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52
Adaptive Multilevel Methods for Edge Element Discretizations of Maxwell's Equations
, 1997
"... . The focus of this paper is on the efficient solution of boundary value problems involving the doublecurl operator. Those arise in the computation of electromagnetic fields in various settings, for instance when solving the electric or magnetic wave equation with implicit timestepping, when tackl ..."
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Cited by 32 (12 self)
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. The focus of this paper is on the efficient solution of boundary value problems involving the doublecurl operator. Those arise in the computation of electromagnetic fields in various settings, for instance when solving the electric or magnetic wave equation with implicit timestepping, when tackling timeharmonic problems or in the context of eddycurrent computations. Their discretization is based on on N'ed'elec's H(curl;\Omega\Gamma7131/59948 edge elements on unstructured grids. In order to capture local effects and to guarantee a prescribed accuracy of the approximate solution adaptive refinement of the grid controlled by a posteriori error estimators is employed. The hierarchy of meshes created through adaptive refinement forms the foundation for the fast iterative solution of the resulting linear systems by a multigrid method. The guiding principle underlying the design of both the error estimators and the multigrid method is the separate treatment of the kernel of the cu...
A POSTERIORI ERROR ESTIMATES FOR MAXWELL EQUATIONS
"... Maxwell equations are posed as variational boundary value problems in the function space H(curl) and are discretized by Nédélec finite elements. In [4], a residual type a posteriori error estimator was proposed and analyzed under certain conditions onto the domain. In the present paper, we prove th ..."
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Cited by 32 (3 self)
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Maxwell equations are posed as variational boundary value problems in the function space H(curl) and are discretized by Nédélec finite elements. In [4], a residual type a posteriori error estimator was proposed and analyzed under certain conditions onto the domain. In the present paper, we prove the reliability of that error estimator on Lipschitz domains. The key is to establish new error estimates for the commuting quasiinterpolation operators introduced recently in [22]. Similar estimates are required for additive Schwarz preconditioning. To incorporate boundary conditions, we establish a new extension result.
EQUILIBRATED RESIDUAL ERROR ESTIMATOR FOR EDGE ELEMENTS
"... 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 simpl ..."
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Cited by 27 (1 self)
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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 curlcurl 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. 1.
Commuting QuasiInterpolation Operators For Mixed Finite Elements
"... Conforming finite elements for H(curl) and H(div) spaces became a main research topics in numerical analysis. The so called de Rham diagram [5, 8, 7, 4] relates the exact sequence of continuous spaces H 1 ! H(curl) ! H(div) ! L 2 to their corresponding discrete counterparts. Up to now, only th ..."
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Cited by 24 (2 self)
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Conforming finite elements for H(curl) and H(div) spaces became a main research topics in numerical analysis. The so called de Rham diagram [5, 8, 7, 4] relates the exact sequence of continuous spaces H 1 ! H(curl) ! H(div) ! L 2 to their corresponding discrete counterparts. Up to now, only the local nodal interpolation operators, and global Fortin operators [3] have been known to fulfill the commuting diagram property. In this paper, new quasilocal, Cl'ementtype operators satisfying the commuting diagram property are introduced. The result, in particular, should help to generalize and simplify existing multigrid theories as well as a posteriori error estimates for Maxwell's equations.
Efficient solvers for nonlinear timeperiodic eddy current problems
 Comp. Vis. Sci
, 2006
"... This work deals with all aspects of the numerical simulation of nonlinear timeperiodic eddy current problems, ranging from the description of the nonlinearity to an efficient solution procedure. Due to the periodicity of the solution, we suggest a truncated Fourier series expansion, i.e. a socalle ..."
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Cited by 14 (11 self)
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This work deals with all aspects of the numerical simulation of nonlinear timeperiodic eddy current problems, ranging from the description of the nonlinearity to an efficient solution procedure. Due to the periodicity of the solution, we suggest a truncated Fourier series expansion, i.e. a socalled multiharmonic ansatz, instead of a costly timestepping scheme. Linearization is done by a Newton iteration, where the preconditioning of the linearized problems is a special issue: Since the matrices are nonsymmetric, we need a special adaptation of a multigrid preconditioner to our problem. Eddy current problems comprise another difficulty that complicates the numerical simulation, namely the formation of extremely thin boundary layers. This challenge is handled by means of adaptive mesh refinement. 1
Equilibrated residual error estimators for Maxwell’s equations
 Johann Radon Institute for Computational and Applied Mathematics (RICAM
, 2006
"... Abstract. 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 wellknown for scalar equations of second order. We simplify and ..."
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Cited by 11 (2 self)
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Abstract. 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 wellknown for scalar equations of second order. We simplify and modify the equilibration such that it can be applied to Maxwell’s equations 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. 1.
Singularities of eddy current problems
 ESAIM: Mathematical Modelling and Numerical Analysis
"... We consider the timeharmonic 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 li ..."
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Cited by 10 (2 self)
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We consider the timeharmonic 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. Key Words Eddy current problem, corner singularity, edge singularity. AMS (MOS) subject classification 35B65, 35R05, 35Q60. 1 Maxwell equations and the eddy current limit Let us consider the model case of an homogeneous conducting body ΩC which we assume to be a threedimensional bounded polyhedral domain with a Lipschitz boundary B. The conductivity σ = σC is constant and positive inside ΩC, while σ vanishes outside ΩC, i.e., σ ≡ 0 in the “air ” (or “empty”) region ΩE = R 3 \ ΩC. For the sake of simplicity we further assume that the boundary B of ΩC is connected (∗). The electric permittivity ε is equal to a positive constant εC inside ΩC and has another value εE in the exterior medium. Similarly, the magnetic permeability µ is equal to µC> 0 in ΩC and to µE> 0 in ΩE. The treatment of piecewise constant σC,εC,µC and µE can be made in a similar manner. 1.1 Maxwell and eddy current problems Let ω>0 be a fixed frequency. The time harmonic Maxwell equations are
An iterative substructuring algorithm for twodimensional problems in H(curl
, 2010
"... Abstract. A domain decomposition algorithm, similar to classical iterative substructuring algorithms, is presented for twodimensional problems in the space H0(curl; Ω). It is defined in terms of a coarse space and local subspaces associated with individual edges of the subdomains into which the dom ..."
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Cited by 7 (4 self)
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Abstract. A domain decomposition algorithm, similar to classical iterative substructuring algorithms, is presented for twodimensional problems in the space H0(curl; Ω). It is defined in terms of a coarse space and local subspaces associated with individual edges of the subdomains into which the domain of the problem has been subdivided. The algorithm differs from others in three basic respects. First, it can be implemented in an algebraic manner that does not require access to individual subdomain matrices or a coarse discretization of the domain; this is in contrast to algorithms of the BDDC, FETI–DP, and classical two–level overlapping Schwarz families. Second, favorable condition number bounds can be established over a broader range of subdomain material properties than in previous studies. Third, we are able to develop theory for quite irregular subdomains and bounds for the condition number of our preconditioned conjugate gradient algorithm, which depend only on a few geometric parameters. The coarse space for the algorithm is based on simple energy minimization concepts, and its dimension equals the number of subdomain edges. Numerical results are presented which confirm the theory and demonstrate the usefulness of the algorithm for a variety of mesh decompositions and distributions of material properties. Key words. domain decomposition, iterative substructuring, H(curl), Maxwell’s equations, preconditioners, irregular subdomain boundaries, discontinuous coefficients
An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures
 Mathematics of Compu20 posted on May 7, 2009, PII S
"... Abstract. An edge element adaptive strategy with error control is developed for wave scattering by biperiodic structures. The unbounded computational domain is truncated to a bounded one by a perfectly matched layer (PML) technique. The PML parameters, such as the thickness of the layer and the medi ..."
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Cited by 6 (3 self)
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Abstract. An edge element adaptive strategy with error control is developed for wave scattering by biperiodic structures. The unbounded computational domain is truncated to a bounded one by a perfectly matched layer (PML) technique. The PML parameters, such as the thickness of the layer and the medium properties, are determined through sharp a posteriori error estimates. Numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive method. 1.