dynamics to reduce numerical dissipation [3,4,25], and in computational wave propagation to reduce numerical dispersion [8,9,12131415. Therefore, it is natural to combine the ideas of DG and staggered mesh to develop a more convincing numerical scheme for computational fluid dynamics. ...... Following [8,14151627,28], we first define the triangulation. For rectangular mesh, we can use the idea in [9]. ...
- ... Discontinuous Galerkin methods have been applied to problems in fluid dynamics and wave propagations with great success, see for example [9, 24, 26, 27, 28, 31, 34, 41, 43, 35]. On the other hand, staggered meshes bring the advantages of reducing numerical dissipation in computational fluid dynamics [2, 3, 30] , and numerical dispersion in computational wave propagation [11, 12, 13, 14, 15, 16, 19]. Combining the ideas of DG methods and staggered meshes, a new class of staggered discontinuous Galerkin (SDG) methods for approximations of the incompressible Navier-Stokes equations was proposed [10]. ...... An analysis of the SDG method for incompressible Navier-Stokes equations is given in [22]. For a more complete discussion on the SDG method, see also [13, 14, 15, 16, 20, 21, 32, 33] and the references therein. In the finite element formulation of IB method in [4], the convection term was neglected and linearized Navier-Stokes equations was considered. ...ArticleFull-text available
- Sep 2016

In this paper, we present a staggered discontinuous Galerkin immersed boundary method (SDG-IBM) for the numerical approximation of fluid-structure interaction. The immersed boundary method is used to model the fluid-structure interaction, while the fluid flow is governed by incompressible Navier-Stokes equations. One advantage of using Galerkin method over the finite difference method with immersed boundary method is that we can avoid approximations of the Dirac Delta function. Another key ingredient of our method is that our solver for incompressible Navier-Stokes equations combines the advantages of discontinuous Galerkin methods and staggered meshes, and results in many good properties, namely local and global conservations and pointwise divergence-free velocity field by a local postprocessing technique. Furthermore, energy stability is improved by a skew-symmetric discretization of the convection term. We will present numerical results to show the performance of the method. - ... Staggered DG (SDG) methods were first introduced in [1][2][3] for wave propagation problems. The idea was subsequently applied to other problems, such as convection-diffusion equations [4], electromagnetic problems [5][6][7][8][9], Stokes equations [10], and multiscale wave simulations [11,12]. Similar to [5], the advantages of using SDG for H(curl) problems are the preservation of the structures of the differential operators, the local conservation property, and the optimal convergence. ...Article
- Apr 2014
- Int J Numer Meth Eng

Convergence theories and a deluxe dual and primal finite element tearing and interconnecting algorithm are developed for a hybrid staggered DG finite element approximation of H(curl) elliptic problems in two dimensions. In addition to the advantages of staggered DG methods, the basis functions of the new hybrid staggered DG method are all locally supported in the triangular elements, and a Lagrange multiplier approach is applied to enforce the global connections of these basis functions. The interface problem on the Lagrange multipliers is further reduced to the resulting problem on the subdomain interfaces, and a dual and primal finite element tearing and interconnecting algorithm with an enriched weight factor is then applied to the resulting problem. Our algorithm is shown to give a condition number bound of C(1 + log(H ∕ h))2, independent of the two parameters, where H ∕ h is the number of triangles across each subdomain. Numerical results are included to confirm our theoretical bounds. Copyright © 2013 John Wiley & Sons, Ltd. - ... Those scalar problems have been thoroughly investigated [5, 35, 7, 24, 2, 12, 3, 10, 13] and sharp results have been recently obtained using the simple variational technique of the T-coercivity. The problems are proved to be of Fredholm type in the classical functional framework if the contrasts (ratios of the values of σ across the interface) are outside some interval, which always contains the value −1. ...In this paper, we study the time-harmonic Maxwell problem with sign-changing permittivity and/or permeability, set in a domain of ℜ3. We prove, using the T-coercivity approach, that the well-posedness of the two canonically associated scalar problems, with Dirichlet and Neumann boundary conditions, implies the well-posedness of the Maxwell problem. This allows us to give simple and sharp criteria, obtained in the study of the scalar cases, to ensure that the Maxwell transmission problem between a classical dielectric material and a negative metamaterial is well-posed.
- ... There are some numerical methods that address the local conservation property, see for example23456. We also refer7891011121314 for successful development of the staggered DG methods in many applications. For fast solutions of the discrete Stokes system, there have been many previous studies based on domain decomposition framework. ...Article
- Dec 2014
- COMPUT MATH APPL

In this paper, a class of FETI-DP preconditioners is developed for a fast solution of the linear system arising from staggered discontinuous Galerkin discretization of the two-dimensional Stokes equations. The discretization has been recently developed and has the distinctive advantages that it is optimally convergent and has a good local conservation property. In order to efficiently solve the linear system, two kinds of FETI-DP preconditioners, namely, lumped and Dirichlet preconditioners, are considered and analyzed. Scalable bounds C(H/h)C(H/h) and C(1+log(H/h))2C(1+log(H/h))2 are proved for the lumped and Dirichlet preconditioners, respectively, with the constant CC depending on the inf–sup constant of the discrete spaces but independent of any mesh parameters. Here H/hH/h stands for the number of elements across each subdomain. Numerical results are presented to confirm the theoretical estimates. - ... Staggered DG spaces on triangular meshes. Following [12, 16, 17, 18], we first define the triangulation. For rectangular mesh, we can use the idea in [13]. ...Discontinuous Galerkin (DG) methods are a cl ass of efficient tools for solving fluid flow problems. There are in the literature many greatly successful DG methods. In this paper, a new staggered DG method for the Stokes system is developed and analyzed. The key feature of our method is that the discrete system preserves the structures of the continuous problem, which results from the use of our new staggered DG spaces. This also provides local and global conservation properties, which are desirable for fluid flow applications. The method is based on the first order mixed formulation involving pressure, velocity, and velocity gradient. The velocity and velocity gradient are approximated by polynomials of the same degree while the choice of polynomial degree for pressure is flexible, namely, the approximation degree for pressure can be chosen as either that of velocity or one degree lower than that of velocity. In any case, stability and optimal convergence of the method are proved. Moreover, a superconvergence result with respect to a discrete H 1-norm for the velocity is proved. Furthermore, a local postprocessing technique is proposed to improve the divergence free property of the velocity approximation and it is proved that the postprocessed velocity retains the original accuracy and is weakly divergence free with respect to pressure test functions. Numerical results are included to validate our theoretical estimates and to present the ability of our method for capturing singular solutions.
- ... ACCEPTED MANUSCRIPT investigated [3, 5, 7, 9, 12, 14, 15, 29, 40] and sharp results have been recently obtained thanks to the simple variational method of the T-coercivity. This technique consists in constructing explicit operators which realize the ...Article
- Feb 2015
- Math Meth Appl Sci

We are interested in finding deformations of the rigid wall of a two-dimensional acoustic waveguide, which are not detectable in the far field, as they produce neither reflection nor conversion of propagative modes. A proof of existence of such invisible deformations has been presented in a previous paper. It combines elements of the asymptotic analysis for small deformations and a fixed-point argument. In the present paper, we give a systematic presentation of the method, and we prove that it works for all frequencies except a discrete set. A particular attention is devoted to the practical implementation of the method. The main difficulty concerns the building of a dual family to given oscillating functions. Advantages and limits of the method are illustrated by several numerical results. Copyright © 2015 John Wiley & Sons, Ltd. - ... Recently, a new class of discontinuous Galekin methods based on a novel type of staggered grid is introduced in Chung & Engquist [8, 9] for the wave equations, in Chung & Lee [12] and Chung & Kim [10] for the curl-curl operator, in Chung, Ciarlet & Yu [14] for Maxwell's equations, in Chung & Lee [13] for the convection-diffusion equation, in Kim, Chung & Lee [21] for the Stokes system and in Chan & Chung [5] for the Burgers equation. Moreover, wave transmission problems in the interface between classical material and meta-material using this kind of method is proposed and analyzed in Chung & Ciarlet [7], and fast solvers have been developed in Chung, Kim & Widlund [11], Kim, Chung & Lee [22] and Kim, Chung & Lee [23]. These methods have the advantages that the structures, such as energy and mass, arising from the partial differential equations are preserved. ...Article
- Sep 2015
- J COMPUT APPL MATH

In this paper, we develop and analyze a new class of spectral element methods for the simulations of elastic wave propagation. The major components of the method are the spatial discretization and the choice of interpolation nodes. The spatial discretization is based on piecewise polynomial approximation defined on staggered grids. The resulting method combines the advantages of both staggered-grid based methods and classical non-staggered-grid based spectral element methods. Our new method is energy conserving and does not require the use of any numerical flux, because of the staggered local continuity of the basis functions. Our new method also uses Radau points as interpolation nodes, and the resulting mass matrix is diagonal, thus time marching is explicit and is very efficient. Moreover, we give a rigorous proof for the optimal convergence of the method. In terms of dispersion, we present a numerical study for the numerical dispersion and show that this error is of very high order. Finally, some numerical convergence tests and applications to unbounded domain problems with perfectly matched layer are shown. - ... Recently, a new class of staggered discontinuous Galerkin methods based on staggered meshes was proposed and analyzed. In particular, the staggered DG (SDG) method has been successfully developed for many wave propagation problems ( Engquist, 2006, 2009; Chung et al., 2013a; Chung and Ciarlet, 2013; Chung and Lee, 2012; Chan et al., 2013) and other applications (Chung et al., 2013b; Kim et al., 2013; Chung and Kim, 2014; Chung et al., 2014a; Kim et al., 2014; Chung et al., 2014b). The SDG method is typically applied to the first order formulation of wave equations, and starts with two sets of irregular, staggered grids, for each of the two unknown functions involved; furthermore, it designs two finite-element spaces on those two sets of staggered grids and carries out integration-by-parts to derive corresponding weak formulations; finally, it applies the standard leap-frog scheme for explicit time stepping. ...Accurate simulation of seismic waves is of critical importance in a variety of geophysical applications. Based on recent works on staggered discontinuous Galerkin methods, we have developed a new method for the simulations of seismic waves, which has energy conservation and extremely low grid dispersion, so that it naturally provided accurate numerical simulations of wave propagation useful for geophysical applications and was a generalization of classical staggered-grid finite-difference methods. Moreover, it could handle with ease irregular surface topography and discontinuities in the subsurface models. Our new method discretized the velocity and the stress tensor on this staggered grid, with continuity imposed on different parts of the mesh. The symmetry of the stress tensor was enforced by the Lagrange multiplier technique. The resulting method was an explicit scheme, requiring the solutions of a block diagonal system and a local saddle point system in each time step, and it was, therefore, very efficient. To tailor our scheme to Rayleigh waves, we developed a mortar formulation of our method. Specifically, a fine mesh was used near the free surface and a coarse mesh was used in the rest of the domain. The two meshes were in general not matching, and the continuity of the velocity at the interface was enforced by a Lagrange multiplier. The resulting method was also efficient in time marching. We also developed a stability analysis of the scheme and an explicit bound for the time step size. In addition, we evaluated some numerical results and found that our method was able to preserve the wave energy and accurately computed the Rayleigh waves. Moreover, the mortar formulation gave a significant speed up compared with the use of a uniform fine mesh, and provided an efficient tool for the simulation of Rayleigh waves.
- ... They have recently been revisited via the T-coercivity approach, see, e.g., Bonnet-BenDhia et al. [2] or Chesnel and Ciarlet [10] and the references therein. Conception of numerical approximations, their well-posedness, and a priori error estimates have been addressed in [10] in the conforming finite element context and in [11] for nonconforming and discontinuous Galerkin context. A posteriori error analysis for problems of type (1.1) has likewise been started recently. ...Article
- May 2015
- ESAIM-MATH MODEL NUM

We present a posteriori error analysis of diffusion problems where the diffusion tensor is not necessarily symmetric and positive definite and can in particular change its sign. We first identify the correct intrinsic error norm for such problems, including both conforming and nonconforming approximations. This involves global dual (residual) norms. Importantly, we show their equivalence with the Hilbertian sums of their localizations. We then design estimators which deliver simultaneously guaranteed error upper bound, local and global error lower bounds, and robustness with respect to the diffusion tensor as well as with respect to the approximation polynomial degree. The estimators are given in a unified setting covering at once conforming, nonconforming, mixed, and discontinuous Galerkin finite element discretizations. Numerical results illustrate the theoretical developments. - ... However, the method in [13] cannot be used directly for the wave equation since it is based on a piecewise constant approximation for pressure, which is not accurate for the wave equation, and the velocity basis functions give mass matrix that is not block diagonal. To derive a new GMsFEM with block diagonal mass matrix and energy conservation, we will use a staggered mesh [6, 7, 10, 11, 12, 8], where it is shown that such idea can give a numerical scheme with block diagonal mass matrix and energy conservation. In addition, this idea can give a smaller dispersion error [4, 9]. ...Article
- Sep 2015
- J COMPUT APPL MATH

Numerical simulations of waves in highly heterogeneous media have important applications, but direct computations are prohibitively expensive. In this paper, we develop a new generalized multiscale finite element method with the aim of simulating waves at a much lower cost. Our method is based on a mixed Galerkin type method with carefully designed basis functions that can capture various scales in the solution. The basis functions are constructed based on some local snapshot spaces and local spectral problems defined on them. The spectral problems give a natural ordering of the basis functions in the snapshot space and allow systematically enrichment of basis functions. In addition, by using a staggered coarse mesh, our method is energy conserving and has block diagonal mass matrix, which are desirable properties for wave propagation. We will prove that our method has spectral convergence, and present numerical results to show the performance of the method. - ... Moreover, it is proved that (see [1, 6]) such method gives smaller dispersion errors, and therefore it is superior for the wave propagation. The staggered idea has also been extended to other problems, see for example [7, 8, 9, 5]. Recently, we have used standard MsFEM basis within staggered methods [2, 16]. ...Numerical modeling of wave propagation in heterogeneous media is important in many applications. Due to the complex nature, direct numerical simulations on the fine grid are prohibitively expensive. It is therefore important to develop efficient and accurate methods that allow the use of coarse grids. In this paper, we present a multiscale finite element method for wave propagation on a coarse grid. The proposed method is based on the Generalized Multiscale Finite Element Method (GMsFEM). To construct multiscale basis functions, we start with two snapshot spaces in each coarse-grid block where one represents the degrees of freedom on the boundary and the other represents the degrees of freedom in the interior. We use local spectral problems to identify important modes in each snapshot space. These local spectral problems are different from each other and their formulations are based on the analysis. To our best knowledge, this is the first time where multiple snapshot spaces and multiple spectral problems are used and necessary for efficient computations. Using the dominant modes from local spectral problems, multiscale basis functions are constructed to represent the solution space locally within each coarse block. These multiscale basis functions are coupled via the symmetric interior penalty discontinuous Galerkin method which provides a block diagonal mass matrix, and, consequently, results in fast computations in an explicit time discretiza- tion. Our methods' stability and spectral convergence are rigorously analyzed. Numerical examples are presented to show our methods' performance. We also test oversampling strategies. In particular, we discuss how the modes from different snapshot spaces can a?ect the proposed methods' accuracy.
- ... problem in metamaterials [11,22]. Due to some interesting phenomena happened in metamaterials such as backward wave propagation and invisibility cloaking, various numerical methods have been developed and analyzed for solving the metamaterial Maxwell's equations in recent years (e.g., [23][24][25][26][27]). ...The perfectly matched layer (PML) is a technique initially proposed by Bérenger for solving unbounded electromagnetic problems with the finite-difference time-domain method. In this work, we first formulate an equivalent PML model from the original Bérenger PML model in the corner region, and then establish its stability. We further develop a discontinuous Galerkin method to solve this PML model, and discrete stability similar to the continuous case is proved. To demonstrate the absorbing property of this PML model, we apply it to simulate wave propagation in metamaterials.
- Article
- Nov 2015
- NUMER MATH-THEORY ME

The staggered discontinuous Galerkin (SDG) method has been recently developed for the numerical approximation of partial differential equations. An important advantage of such methodology is that the numerical solution automatically satisfies some conservation properties which are also satisfied by the exact solution. In this paper, we will consider the numerical approximation of the inviscid Burgers equation by the SDG method. For smooth solutions, we prove that our SDG method has the properties of mass and energy conservation. It is well-known that extra care has to be taken at locations of shocks and discontinuities. In this respect, we propose a local total variation (TV) regularization technique to suppress the oscillations in the numerical solution. This TV regularization is only performed locally where oscillation is detected, and is thus very efficient. Therefore, the resulting scheme will preserve the mass and energy away from the shocks and the numerical solution is regularized locally near shocks. Detailed description of the method and numerical results are presented. - Article
- Mar 2017
- COMPUT MATH APPL

Simulation of electromagnetic wave propagation in metamaterials leads to more complicated time domain Maxwell’s equations than the standard Maxwell’s equations in free space. In this paper, we develop and analyze a non-dissipative discontinuous Galerkin (DG) method for solving the Maxwell’s equations in Drude metamaterials. Previous discontinuous Galerkin methods in the literature for electromagnetic wave propagation in metamaterials were either non-dissipative but sub-optimal, or dissipative and optimal. Our method uses a different and simple choice of numerical fluxes, achieving provable non-dissipative stability and optimal error estimates simultaneously. We prove the stability and optimal error estimates for both semi- and fully discrete DG schemes, with the leap-frog time discretization for the fully discrete case. Numerical results are given to demonstrate that the DG method can solve metamaterial Maxwell’s equations effectively. - Article
- Feb 2013
- J COMPUT PHYS

In this paper, a new type of staggered discontinuous Galerkin methods for the three dimensional Maxwell’s equations is developed and analyzed. The spatial discretization is based on staggered Cartesian grids so that many good properties are obtained. First of all, our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the Gauss law. Moreover, the mass matrices are diagonal, thus time marching is explicit and is very efficient. Our method is high order accurate and the optimal order of convergence is rigorously proved. It is also very easy to implement due to its Cartesian structure and can be regarded as a generalization of the classical Yee’s scheme as well as the quadrilateral edge finite elements. Furthermore, a superconvergence result, that is the convergence rate is one order higher at interpolation nodes, is proved. Numerical results are shown to confirm our theoretical statements, and applications to problems in unbounded domains with the use of PML are presented. A comparison of our staggered method and non-staggered method is carried out and shows that our method has better accuracy and efficiency. - Article
- Oct 2017
- NUMER MATH

Transmission problems with sign-changing coefficients occur in electromagnetic theory in the presence of negative materials surrounded by classical materials. For general geometries, establishing Fredholmness of these transmission problems is well-understood thanks to the \(\mathtt {T}\)-coercivity approach. Moreover, for a plane interface, there exist meshing rules that guarantee an optimal convergence rate for the finite element approximation. We propose here a new treatment at the corners of the interface which allows to design meshing rules for an arbitrary polygonal interface and then recover standard error estimates. This treatment relies on the use of simple geometrical transforms to define the meshes. Numerical results illustrate the importance of this new design. - We show, in the framework of steady-state diffusion boundary-value problems, that the staggered discontinuous Galerkin (SDG) method [SIAM J. Numer. Anal., 47 (2009), pp. 3820–3848] can be obtained from a hybridizable discontinuous Galerkin (HDG) method [SIAM J. Numer. Anal., 47 (2009), pp. 1319–1365] by setting its stabilization function to zero at some suitably chosen element faces and by letting it go to infinity at all the remaining others. We then show that this point of view allows the SDG method to immediately acquire new properties all inherited from the HDG methods, namely, their efficient implementation (by hybridization), their postprocessings, and their superconvergence properties.
- Article
- Jul 2017
- J SCI COMPUT

In this paper, we develop a new mass conservative numerical scheme for the simulations of a class of fluid–structure interaction problems. We will use the immersed boundary method to model the fluid–structure interaction, while the fluid flow is governed by the incompressible Navier–Stokes equations. The immersed boundary method is proven to be a successful scheme to model fluid–structure interactions. To ensure mass conservation, we will use the staggered discontinuous Galerkin method to discretize the incompressible Navier–Stokes equations. The staggered discontinuous Galerkin method is able to preserve the skew-symmetry of the convection term. In addition, by using a local postprocessing technique, the weakly divergence free velocity can be used to compute a new postprocessed velocity, which is exactly divergence free and has a superconvergence property. This strongly divergence free velocity field is the key to the mass conservation. Furthermore, energy stability is improved by the skew-symmetric discretization of the convection term. We will present several numerical results to show the performance of the method. - Article
- Apr 2017
- J COMPUT PHYS

The propagation of electromagnetic waves in general media is modeled by the time-dependent Maxwell's partial differential equations (PDEs), coupled with constitutive laws that describe the response of the media. In this work, we focus on nonlinear optical media whose response is modeled by a system of first order nonlinear ordinary differential equations (ODEs), which include a single resonance linear Lorentz dispersion, and the nonlinearity comes from the instantaneous electronic Kerr response and the residual Raman molecular vibrational response. To design efficient, accurate, and stable computational methods, we apply high order discontinuous Galerkin discretizations in space to the hybrid PDE-ODE Maxwell system with several choices of numerical fluxes, and the resulting semi-discrete methods are shown to be energy stable. Under some restrictions on the strength of the nonlinearity, error estimates are also established. When we turn to fully discrete methods, the challenge to achieve provable stability lies in the temporal discretizations of the nonlinear terms. To overcome this, novel strategies are proposed to treat the nonlinearity in our model within the framework of the second-order leap-frog and implicit trapezoidal time integrators. The performance of the overall algorithms are demonstrated through numerical simulations of kink and antikink waves, and third-harmonic generation in soliton propagation. - Article
- Aug 2016
- J SCI COMPUT

Staggered grid techniques are attractive ideas for flow problems due to their more enhanced conservation properties. Recently, a staggered discontinuous Galerkin method is developed for the Stokes system. This method has several distinctive advantages, namely high order optimal convergence as well as local and global conservation properties. In addition, a local postprocessing technique is developed, and the postprocessed velocity is superconvergent and pointwisely divergence-free. Thus, the staggered discontinuous Galerkin method provides a convincing alternative to existing schemes. For problems with corner singularities and flows in porous media, adaptive mesh refinement is crucial in order to reduce the computational cost. In this paper, we will derive a computable error indicator for the staggered discontinuous Galerkin method and prove that this indicator is both efficient and reliable. Moreover, we will present some numerical results with corner singularities and flows in porous media to show that the proposed error indicator gives a good performance. - ThesisFull-text available
- Dec 2015

Dans cette thèse, nous nous intéressons à la propagation d’ondes électromagnétiques dans des structures plasmoniques, composées d’un diélectrique et d’un métal. Les métaux exhibent aux fréquences optiques des propriétés électromagnétiques inhabituelles comme une permittivité diélectrique négative, alors que les diélectriques possèdent une permittivité positive. Ce changement de signe de permittivité a pour conséquence la propagation d’ondes de surface (plasmons de surface) à l’interface métal-diélectrique. Cette thèse concerne le cas où cette interface présente des coins. Des études théoriques ont été menées ces dernières années, combinant la méthode de la T-coercivité et l’analyse des singularités de coins. En particulier, il a été mis en évidence l’existence de deux régimes, selon les paramètres du problème (fréquence, matériau, géométrie). L’objectif de cette thèse est de développer, dans le cas bidimensionnel, une méthode numérique stable pour chacun de ces deux régimes, en apportant un traitement spécifique aux coins. Dans le premier régime (où les solutions appartiennent à l’espace "d’énergie classique"), nous développons des règles de maillages adaptées à la géométrie de l’interface pour garantir la convergence optimale des méthodes d’approximation par éléments finis : on parle de maillages T-conformes. Dans le second régime (où les solutions ne sont plus d’énergie finie), nous proposons une méthode numérique originale utilisant des PMLs (Perfectly Matched Layers) aux coins pour capturer les singularités, appelées ondes de trou noir car elles transportent de l’énergie absorbée par les coins. Nous appliquons ces techniques numériques à deux problèmes physiques : la diffraction par une onde plane d’une inclusion métallique polygonale, et la détermination des modes guidés d’un guide d’ondes plasmonique à section polygonale. Pour le problème de diffraction, nous montrons que les coins de l’inclusion métallique peuvent absorber de l’énergie, transportée par les ondes de trou noir, et nous calculons numériquement l’énergie absorbée par chaque coin. L’étude des modes guidés du guide plasmonique quant à elle s’écrit sous la forme d’un problème de théorie spectrale non classique. En présence d’ondes de trou noir, les valeurs propres associées aux modes guidés sont plongées dans le spectre essentiel. Pour les dévoiler, on utilise à nouveau des PMLs aux coins, ce qui revient à calculer les valeurs propres d’un opérateur étendu dont le spectre est discret. - Article
- Jun 2014
- APPL MATH COMPUT

In this paper, we present the first a-posteriori error analysis for the staggered discontinuous Galerkin (SDG) method. Specifically, we consider the approximation of the time-harmonic Maxwell's equations by a SDG method, and prove that our residual based a-posteriori error indicator is both reliable and efficient. We validate the performance of the indicator within an adaptive mesh refinement procedure and show its asymptotic exactness for a range of test problems. - Article
- Jan 2016
- INT J NUMER ANAL MOD

Since the successful construction of the so-called double negative metamaterials in 2000, there has been a growing interest in studying metamaterials across many disciplinaries. In this paper, we present a survey of recent progress in metamaterials and its applications from the mathematical point of view. Due to the great amount of papers published in this area, here we mainly discuss those issues interested to us. Our main goal is to attract more mathematicians to study this fascinating subject. - Article
- Mar 2016
- COMPUT MATH APPL

A mortar formulation is developed and analyzed for a class of staggered discontinuous Galerkin (SDG) methods applied to second order elliptic problems in two dimensions. The computational domain consists of nonoverlapping subdomains and a triangulation is provided for each subdomain, which need not conform across subdomain interfaces. This feature allows a more flexible design of discrete models for problems with complicated geometries, shocks, or singular points. A mortar matching condition is enforced on the solutions across the subdomain interfaces by introducing a Lagrange multiplier space. Moreover, optimal convergence rates in both and discrete energy norms are proved. Numerical results are presented to show the performance of the method. - Article
- Jan 2014
- J COMPUT MATH

It is well-known that artificial boundary conditions are crucial for the efficient and accurate computations of wavefields on unbounded domains. In this paper, we investigate stability analysis for the wave equation coupled with the first and the second order absorbing boundary conditions. The computational scheme is also developed. The approach allows the absorbing boundary conditions to be naturally imposed, which makes it easier for us to construct high order schemes for the absorbing boundary conditions. A third-order Lagrange finite element method with mass lumping is applied to obtain the spatial discretization of the wave equation. The resulting scheme is stable and is very efficient since no matrix inversion is needed at each time step. Moreover, we have shown both abstract and explicit conditional stability results for the fully-discrete schemes. The results are helpful for designing computational parameters in computations. Numerical computations are illustrated to show the efficiency and accuracy of our method. In particular, essentially no boundary reflection is seen at the artificial boundaries. - Article
- Jan 2015
- INT J NUMER ANAL MOD

In this paper, we develop and analyze a preconditioning technique and an iterative solver for the linear systems resulting from the discretization of second order elliptic problems by the symmetric interior penalty discontinuous Galerkin methods. The main ingredient of our approach is a stable decomposition of the piecewise polynomial discontinuous finite element space of arbitrary order into a linear conforming space and a space containing high frequency components. To derive such decomposition, we introduce a novel interpolation operator which projects piece-wise polynomials of arbitrary order to continuous piecewise linear functions. We prove that this operator is stable which allows us to derive the required space decomposition easily. Moreover, we prove that both the condition number of the preconditioned system and the convergent rate of the iterative method are independent of the mesh size. Numerical experiments are also shown to confirm these theoretical results. - Article
- May 2015
- INT J NUMER ANAL MOD

We consider time domain formulations of Maxwell’s equations for the Lorentz model for metamaterials. The field equations are considered in two different forms which have either six or four unknown vector fields. In each case we use arguments tuned to the physical laws to derive data-stability estimates which do not require Gronwall’s inequality. The resulting estimates are, in this sense, sharp. We also give fully discrete formulations for each case and extend the sharp data-stability to these. Since the physical problem is linear it follows (and we show this with examples) that this stability property is also reflected in the constants appearing in the a priori error bounds. By removing the exponential growth in time from these estimates we conclude that these schemes can be used with confidence for the long-time numerical simulation of Lorentz metamaterials. - Article
- Mar 2018
- J SCI COMPUT

Staggered grid techniques have been applied successfully to many problems. A distinctive advantage is that physical laws arising from the corresponding partial differential equations are automatically preserved. Recently, a staggered discontinuous Galerkin (SDG) method was developed for the convection–diffusion equation. In this paper, we are interested in solving the steady state convection–diffusion equation with a small diffusion coefficient \(\epsilon \). It is known that the exact solution may have large gradient in some regions and thus a very fine mesh is needed. For convection dominated problems, that is, when \(\epsilon \) is small, exact solutions may contain sharp layers. In these cases, adaptive mesh refinement is crucial in order to reduce the computational cost. In this paper, a new SDG method is proposed and the proof of its stability is provided. In order to construct an adaptive mesh refinement strategy for this new SDG method, we derive an a-posteriori error estimator and prove its efficiency and reliability under a boundedness assumption on \(h/\epsilon \), where h is the mesh size. Moreover, we will present some numerical results with singularities and sharp layers to show the good performance of the proposed error estimator as well as the adaptive mesh refinement strategy. - PreprintFull-text available
- Jun 2018

In this paper, we present an embedded staggered discontinuous Galerkin method for the convection-diffusion equation. The new method combines the advantages of staggered discontinuous Galerkin (SDG) and embedded discontinuous Galerkin (EDG) method, and results in many good properties, namely local and global conservations, free of carefully designed stabilization terms or flux conditions and high computational efficiency. In applying the new method to convection-dominated problems, the method provides optimal convergence in potential and suboptimal convergence in flux, which is comparable to other existing DG methods, and achieves $L^2$ stability by making use of a skew-symmetric discretization of the convection term, irrespective of diffusivity. We will present numerical results to show the performance of the method.

- Article
- Jul 2010
- J COMPUT APPL MATH

Some electromagnetic materials present, in a given frequency range, an effective dielectric permittivity and/or magnetic permeability which are negative. We are interested in the reunion of such a “negative” material and a classical one. More precisely, we consider here a scalar model problem for the simulation of a wave transmission between two such materials. This model is governed by a Helmholtz equation with a weight function in the Δ principal part which takes positive and negative real values. Introducing additional unknowns, we have already proposed in Bonnet-Ben Dhia et al. (2006) [1] some new variational formulations of this problem, which are of Fredholm type provided the absolute value of the contrast of permittivities is large enough, and therefore suitable for a finite element discretization. We prove here that, under similar conditions on the contrast, the natural variational formulation of the problem, although not “coercive plus compact”, is nonetheless suitable for a finite element discretization. This leads to a numerical approach which is straightforward, less costly than the previous ones, and very accurate. - Jan 2010
- 1912-1919

- Appl
- Math

Appl. Math., 234 (2010), pp. 1912–1919. Corrigendum 234 (2010), p. 2616.- Article
- Dec 1972

This chapter focuses on variational crimes in the finite element method. The finite element method is nearly a special case of the Rayleigh-Ritz technique. The convenience and effectiveness of the finite element technique is regarded as conclusively established; it has brought a revolution in the calculations of structural mechanics, and other applications are rapidly developing. The chapter reviews the modifications of the Ritz procedure which have been made to achieve an efficient finite element system. On a regular mesh one could regard the system of Ritz-finite element equations KQ = F as a finite difference scheme, and then the patch test would be equivalent to the formal consistency of the difference equations with the correct differential equation. The chapter discusses the convergence theory for non-conforming elements. It is impossible for a polynomial to satisfy a condition like u = 0 on a general curved boundary. Therefore, some alteration in the boundary condition will be necessary. The most important possibility is to change the domain. - Article
- Feb 2013
- NUMER MATH

To solve variational indefinite problems, one uses classically the Banach–Nečas–Babuška theory. Here, we study an alternate theory to solve those problems: T-coercivity. Moreover, we prove that one can use this theory to solve the approximate problems, which provides an alternative to the celebrated Fortin lemma. We apply this theory to solve the indefinite problem $\text{ div}\sigma \nabla u=f$ set in $H^1_0$ , with $\sigma $ exhibiting a sign change. - Article
- Aug 2008
- COMPUT METHOD APPL M

In this paper, we develop a Crank–Nicolson mixed finite element method for modeling wave propagation in negative-index materials (NIMs). The NIMs model is formed as a time-dependent system involving four dependent vector variables: the electric and magnetic fields, and the induced electric and magnetic currents. Optimal error estimates for all four variables are proved for Nédélec edge elements. Numerical examples are presented to show the exotic properties for wave propagation in NIMs. - Book
- Jan 1992

I Theoretical Foundations.- 1 Finite Element Interpolation.- 2 Approximation in Banach Spaces by Galerkin Methods.- II Approximation of PDEs.- 3 Coercive Problems.- 4 Mixed Problems.- 5 First-Order PDEs.- 6 Time-Dependent Problems.- III Implementation.- 7 Data Structuring and Mesh Generation.- 8 Quadratures, Assembling, and Storage.- 9 Linear Algebra.- 10 A Posteriori Error Estimates and Adaptive Meshes.- IV Appendices.- A Banach and Hilbert Spaces.- A.1 Basic Definitions and Results.- A.2 Bijective Banach Operators.- B Functional Analysis.- B.1 Lebesgue and Lipschitz Spaces.- B.2 Distributions.- B.3 Sobolev Spaces.- Nomenclature.- References.- Author Index. - In this paper, we present several extensions of theoretical tools for the analysis of discontinuous Galerkin (DG) method beyond the linear case. We define broken Sobolev spaces for Sobolev indices in [1, ∞), and we prove generalizations of many techniques of classical analysis in Sobolev spaces. Our targeted application is the convergence analysis for DG discretizations of energy minimization problems of the calculus of variations. Our main tool in this analysis is a theorem which permits the extraction of a ‘weakly’ converging subsequence of a family of discrete solutions and which shows that any ‘weak limit’ is a Sobolev function. As a second application, we compute the optimal embedding constants in broken Sobolev–Poincaré inequalities.
- Article
- Apr 2012
- J Numer Math

This paper is concerned with the staggered discontinuous Galerkin method for convection-diffusion equations. Over the past few decades, staggered type methods have been applied successfully to many problems, such as wave propagation and fluid flow problems. A distinctive feature of these methods is that the physical laws arising from the corresponding partial differential equations are automatically preserved. Nevertheless, staggered methods for convection-diffusion equations are rarely seen in literature. It is thus the main goal of this paper to develop and analyze a class of staggered numerical schemes for the approximation of convection-diffusion equations. We will prove that our new method preserves the underlying physical laws in some discrete sense.Moreover, the stability and convergence of the method are proved. Numerical results are shown to verify the theoretical estimates. - In this paper, we develop and analyze a new class of discontinuous Galerkin (DG) methods for the acoustic wave equation in mixed form. Traditional mixed finite element (FE) methods produce energy conserving schemes, but these schemes are implicit, making the time-stepping inefficient. Standard DG methods give explicit schemes, but these approaches are typically dissipative or suboptimally convergent, depending on the choice of numerical fluxes. Our new method can be seen as a compromise between these two kinds of techniques, in the way that it is both explicit and energy conserving, locally and globally. Moreover, it can be seen as a generalized version of the Raviart-Thomas FE method and the finite volume method. Stability and convergence of the new method are rigorously analyzed, and we have shown that the method is optimally convergent. Furthermore, in order to apply the new method for unbounded domains, we apply our new method with the first order absorbing boundary condition. The stability of the resulting numerical scheme is analyzed.
- Some electromagnetic materials have, in a given frequency range, an effective dielectric permittivity and/or a magnetic permeability which are real-valued negative parameters when dissipation is neglected. They are usually called metamaterials. We study a scalar wave transmission problem between a classical dielectric material and a metamaterial, set in an open, bounded subset of R^d, with d = 2, 3. Our aim is to characterize occurences where the problem is well-posed within the Fredholm (or coercive + compact) framework. For that, we build some criteria, based on the geometry of the interface between the dielectric and the metamaterial. The proofs rely on localization techniques, together with the theory of T-coercivity introduced by the first and third authors and co-worker. In particular, we establish the optimality of the criteria, when the dielectric permittivity is piecewise constant.
- Article
- Sep 2010
- J COMPUT APPL MATH

We perform the a posteriori error analysis of residual type of transmission problem with sign changing coefficients. According to Bonnet-BenDhia et al. (2010) [9], if the contrast is large enough, the continuous problem can be transformed into a coercive one. We further show that a similar property holds for the discrete problem for any regular meshes, extending the framework from Bonnet-BenDhia et al. [9]. The reliability and efficiency of the proposed estimator are confirmed by some numerical tests. - Article
- Jan 1972

"Proceedings of a symposium held at the University of Maryland Baltimore, Maryland June 26-30, 1972" Sponsored by the Division of Mathematics University of Maryland, Baltimore Country Campus, and the U. S. Office Naval Research Incluye bibliografía - We have developed and analyzed a new class of discontinuous Galerkin methods (DG) which can be seen as a compromise between standard DG and the finite element (FE) method in the way that it is explicit like standard DG and energy conserving like FE. In the literature there are many methods that achieve some of the goals of explicit time marching, unstructured grid, energy conservation, and optimal higher order accuracy, but as far as we know only our new algorithms satisfy all the conditions. We propose a new stability requirement for our DG. The stability analysis is based on the careful selection of the two FE spaces which verify the new stability condition. The convergence rate is optimal with respect to the order of the polynomials in the FE spaces. Moreover, the convergence is described by a series of numerical experiments.