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

Wang, Lixiu; Zhang, Qian; Sun, Jiguang; Zhang, Zhimin

doi:10.1051/m2an/2022027

Cited by 7 publications

(3 citation statements)

References 47 publications

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“…As a consequence, the superconvergent result can be used to construct an asymptotically exact a posteriori error estimator for the lowest order curl-curl element. It should be pointed out that although there is a residual-type a posteriori error estimator for the quad-curl problem [17], to our knowledge, there is no result for the recovery-type a posteriori error estimator for the problem. Therefore, the superconvergence result and recovery-type a posteriori error estimator provided in this paper are new for the quad-curl problem.…”

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confidence: 88%

Polynomial preserving recovery and a posteriori error estimates for the two-dimensional quad-curl problem

Zhang

2023

DCDS-B

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We analyze superconvergence property of the lowest order curl-curl conforming finite element method based on polynomial preserving recovery (PPR) for the two-dimensional quad-curl problem on triangular meshes. We observe that the linear interpolation of <inline-formula><tex-math id="M1">\begin{document}$ \nabla \times \boldsymbol u_h $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M2">\begin{document}$ \boldsymbol u_h $\end{document}</tex-math></inline-formula> is the numerical solution) can be written as a linear combination of solutions of two discrete Poisson equations obtained by the usual linear finite element method. Therefore, the superconvergence analysis of the quad-curl problem can be attributed to the analysis of the Poisson equation. Then, with the help of the existing superconvergence results for the Poisson equation, we prove that recovered <inline-formula><tex-math id="M3">\begin{document}$ \nabla \times \nabla \times \boldsymbol u_h $\end{document}</tex-math></inline-formula> (by applying PPR to <inline-formula><tex-math id="M4">\begin{document}$ \nabla \times \boldsymbol u_h $\end{document}</tex-math></inline-formula>) is superconvergent to <inline-formula><tex-math id="M5">\begin{document}$ \nabla \times \nabla \times \boldsymbol u $\end{document}</tex-math></inline-formula>. Based on this superconvergent result, we derive an asymptotically exact a posteriori error estimator. Numerical tests are provided to demonstrate effectiveness of the proposed method and confirm our theoretical findings.

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confidence: 88%

Polynomial preserving recovery and a posteriori error estimates for the two-dimensional quad-curl problem

Zhang

2023

DCDS-B

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“…In addition, even for the lowest order H(curl 2 )-conforming element, the construction of the Lagrange type basis functions are very complicated, which prevents the implement for higher-order elements. In another recent work [26], in order to solve quad-curl eigenvalue problems, a family of H(curl 2 )-conforming finite elements over triangles are constructed using complete polynomials of total degree ≥ 4 with at least 30 DOFs on each element. More recently, by introducing continuous and discrete de Rham complexes with high order Sobolev spaces, Hu et al discovered in [11] that the simplest rectangular finite element possess only 8 DOFs.…”

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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“…Many numerical methods have been proposed for problems involving those two fourth-order operators. We refer to [3,20,[29][30][31][32] for conforming finite element methods and [8,10,11,19,24,[26][27][28][33][34][35] for other methods. However, unlike the biharmonic operator and the quad-curl operator, very limited work has been done…”

Section: Introductionmentioning

confidence: 99%

Three families of grad-div-conforming finite elements

Zhang¹,

Zhang²

2020

Preprint

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Several smooth finite element de Rham complexes are constructed in threedimensional space, which yield three families of grad-div conforming finite elements. The simplest element has only 8 degrees of freedom (DOFs) for a tetrahedron and 14 DOFs for a cuboid. These elements naturally lead to conforming approximations to quad-div problems. Numerical experiments for each family validate the correctness and efficiency of the elements for solving the quad-div problem.

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

Cited by 7 publications

References 47 publications

Polynomial preserving recovery and a posteriori error estimates for the two-dimensional quad-curl problem

Polynomial preserving recovery and a posteriori error estimates for the two-dimensional quad-curl problem

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

Three families of grad-div-conforming finite elements

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