POEMS: Propagation des Ondes : Etude Mathématique et Simulationan

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In 1994 Bérenger showed how to construct a perfectly matched absorbing layer for the Maxwell system in rectilinear coordinates. This layer absorbs waves of any wavelength and any frequency without reflection and thus can be used to artificially terminate the domain of scattering calculations. In this paper we show how to derive and implement the Bérenger layer in curvilinear coordinates (in two space dimensions). We prove that an infinite layer of this type can be used to solve time harmonic scattering problems. We also show that the truncated Bérenger problem has a solution except at a discrete set of exceptional frequencies (which might be empty). Finally numerical results show that the curvilinear layer can produce accurate solutions in the time and frequency domain.

This paper presents the Cholesky factor--alternating direction implicit (CF--ADI) algorithm, which generates a low rank approximation to the solution X of the Lyapunov equation AX+XAT=-BBT. The coefficient matrix A is assumed to be large, and the rank of the right-hand side -BBT is assumed to be much smaller than the size of A. The CF--ADI algorithm requires only matrix-vector products and matrix-vector solves by shifts of A. Hence, it enables one to take advantage of any sparsity or structure in A. This paper also discusses the approximation of the dominant invariant subspace of the solution X. We characterize a group of spanning sets for the range of X. A connection is made between the approximation of the dominant invariant subspace of X and the generation of various low order Krylov and rational Krylov subspaces. It is shown by numerical examples that the rational Krylov subspace generated by the CF--ADI algorithm, where the shifts are obtained as the solution of a rational minimax problem, often gives the best approximation to the dominant invariant subspace of X.

In this article, we construct new higher order finite element spaces for the approximation of the two-dimensional (2D) wave equation. These elements lead to explicit methods after time discretization through the use of appropriate quadrature formulas which permit mass lumping. These formulas are constructed explicitly. Error estimates are provided for the corresponding semidiscrete problem. Finally, higher order finite difference time discretizations are proposed and various numerical results are shown.

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This paper deals with the scattering of a monochromatic electromagnetic wave by a perfect conductor surrounded by a locally inhomogeneous medium. The direct numerical solution of this problem by a finite-element method requires special edge elements. The aim of the present paper is to give an equivalent formulation of the problem well suited for both easy theoretical investigation and numerical implementation. Following a well-known idea, this formulation is obtained by adding a regularizing term such as “grad div” in the time-harmonic Maxwell equations, which leads us to solve an elliptic problem similar to the vector Helmholtz equation instead of Maxwell’s equation. The numerical treatment of this new formulation requires only standard Lagrange finite elements. A unified approach, which is valid for the equations satisfied by either the electric or the magnetic field, is presented. It applies for a conductor with a Lipschitz-continuous boundary surrounded by a dissipative or nondissipative medium whose electromagnetic coefficients (permittivity and permeability) may be irregular. A family of scattering problems is defined, that is, the classical problem (which follows from Maxwell’s equations) and the so-called “regularized problem” obtained by adding a regularizing term in Maxwell’s equations. These problems are shown to be well posed and to have the same solution. An integral representation technique is described.

In this paper, we propose and analyze perfectly matched absorbing layers for a problem of time-harmonic acoustic waves propagating in a duct in the presence of a uniform flow. The absorbing layers are designed for the pressure field, satisfying the convected scalar Helmholtz equation. A difficulty, compared to the Helmholtz equation, comes from the presence of so-called inverse upstream modes which become unstable, instead of evanescent, with the classical Bérenger's perfectly matched layers (PMLs). We investigate here a PML model, recently introduced for time-dependent problems, which makes all outgoing waves evanescent. We then analyze the error due to the truncation of the domain and prove that the convergence is exponential with respect to the size of the layers for both the classical and the new PML models. Numerical validations are finally presented.

In this work, we investigate the Perfectly Matched Layers (PML) introduced by Bérenger [3] for designing efficient numerical absorbing layers in electromagnetism. We make a mathematical analysis of this model, first via a modal analysis with standard Fourier techniques, then via energy techniques. We obtain uniform in time stability results (that make precise some results known in the literature) and state some energy decay results that illustrate the absorbing properties of the model. This last technique allows us to prove the stability of the Yee's scheme for discretizing PML's.

Hodge decompositions of tangential vector fields defined on piecewise regular manifolds are provided. The first step is the study of L2 tangential fields and then the attention is focused on some particular Sobolev spaces of order $-{1\over 2}$\nopagenumbers\end. In order to reach this goal, it is required to properly define the first order differential operators and to investigate their properties. When the manifold Γ is the boundary of a polyhedron Ω, these spaces are important in the analysis of tangential trace mappings for vector fields in H(curl, Ω) on the whole boundary or on a part of it. By means of these Hodge decompositions, one can then provide a complete characterization of these trace mappings: general extension theorems, from the boundary, or from a part of it, to the inside; definition of suitable dualities and validity of integration by parts formulae. Copyright © 2001 John Wiley & Sons, Ltd.

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Some electromagnetic materials have, in a given frequency range, an effective dielectric permittivity and/or a magnetic permeability which are real-valued negative coefficients when dissipation is neglected. They are usually called metamaterials. We study a scalar transmission problem between a classical dielectric material and a metamaterial, set in an open, bounded subset of Rd, 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 combine simple geometrical arguments with the approach of T-coercivity, introduced by the first and third authors and co-worker. Furthermore, the use of localization techniques allows us to derive well-posedness under conditions that involve the knowledge of the coefficients only near the interface. When the coefficients are piecewise constant, we establish the optimality of the criteria.

Abstract The need for numerical schemes for wave problems in large and unbounded domains appears in various applications, including modeling of pressure waves in arteries and other problems in biomedical engineering. Two powerful methods to handle such problems via domain truncation are the use of high‐order absorbing boundary conditions (ABCs) and perfectly matched layers (PMLs). A numerical study is presented to compare the performance of these two types of methods, for two‐dimensional problems governed by the Helmholtz equation. The high‐order ABCs employed here are of the Hagstrom–Warburton type; they are adapted and applied to the frequency domain for the first time. Four PMLs are examined, with linear, quadratic, constant and unbounded decay functions. Two planar configurations are considered: a waveguide and a quarter plane. In the latter case, special corner conditions are developed and used in conjunction with the ABC. One of the main conclusions from the ABC‐PML comparison is that in the high‐accuracy regime, the ABC scheme and the unbounded PML are equally effective. Copyright © 2010 John Wiley & Sons, Ltd.

We derive different classes of generalized impedance boundary conditions for the scattering problem from strongly absorbing obstacles. Compared to existing works, our construction is based on an asymptotic development of the solution with respect to the medium absorption. Error estimates are derived to validate the accuracy of each condition.

We construct and analyze a new family of quadrangular (in two dimensions) or cubic (in three dimensions) mixed finite elements for the approximation of elastic wave equations. Our elements lead to explicit schemes (via mass lumping), after time discretization, including in the case of anisotropic media. Error estimates are given for these new elements.

In this Note, we investigate the so-called interior transmission problem using the T -coercivity approach. In particular, we prove that this problem, which appears when one is interested in the reconstruction of the support of an inclusion embedded in a homogeneous medium, is of Fredholm type and that so-called transmission eigenvalues form at most a discrete set. Our approach treats cases where the difference between the inclusion index and the background index can change sign, which are not covered by other techniques that can be found in the literature. We also provide Faber–Krahn type inequalities associated with this general case.

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This paper presents the Cholesky factor--alternating direction implicit (CF--ADI) algorithm, which generates a low-rank approximation to the solution X of the Lyapunov equation AX+XAT = -BBT. The coefficient matrix A is assumed to be large, and the rank of the right-hand side -BBT is assumed to be much smaller than the size of A. The CF--ADI algorithm requires only matrix-vector products and matrix-vector solves by shifts of A. Hence, it enables one to take advantage of any sparsity or structure in A. This paper also discusses the approximation of the dominant invariant subspace of the solution X. We characterize a group of spanning sets for the range of X. A connection is made between the approximation of the dominant invariant subspace of X and the generation of various low-order Krylov and rational Krylov subspaces. It is shown by numerical examples that the rational Krylov subspace generated by the CF--ADI algorithm, where the shifts are obtained as the solution of a rational minimax problem, often gives the best approximation to the dominant invariant subspace of X.

In this paper, we present a mixed formulation of a spectral element approximation of the linear elasticity system. After studying the main features of this approach, we construct perfectly matched layers (PMLs) for modeling unbounded domains. Then, algorithmic issues are discussed and numerical results are given.

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