Convergence of adaptive edge finite element methods for H(curl)‐elliptic problems

Zhong, Liuqiang; Shu, Shi; Chen, Long; Xu, Jinchao

doi:10.1002/nla.694

Cited by 16 publications

(14 citation statements)

References 22 publications

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“…Zhong et al . 13 prove convergence of the adaptive edge finite element method for three‐dimensional H (curl)‐elliptic problems discretized with Nédélec edge elements (first or second families) of any order. The complexity of the method is shown numerically to be quasi‐optimal.…”

mentioning

confidence: 94%

Multigrid methods

Falgout¹

2010

Numerical Linear Algebra App

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confidence: 94%

Multigrid methods

Falgout¹

2010

Numerical Linear Algebra App

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“…The relevant research in the case of Maxwell's equations, which dated back to Beck et al [10], has also reached a mature level nowadays (cf. [34,53,67,68]). However, the theory of AFEMs for PDE-constrained optimization problems is still a work in progress.…”

Section: Introductionmentioning

confidence: 99%

An adaptive edge element method and its convergence for an electromagnetic constrained optimal control problem

Zou

2021

ESAIM: M2AN

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In this work, an adaptive edge element method is developed for an H(curl)-elliptic constrained optimal control problem. We use the lowest-order Nedelec’s edge elements of first family and the piecewise (element-wise) constant functions to approximate the state and the control, respectively, and propose a new adaptive algorithm with error estimators involving both residual-type error estimators and lower-order data oscillations. By using a local regular decomposition for H(curl)-functions and the standard bubble function techniques, we derive the a posteriori error estimates for the proposed error estimators. Then we exploit the convergence properties of the orthogonal L^2- projections and the mesh-size functions to demonstrate that the sequences of the discrete states and controls generated by the adaptive algorithm converge strongly to the exact solutions of the state and control in the energy norm and L^2 -norm, respectively, by first achieving the strong convergence towards the solution to a limiting control problem. Three-dimensional numerical experiments are also presented to confirm our theoretical results and the quasi-optimality of the adaptive edge element method.

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“…One of the key technical tools, a Helmholtz decomposition, used in this proving mechanism, relies on f being in H(div), and fails if f / ∈ H(div). In [12], the assumption that f ∈ H(div) is weakened to f being in the piecewise H(div) space with respect to the triangulation, at the same time, the divergence residual and norm jump are modified to incorporate this relaxation. Another drawback of using Helmholtz decomposition on the error is that it introduces the assumption of the coefficients' quasi-monotonicity into the proof pipeline.…”

Section: Introductionmentioning

confidence: 99%

Robust a posteriori error estimation for finite element approximation to (curl) problem

Cai

Cao

Falgout

2016

Computer Methods in Applied Mechanics and Engineering

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In this paper, we introduce a novel a posteriori error estimator for the conforming finite element approximation to the H(curl) problem with inhomogeneous media and with the right-hand side only in L 2 . The estimator is of the recovery type. Independent with the current approximation to the primary variable (the electric field), an auxiliary variable (the magnetizing field) is recovered in parallel by solving a similar H(curl) problem. An alternate way of recovery is presented as well by localizing the error flux. The estimator is then defined as the sum of the modified element residual and the residual of the constitutive equation defining the auxiliary variable. It is proved that the estimator is approximately equal to the true error in the energy norm without the quasi-monotonicity assumption. Finally, we present numerical results for several H(curl) interface problems.

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Convergence of adaptive edge finite element methods for H(curl)‐elliptic problems

Cited by 16 publications

References 22 publications

Multigrid methods

Multigrid methods

An adaptive edge element method and its convergence for an electromagnetic constrained optimal control problem

Robust a posteriori error estimation for finite element approximation to (curl) problem

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