hp-adaptive finite elements in electromagnetics

Vardapetyan, L.; Demkowicz, Leszek

doi:10.1016/s0045-7825(98)00161-3

Cited by 67 publications

(43 citation statements)

References 21 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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“…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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“…And how to rescale E 3 ? Following the logic of our previous papers [14] and [33], we keep (3.10) 3 , which explicitly involves a weak divergence condition on E , and turn it into a constraint on (E , E 3 ) by rescaling E 3 as follows:…”

Section: Theoremmentioning

confidence: 99%

“…Since ω 2 is not in the spectrum of ∇ × 1 µ ∇×, the bounded sesquilinear form associated with the system (4.26) satisfies the inf − sup conditions; see for example [14] and [33], where the stability constant is computed. Therefore, a unique solution to this system exists and is bounded by (G + J ), (…”

Section: Analysis Of the Continuous Problemmentioning

confidence: 99%

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Full-wave analysis of dielectric waveguides at a given frequency

Vardapetyan

Demkowicz

2002

Math. Comp.

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Abstract. New variational formulation to compute propagation constants is proposed. Based on it, vector finite element method is proved to exclude spurious modes provided finite elements possess discrete compactness property. Convergence analysis is conducted in the framework of collectively compact operators. Reported theoretical results apply to a wide class of vector finite elements including two families of Nedelec and their generalization, the hpedge elements. Numerical experiments fully support theoretical estimates for convergence rates.

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hp-adaptive finite elements in electromagnetics

Cited by 67 publications

References 21 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

Full-wave analysis of dielectric waveguides at a given frequency

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