Residual based a posteriori error estimators for eddy current computation

Beck, R. M.; Hiptmair, Ralf; Hoppe, Ronald H. W.; Wohlmuth, Barbara

doi:10.1051/m2an:2000136

Cited by 205 publications

(223 citation statements)

References 49 publications

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“…The reliability follows from Lemma 12, the exactness of the second distributional de Rham sequence (3.17), and Theorem 10. The efficiency estimate follows from the stability of the right inverse, Lemma 9, and the efficiency of the residual error estimator analyzed in [6]. …”

Section: Equilibration In 3dmentioning

confidence: 99%

Equilibrated residual error estimator for edge elements

2007

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Abstract. Reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem. A cheap computation of such functions via equilibration is well known for scalar equations of second order. We simplify and modify the equilibration such that it can be applied to the curl-curl equation and edge elements. The construction is more involved for edge elements since the equilibration has to be performed on subsets with different dimensions. For this reason, Raviart-Thomas elements are extended in the spirit of distributions.

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Section: Equilibration In 3dmentioning

confidence: 99%

Equilibrated residual error estimator for edge elements

2007

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“…The condition at infinity takes the form (5)- (6) are clearly obtained from (1)-(2) by setting ε to zero. The gauge conditions (7) can also be obtained from (3): Since iωε + σ is equal to the two non-zero constants iωε C + σ in Ω C and iωε E in Ω E , (3) implies that div E C = 0 in Ω C , div E E = 0 in Ω E and (by a result similar to Lem.…”

mentioning

confidence: 99%

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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“…There are several techniques to obtain explicit bounds for the unknown constant term (see e. g. [12]), but in most applications the estimates are somewhat pessimistic, hence the resulting estimators tend to be unrealistic and fail to detect the more subtle nuances of the specific problem. Several applications of adaptive methods with an explicit error estimation technique for the Maxwell equations can be found in [8,10,13,27,28].…”

Section: Introductionmentioning

confidence: 99%

“…Recently, implicit a posteriori error estimation methods have attracted attention in the literature. It is worth noting the pioneering articles [1,2,17] and applications of implicit error estimation techniques for the Maxwell equations can be found in [9,11,16,26,29].…”

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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Residual based a posteriori error estimators for eddy current computation

Cited by 205 publications

References 49 publications

Equilibrated residual error estimator for edge elements

Equilibrated residual error estimator for edge elements

Singularities of eddy current problems

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

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