# An Equilibrated a Posteriori Error Estimator for Arbitrary-Order Nédélec Elements for Magnetostatic Problems

@article{Gedicke2020AnEA, title={An Equilibrated a Posteriori Error Estimator for Arbitrary-Order N{\'e}d{\'e}lec Elements for Magnetostatic Problems}, author={Joscha Gedicke and Sjoerd Geevers and Ilaria Perugia}, journal={Journal of Scientific Computing}, year={2020}, volume={83} }

We present a novel a posteriori error estimator for Nédélec elements for magnetostatic problems that is constant-free, i.e. it provides an upper bound on the error that does not involve a generic constant. The estimator is based on equilibration of the magnetic field and only involves small local problems that can be solved in parallel. Such an error estimator is already available for the lowest-degree Nédélec element (Braess and Schöberl in Math Comput 77(262):651-672, 2008) and requires…

## 7 Citations

Adaptive virtual element methods with equilibrated flux

- Mathematics, Computer ScienceArXiv
- 2020

An hp-adaptive virtual element method based on the hypercircle method of Prager and Synge for the approximation of solutions to diffusion problems is presented and a reliable and efficient a posteriori error estimator is introduced, computed by solving an auxiliary global mixed problem.

Stable broken H(curl) polynomial extensions and p-robust quasi-equilibrated a posteriori estimators for Maxwell's equations

- Computer Science, MathematicsArXiv
- 2020

Stability is shown in the sense that the minimizer over piecewise polynomial spaces with prescribed tangential component jumps across faces and prescribed piecewise curl in elements are subordinate in the broken energy norm to the minimizers over the broken H(curl) space with the same prescriptions.

A simple equilibration procedure leading to polynomial-degree-robust a posteriori error estimators for the curl-curl problem

- Computer Science, MathematicsArXiv
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Two a posteriori error estimators for Nédélec finite element discretizations of the curl–curl problem are introduced, which are reliable and efficient, and the error estimates are polynomial-degree-robust.

Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem

- Computer Science, MathematicsMath. Comput.
- 2022

Stability in the sense that the minimizers over piecewise polynomial spaces with prescribed tangential component jumps across faces and prescribed piecewise curl in elements are subordinate in the broken energy norm to the minimizer over the broken patchwise equilibration is shown.

$p$-robust equilibrated flux reconstruction in ${\boldsymbol H}(\mathrm{curl})$ based on local minimizations. Application to a posteriori analysis of the curl-curl problem

- Computer Science, MathematicsArXiv
- 2021

A divergence-free decomposition of a divergence- free H(div)-conforming piecewise polynomial, relying on over-constrained minimizations in Raviart–Thomas spaces, is the key ingredient.

A polynomial-degree-robust a posteriori error estimator for Nédélec discretizations of magnetostatic problems

- Computer Science, MathematicsSIAM J. Numer. Anal.
- 2021

An equilibration-based a posteriori error estimator for Nedelec element discretizations of the magnetostatic problem is proven to be reliable, with reliability constant 1, and efficient, with an efficiency constant that is independent of the polynomial degree of the approximation.

Adaptive virtual element methods with equilibrated fluxes

- Mathematics, Computer ScienceApplied Numerical Mathematics
- 2021

An hp-adaptive virtual element method based on the hypercircle method of Prager and Synge for the approximation of solutions to diffusion problems is presented and a reliable and efficient a posteriori error estimator is introduced, computed by solving an auxiliary global mixed problem.

## References

SHOWING 1-10 OF 28 REFERENCES

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

- Mathematics
- 2016

Abstract 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…

Two guaranteed equilibrated error estimators for Harmonic formulations in eddy current problems

- Mathematics, Computer ScienceComput. Math. Appl.
- 2019

A comparison of this estimator with an equilibrated error estimator already developed through a complementary problem points out the advantages and drawbacks of these two estimators.

A guaranteed equilibrated error estimator for the $\mathbf{A}-\varphi$ and $\mathbf{T}-\Omega$ magnetodynamic harmonic formulations of the Maxwell system

- Mathematics
- 2016

In this paper, a guaranteed equilibrated error estimator is proposed for the harmonic magnetodynamic formulation of the Maxwell's system. This system is recast in two classical potential…

A guaranteed equilibrated error estimator for the A – Φ and T – Ω magnetodynamic harmonic formulations of the Maxwell system

- 2018

In this paper, a guaranteed equilibrated error estimator is proposed for the harmonic magnetodynamic formulation of the Maxwell’s system. This system is recast in two classical potential…

Residual and equilibrated error estimators for magnetostatic problems solved by finite element method

- MathematicsIEEE Transactions on Magnetics
- 2013

In finite element computations, the choice of the mesh is crucial to obtain accurate solutions. In order to evaluate the quality of the mesh, a posteriori error estimators can be used. In this paper,…

Residual based a posteriori error estimators for eddy current computation

- Mathematics
- 2000

We consider H (curl ;Ω)-elliptic problems that have been discretized by means of Nedelec's edge elements on tetrahedral meshes. Such problems occur in the numerical computation of eddy currents. From…

Equilibrated residual error estimator for edge elements

- Computer Science, MathematicsMath. Comput.
- 2008

This work simplifies and modify the equilibration of Raviart-Thomas elements such that it can be applied to the curl-curl equation and edge elements and extended in the spirit of distributions.

Guaranteed Error Bounds for Conforming Approximations of a Maxwell Type Problem

- Mathematics
- 2010

This paper is concerned with computable error estimates for approximations to a boundary-value problem
$$\mathrm{curl}\ {\mu }^{-1}\mathrm{curl}\ u + {\kappa }^{2}u = j\quad \textrm{ in }\Omega…

Hierarchical Error Estimator for Eddy Current Computation

- Mathematics
- 2000

We consider the quasi-magnetostatic eddy current problem discretized by means of lowest order -conforming finite elements (edge elements) on tetrahedral meshes. Bounds for the discretization error in…

About the gauge conditions arising in Finite Element magnetostatic problems

- Mathematics, Computer ScienceComput. Math. Appl.
- 2019

This paper deals with some magnetostatic models considered in vector potential formulations and solved by a Finite Element solver, and shows the equivalence between some of these choices for several kinds of boundary conditions.