A new error analysis of a mixed finite element method for the quad-curl problem

Wang, Chao; Sun, Zhengjia; Cui, Jintao

doi:10.1016/j.amc.2018.12.027

Cited by 12 publications

(6 citation statements)

References 27 publications

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“…In [23], we also studied a mixed finite element method for this example. In order to provide a comparison in terms of efficiency, we plot the convergent error of u − u h H(curl,Ω) / u H(curl,Ω) with computational time for the C 0 -IP method and the mixed method in Figure 2.…”

Section: Numerical Resultsmentioning

confidence: 99%

A Quadratic C0 Interior Penalty Method for the Quad-Curl Problem

Sun

Gao

Wang

et al. 2020

Mathematical Modelling and Analysis

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In this paper we study the C0 interior penalty method for a quad-curl problem arising from magnetohydrodynamics model on bounded polygons or polyhedrons. We prove the well-posedness of the numerical scheme and then derive the optimal error estimates in a discrete energy norm. A post-processing procedure that can produce C1 approximations is also presented. The performance of the method is illustrated by numerical experiments.

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Section: Numerical Resultsmentioning

confidence: 99%

A Quadratic C0 Interior Penalty Method for the Quad-Curl Problem

Sun

Gao

Wang

et al. 2020

Mathematical Modelling and Analysis

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“…The corresponding quad-curl eigenvalue problem plays a fundamental role in the analysis and computation of the electromagnetic interior transmission eigenvalues [27,32,33]. Various numerical methods have been proposed for the quad-curl source problem, see, e.g., [6,7,13,22,35,36,38,41,42,45]. However, there exist only a few results on the numerical methods for the quad-curl eigenvalue problem.…”

Section: Introductionmentioning

confidence: 99%

A priori and a posteriori error estimates for the quad-curl eigenvalue problem

et al. 2022

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Abstract. In this paper, we consider a priori and a posteriori error estimates of the H (curl 2 )-conforming finite element when solving the quad-curl eigenvalue problem. An a priori estimate of eigenvalues with convergence order 2( s −1) is obtained if the corresponding eigenvector u ∈ H s −1 (Ω) and ∇ × u ∈ H s (Ω). For the a posteriori estimate, by analyzing the associated source problem, we obtain lower and upper bounds for the errors of eigenvectors in the energy norm and upper bounds for the errors of eigenvalues. Numerical examples are presented for validation.

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“…In contrast to second-order curl problems, limited work has been done on numerical methods for quad-curl problems. Initially, numerical methods with various nonconformity/mix techniques, such as nonconforming finite element methods [32], discontinuous Galerkin methods [10], weak Galerkin methods [21], mixed finite element methods [19,20,14,25,31,30,23], and the Hodge decomposition method [4,3], were proposed to solve quad-curl problems as well as their related eigenvalue problems. Indeed, H(curl 2 )-conforming methods were unavailable for quad-curl problems until recently.…”

Section: Introductionmentioning

confidence: 99%

H($\text{curl}^2$)-conforming quadrilateral spectral element method for quad-curl problems

Wang¹,

Shan²,

Li³

et al. 2021

Preprint

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In this paper, we propose an H(curl 2 )-conforming quadrilateral spectral element method to solve quadcurl problems. Starting with generalized Jacobi polynomials, we first introduce quasi-orthogonal polynomial systems for vector fields over rectangles. H(curl 2 )-conforming elements over arbitrary convex quadrilaterals are then constructed explicitly in a hierarchical pattern using the contravariant transform together with the bilinear mapping from the reference square onto each quadrilateral. It is astonishing that both the simplest rectangular and quadrilateral spectral elements possess only 8 degrees of freedom on each physical element. In the sequel, we propose our H(curl 2 )-conforming quadrilateral spectral element approximation based on the mixed weak formulation to solve the quad-curl equation and its eigenvalue problem. Numerical results show the effectiveness and efficiency of our method.

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A new error analysis of a mixed finite element method for the quad-curl problem

Cited by 12 publications

References 27 publications

A Quadratic C0 Interior Penalty Method for the Quad-Curl Problem

A Quadratic C0 Interior Penalty Method for the Quad-Curl Problem

A priori and a posteriori error estimates for the quad-curl eigenvalue problem

H($\text{curl}^2$)-conforming quadrilateral spectral element method for quad-curl problems

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