Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements

Demkowicz, Leszek; Vardapetyan, L.

doi:10.1016/s0045-7825(97)00184-9

Cited by 139 publications

(102 citation statements)

References 22 publications

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“…Extensions to more general h-version elements, including those introduced by Demkowicz and Vardapetyan in [9,29], or the second family of edge elements on tetrahedra [25] can be proved in the same way.…”

Section: Details Of the Finite Element Methodsmentioning

confidence: 96%

Discrete compactness and the approximation of Maxwell's equations in $\mathbb{R}^3$

Monk

Demkowicz

2000

Math. Comp.

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Abstract. We analyze the use of edge finite element methods to approximate Maxwell's equations in a bounded cavity. Using the theory of collectively compact operators, we prove h-convergence for the source and eigenvalue problems. This is the first proof of convergence of the eigenvalue problem for general edge elements, and it extends and unifies the theory for both problems. The convergence results are based on the discrete compactness property of edge element due to Kikuchi. We extend the original work of Kikuchi by proving that edge elements of all orders possess this property.

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Section: Details Of the Finite Element Methodsmentioning

confidence: 96%

Discrete compactness and the approximation of Maxwell's equations in $\mathbb{R}^3$

Monk

Demkowicz

2000

Math. Comp.

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“…The interior penalty method proposed in this article is based on a mixed formulation of (1)-(2) already used in the hp-approaches of [1,8], as well as in the mortar approach [6]. To this end, we decompose the field E as…”

Section: Mixed Formulationmentioning

confidence: 99%

“…We notice that the form a is bilinear, continuous and coercive on the kernel of b, and b is bilinear, continuous, and satisfies the inf-sup condition; see, e.g., [8,18,24]. Hence, problem (6)- (9) is well-posed (provided that k 2 is not an interior Maxwell eigenvalue) and there is a positive constant C, depending on Ω and k 2 , such that…”

Section: Mixed Formulationmentioning

confidence: 99%

Mixed discontinuous Galerkin approximation of the Maxwell operator: The indefinite case

Houston¹,

Perugia²,

Schneebeli³

et al. 2005

ESAIM: M2AN

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Abstract. We present and analyze an interior penalty method for the numerical discretization of the indefinite time-harmonic Maxwell equations in mixed form. The method is based on the mixed discretization of the curl-curl operator developed in [Houston et al., J. Sci. Comp. 22 (2005) 325-356] and can be understood as a non-stabilized variant of the approach proposed in [Perugia et al., Comput. Methods Appl. Mech. Engrg. 191 (2002) 4675-4697]. We show the well-posedness of this approach and derive optimal a priori error estimates in the energy-norm as well as the L 2 -norm. The theoretical results are confirmed in a series of numerical experiments.

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“…One way of imposing this constraint is to use mixed formulations, where new unknowns are introduced as Lagrange multipliers (see, e.g., Chen, Du and Zou [21], Demkowicz and Vardapetyan [29,45] and the references therein). Other approaches consist in regularizing the formulation in Ω 0 by adding suitable terms containing the divergence of E, giving rise to formulations in the variable E only (see, e.g., Alonso and Valli [2] where an iteration-by-subdomain procedure is studied, using edge elements in the conducting region Ω \ Ω 0 and continuous elements in Ω 0 ).…”

Section: Introductionmentioning

confidence: 99%

The $hp$-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations

2002

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Abstract. The local discontinuous Galerkin method for the numerical approximation of the time-harmonic Maxwell equations in a low-frequency regime is introduced and analyzed. Topologically nontrivial domains and heterogeneous media are considered, containing both conducting and insulating materials. The presented method involves discontinuous Galerkin discretizations of the curl-curl and grad-div operators, derived by introducing suitable auxiliary variables and so-called numerical fluxes. An hp-analysis is carried out and error estimates that are optimal in the meshsize h and slightly suboptimal in the approximation degree p are obtained. IntroductionIn this paper, we propose and analyze an hp-local discontinuous Galerkin (LDG) method for the low-frequency time-harmonic Maxwell equations in heterogeneous media, containing both conducting and insulating materials: Find the complex field E that satisfiesin Ω 0 ⊂ Ω, (1.2) together with suitable boundary conditions (see Alonso and Valli [2] and Alonso [1]). Here, the field E is related to the electric field E by the identity E(x, t) = Re(E(x)e iωt ), where ω = 0 is a given frequency. The parameter µ = µ(x) is the magnetic permeability, ε = ε(x) the electric permittivity, J s is the phasor associated with a given current density, and σ = σ(x) is the electric conductivity, which is zero in the subdomain Ω 0 occupied by insulating materials. We remark that the electric field-based formulation in (1.1)-(1.2) is only one of several field-and potential-based formulations proposed in the literature for the solution of eddy current problems The main motivation for using discontinuous Galerkin (DG) methods for the numerical approximation of the above problem is that these methods, being based on discontinuous finite element spaces, can easily handle meshes with hanging nodes, elements of general shape, and local spaces of different types. Thus, they are ideally suited for hp-adaptivity and multiphysics or multimaterial problems. This flexibility in the mesh design is not shared in a straightforward way by standard edge or face elements commonly used in computational electromagnetics. Indeed, these elements are designed to enforce the continuity of either the tangential or the normal components of the fields across interelement boundaries (see, e.g., Nédéléc [37, 38], Bossavit [10, 11], and Monk [36]). This makes the handling of nonmatching grids and high-order approximations rather inconvenient from an implementational point of view. Nevertheless, efficient hp-adaptive edge element methods have been developed recently by Demkowicz and Vardapetyan [29,45].There are several possibilities for dealing with the divergence-free constraint (1.2) in the subregion Ω 0 . One way of imposing this constraint is to use mixed formulations, where new unknowns are introduced as Lagrange multipliers (see, e.g., Chen, Du and Zou [21], Demkowicz and Vardapetyan [29,45] and the references therein). Other approaches consist in regularizing the formulation in Ω 0 by adding suitable terms contai...

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Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements

Cited by 139 publications

References 22 publications

Discrete compactness and the approximation of Maxwell's equations in $\mathbb{R}^3$

Discrete compactness and the approximation of Maxwell's equations in $\mathbb{R}^3$

Mixed discontinuous Galerkin approximation of the Maxwell operator: The indefinite case

The $hp$-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations

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