On traces for H(curl,Ω) in Lipschitz domains

Buffa, Annalisa; Costabel, Martin; Sheen, Dongwoo

doi:10.1016/s0022-247x(02)00455-9

Cited by 339 publications

(430 citation statements)

References 20 publications

(46 reference statements)

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“…In addition, since curl Γ π t = div Γ γ t and div Γ (γ t (W )) = γ n (curl(W )) ∈ H −1/2 (Γ) for each W ∈ H(curl; Ω m ) (see [7]), we deduce that for each W ∈ X m h there holds…”

Section: Well-posedness Of the Discrete Problemmentioning

confidence: 93%

“…In this section we proceed analogously to [7] (see also [12]) and employ suitable decompositions of H Γ (curl; Ω m ) and H(div; Ω s ) to prove that (4.10) becomes a compact perturbation of a well-posed problem. In particular, the splitting of H(div; Ω s ) is defined in terms of an elasticity problem in Ω s with Neumann boundary conditions, whereas a well-known result on divergence-free potential vectors is the basis of the splitting of H Γ (curl; Ω m ).…”

Section: Analysis Of the Continuous Variational Formulationmentioning

confidence: 99%

“…In fact, it is clear that curl(W ) is a divergence-free element of H(div; Ω m ). Next, in order to show that γ n (curl(W )), 1 Σ = 0 we recall that γ n (curl(W )) = div Σ (γ t (W )) and that the adjoint of div Σ is −∇ Σ (see [5], [7]). It follows that…”

Section: 1mentioning

confidence: 99%

“…A more precise result is given by the following theorem (see [7]) (we refer to [5], [7] for the definition of the differential operators div Σ and curl Σ on piecewise smooth Lipschitz boundaries). In addition, exchanging the roles of u and v in (2.3), we find that (2.4) γ t (u), π t (v) t,Σ = − γ t (v), π t (u) t,Σ ∀ u, v ∈ H(curl; Ω).…”

Section: Introduction a Successful Strategy Has Been Developed Inmentioning

confidence: 99%

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A Coupled Mixed Finite Element Method for the Interaction Problem between an Electromagnetic Field and an Elastic Body

Gatica¹,

Hsiao²,

Meddahi³

2010

SIAM J. Numer. Anal.

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Abstract. This paper deals with the coupled problem arising from the interaction of a timeharmonic electromagnetic field with a three-dimensional elastic body. More precisely, we consider a suitable transmission problem holding between the solid and a sufficiently large annular region surrounding it, and aim to compute both the magnetic component of the scattered wave and the stresses that take place in the obstacle. To this end, we assume Voigt's model, which allows interaction only through the boundary of the body, and employ a dual-mixed variational formulation in the solid medium. As a consequence, one of the two transmission conditions becomes essential, whence it is enforced weakly through the introduction of a Lagrange multiplier. An abstract framework developed recently, which is based on regular decompositions of the spaces involved, is applied next to show that our coupled variational formulation is well-posed. In addition, we define the corresponding Galerkin scheme by using PEERS in the solid and using the edge finite elements of Nédélec in the electromagnetic region. Then, we prove that the resulting coupled mixed finite element scheme is uniquely solvable and convergent. Moreover, optimal a priori error estimates are derived in the usual way. Finally, some numerical results illustrating the analysis and the good performance of the method are also reported. Key words. Maxwell equations, edge finite elements, elastodynamics equations, PEERS AMS subject classifications. 65N30, 65N12, 65N15, 74F10, 74B05, 35J05DOI. 10.1137/090754212 [4] to analyze, at the continuous and discrete levels, a class of variational formulations defined by noncoercive bilinear (or sesquilinear) forms. More precisely, though the analysis in [4] was originally motivated by the study of Maxwell equations, the author succeeded in setting up the corresponding technique in a quite general framework. In fact, the key issue is the utilization of a Helmholtz-type decomposition of the main unknown, which allows us to reveal hidden compactness properties of the formulation, and hence the classical results connecting Fredholm alternative and projection methods (see, e.g., [18], [21]) can be applied straightforwardly. Introduction. A successful strategy has been developed inThe method from [4] was applied recently in [12] and [14] to deal with a timeharmonic fluid-solid interaction problem posed in the plane. The model consists of an elastic body occupying a region Ω s , which is subject to a given incident acoustic wave that travels in the fluid surrounding it. In [12] the fluid is supposed to occupy an annular region Ω f , and a Robin boundary condition imitating the behavior of the

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Section: Well-posedness Of the Discrete Problemmentioning

confidence: 93%

Section: Analysis Of the Continuous Variational Formulationmentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

Section: Introduction a Successful Strategy Has Been Developed Inmentioning

confidence: 99%

See 2 more Smart Citations

A Coupled Mixed Finite Element Method for the Interaction Problem between an Electromagnetic Field and an Elastic Body

Gatica¹,

Hsiao²,

Meddahi³

2010

SIAM J. Numer. Anal.

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“…In this section we introduce continuously differentiable function spaces and Lebesgue integrable function spaces for vector-valued functions and use them to define the function spaces for the Maxwell's equations in the domain and on the boundary. The function spaces on the boundary have been studied in [9,10,6] for piecewise smooth boundaries and in [12] for Lipschitz boundaries. [7] is a summary of all these work.…”

Section: Function Spacesmentioning

confidence: 99%

Boundary element approximation for Maxwell's eigenvalue problem

Wieners

Xin

2013

Math Methods in App Sciences

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We introduce a new method for computing eigenvalues of the Maxwell operator with boundary finite elements. On bounded domains with piecewise constant material coefficients, the Maxwell solution for fixed wave number can be represented by boundary integrals, which allows to reduce the eigenvalue problem to a nonlinear problem for determining the wave number along with boundary and interface traces. A Galerkin discretization yields a smooth nonlinear matrix eigenvalue problem that is solved by Newton's method or, alternatively, the contour integral method. Several numerical results including an application to the band structure computation of a photonic crystal illustrate the efficiency of this approach. Copyright © 2013 John Wiley & Sons, Ltd.

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Finite Element Methods for M axwell's Equations

Demkowicz

2017

Encyclopedia of Computational Mechanics Second Edition

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The chapter reviews fundamentals of finite element discretizations of time‐harmonic Maxwell equations. The chapter includes a short introduction to Maxwell equations, derivation of various variational formulations, discussion on Bubnov‐ and Petrov–Galerkin discretizations, three fundamental Nédélec's elements in context of exact sequences, projection‐based interpolation, and the commuting diagram property. The presentation is biased toward conforming methods and higher order discretizations.

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On traces for H(curl,Ω) in Lipschitz domains

Cited by 339 publications

References 20 publications

A Coupled Mixed Finite Element Method for the Interaction Problem between an Electromagnetic Field and an Elastic Body

A Coupled Mixed Finite Element Method for the Interaction Problem between an Electromagnetic Field and an Elastic Body

Boundary element approximation for Maxwell's eigenvalue problem

Finite Element Methods for M axwell's Equations

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