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We introduce two a posteriori error estimators for Nédélec finite element discretizations of the curl-curl problem. These estimators pertain to a new Prager–Synge identity and an associated equilibration procedure. They are reliable and efficient, and the error estimates are polynomial-degree-robust. In addition, when the domain is convex, the reliability constants are fully computable. The proposed error estimators are also cheap and easy to implement, as they are computed by solving divergence-constrained minimization problems over edge patches. Numerical examples highlight our key findings, and show that both estimators are suited to drive adaptive refinement algorithms. Besides, these examples seem to indicate that guaranteed upper bounds can be achieved even in non-convex domains. [ABSTRACT FROM AUTHOR]

We study extensions of piecewise polynomial data prescribed in a patch of tetrahedra sharing an edge. We show 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 minimizers over the broken \boldsymbol H(\boldsymbol {\operatorname {curl}}) space with the same prescriptions. Our proofs are constructive and yield constants independent of the polynomial degree. We then detail the application of this result to the a posteriori error analysis of the curl–curl problem discretized with Nédélec finite elements of arbitrary order. The resulting estimators are reliable, locally efficient, polynomial-degree-robust, and inexpensive. They are constructed by a broken patchwise equilibration which, in particular, does not produce a globally \boldsymbol H(\boldsymbol {\operatorname {curl}})-conforming flux. The equilibration is only related to edge patches and can be realized without solutions of patch problems by a sweep through tetrahedra around every mesh edge. The error estimates become guaranteed when the regularity pick-up constant is explicitly known. Numerical experiments illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

A bound is obtained for the condition number of a BDDC algorithm for problems posed in H(curl) in two dimensions, where the subdomains are only assumed to be uniform in the sense of Peter Jones. For the primal variable space, a continuity constraint for the tangential average over each interior subdomain edge is imposed. For the averaging operator, a new technique named deluxe scaling is used. Our optimal bound is independent of jumps in the coefficients across the interface between the subdomains and depends only on a few geometric parameters of the decomposition. Numerical results that verify the result are shown, including some with subdomains with fractal edges and others obtained by a mesh partitioner. [ABSTRACT FROM AUTHOR]

We consider a standard Adaptive Edge Finite Element Method (AEFEM) based on arbitrary order Nédélec edge elements, for three-dimensional indefinite time-harmonic Maxwell equations. We prove that the AEFEM gives a contraction for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops provided the initial mesh is fine enough. Using the geometric decay, we show that the AEFEM yields the best possible decay rate of the error plus oscillation in terms of the number of degrees of freedom. The main technical contribution of the paper is the establishment of a quasi-orthogonality and a localized a posteriori error estimator. [ABSTRACT FROM AUTHOR]

We consider a standard Adaptive Edge Finite Element Method (AEFEM) based on arbitrary order Nedelec edge elements, for three-dimensional indefinite time-harmonic Maxwell equations. We prove that the AEFEM gives a contraction for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops provided the initial mesh is fine enough. Using the geometric decay, we show that the AEFEM yields the best possible decay rate of the error plus oscillation in terms of the number of degrees of freedom. The main technical contribution of the paper is the establishment of a quasi-orthogonality and a localized a posteriori error estimator. [ABSTRACT FROM AUTHOR]

The article discusses the development of an edge element adaptive strategy with error control for wave scattering by biperiodic structures. It determines the perfectly matched layer (PML) through sharp a posteriori error such as the thickness of the layer and the medium properties. Moreover, to illustrate the competitive behavior of the proposed adaptive method, numerical experiments are presented.

The article focuses on the analysis of the numerical solution of the time-harmonic Maxwell equations through a posterior error estimation technique. It states that a weak formulation for the error in each element solved with a finite element method was considered in obtaining local error estimates for the Maxwell equations. It mentions that the Maxwell equations has perfectly conducting boundaries discretized with Nédélec edge elements.

The article focuses on the modification conducted on equilibrated residual error estimator that is applicable to edge elements. It states the establishment of posteriori error estimators with like properties for edge elements and the equations of magnetostatics. It states the provision of ideas for acheiving estimates that does not have generic constants. Furthermore, the introduction of the distributional finite element de Rham sequence is mentioned.

The article focuses on the time harmonic electromagnetic scattering problem which was solved through an adaptive perfectly matched layer (PML) technique. It mentions the application of a posteriori error estimates to determine the PML parameters which was used for solving the electromagnetic scattering problem. It states that a complete numerical strategy was provided through the adaptive PML techniques.

The article focuses on proving the reliability of error estimator on Lipschitz domains to establish new error estimator for quasi-interpolation operators. It says that Maxwell equations are differential equations that describe electro-magnetic phenomena which numerical treatment by finite element methods is new. It mentions that Maxwell equations require vector valued function space for numerical analysis. Moreover, the reliability estimate was proven under Helmholtz decomposition's assumption.