Singularities of Maxwell interface problems

Costabel, Martin; Dauge, Monique; Nicaise, Serge

doi:10.1051/m2an:1999155

Cited by 233 publications

(301 citation statements)

References 11 publications

(24 reference statements)

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“…Moreover, considering a one parameter family of Maxwell problems along the lines of (12)-(13), we want to follow the singularities as δ → 0. The "standard" regularity and singularity results for the Maxwell interface problem from [9,15,17] can be adapted for δ > 0, but not for the limit δ = 0.…”

Section: Outline Of the Papermentioning

confidence: 99%

“…To our knowledge this coupling phenomenon seems to be new. As in [15,17] our technique relies on a regularized formulation of the problem and on the use of Mellin transformation.…”

Section: Outline Of the Papermentioning

confidence: 99%

“…We describe in Section 5 the corner and edge singularities for our eddy current problem (case when δ = 0, the case δ > 0 being already investigated in [17]). Section 6 is devoted to the regularity of the solution of the eddy current problem in terms of standard Sobolev spaces, we further give two different decompositions into a regular part and a singular one.…”

Section: Outline Of the Papermentioning

confidence: 99%

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Singularities of eddy current problems

2003

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Abstract.We consider the time-harmonic eddy current problem in its electric formulation where the conductor is a polyhedral domain. By proving the convergence in energy, we justify in what sense this problem is the limit of a family of Maxwell transmission problems: Rather than a low frequency limit, this limit has to be understood in the sense of Bossavit [11]. We describe the singularities of the solutions. They are related to edge and corner singularities of certain problems for the scalar Laplace operator, namely the interior Neumann problem, the exterior Dirichlet problem, and possibly, an interface problem. These singularities are the limit of the singularities of the related family of Maxwell problems. Mathematics Subject Classification. 35B65, 35R05, 35Q60.Received: May 20, 2003. Maxwell equations and the eddy current limitLet us consider the model case of an homogeneous conducting body Ω C which we assume to be a threedimensional bounded polyhedral domain with a Lipschitz boundary B. The conductivity σ = σ C is constant and positive inside Ω C , while σ vanishes outside Ω C , i.e., σ ≡ 0 in the "air" (or "empty") region Ω E = R 3 \ Ω C . For the sake of simplicity we further assume that the boundary B of Ω C is connected ( * ) . The electric permittivity ε is equal to a positive constant ε C inside Ω C and has another value ε E in the exterior medium. Similarly, the magnetic permeability µ is equal to µ C > 0 in Ω C and to µ E > 0 in Ω E . The treatment of piecewise constant σ C , ε C , µ C and µ E can be made in a similar manner. Maxwell and eddy current problemsLet ω > 0 be a fixed frequency. The time harmonic Maxwell equations are

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Section: Outline Of the Papermentioning

confidence: 99%

“…To our knowledge this coupling phenomenon seems to be new. As in [15,17] our technique relies on a regularized formulation of the problem and on the use of Mellin transformation.…”

Section: Outline Of the Papermentioning

confidence: 99%

Section: Outline Of the Papermentioning

confidence: 99%

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Singularities of eddy current problems

2003

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“…We fix non-homogeneous Dirichlet boundary conditions on Γ accordingly. It is easy to see (see for instance [11]) that α is the root of the transcendental equation…”

Section: S(x Y) = R α φ(θ)mentioning

confidence: 99%

Adaptive finite element methods for elliptic problems: Abstract framework and applications

Nicaise

Cochez-Dhondt

2010

ESAIM: M2AN

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Abstract. We consider a general abstract framework of a continuous elliptic problem set on a Hilbert space V that is approximated by a family of (discrete) problems set on a finite-dimensional space of finite dimension not necessarily included into V . We give a series of realistic conditions on an error estimator that allows to conclude that the marking strategy of bulk type leads to the geometric convergence of the adaptive algorithm. These conditions are then verified for different concrete problems like convectionreaction-diffusion problems approximated by a discontinuous Galerkin method with an estimator of residual type or obtained by equilibrated fluxes. Numerical tests that confirm the geometric convergence are presented.Mathematics Subject Classification. 65N30, 65N15, 65N50.

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“…Due to the complexity of the domains the solution of the Maxwell equations has limited regularity, such as singularities near corners and non-convex edges [15], and efficient solution methods require adaptive techniques in order to capture detailed structures.…”

Section: Introductionmentioning

confidence: 99%

Adaptive finite element techniques for the Maxwell equations using implicit a posteriori error estimates

Harutyunyan

Izsák

Vegt

et al. 2008

Computer Methods in Applied Mechanics and Engineering

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For the adaptive solution of the Maxwell equations on three-dimensional domains with Nédélec edge finite element methods, we consider an implicit a posteriori error estimation technique. On each element of the tessellation an equation for the error is formulated and solved with a properly chosen local finite element basis. We show that the discrete bilinear form of the local problems satisfies an inf-sup condition ensuring the well posedness of the error equations. An adaptive algorithm is developed based on the estimated error. We show that the method accurately predicts the regions in the domain with a larger error. The performance of the method is tested on various problems on non-convex domains with non-smooth boundaries. The numerical results show an accurate approximation of the true error. On the meshes generated adaptively with the help of the implicit a posteriori error estimation technique an error is obtained which is smaller than on globally refined meshes. Moreover, the convergence of the error on the locally adapted meshes is faster than that on the globally refined mesh.

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Singularities of Maxwell interface problems

Cited by 233 publications

References 11 publications

Singularities of eddy current problems

Singularities of eddy current problems

Adaptive finite element methods for elliptic problems: Abstract framework and applications

Adaptive finite element techniques for the Maxwell equations using implicit a posteriori error estimates

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