Nonconforming finite element Stokes complexes in three dimensions

Huang, Xuehai

doi:10.48550/arxiv.2007.14068

Cited by 8 publications

(30 citation statements)

References 35 publications

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“…Discrete complexes in three dimensions. First recall a nonconforming discretization of the following Stokes complex in three dimensions [26] (5.1) 0GGGAH 1 0 (Ω)…”

Section: Lemma 41 [36] There Exists An Operator πmentioning

confidence: 99%

“…. The space of the shape functions of the H(grad curl) nonconforming element proposed in [26] is P 0 (K; R 3 ) ⊕ x ∧ P 1 (K; R 3 ), and the local degrees of freedom are given by e v • t e ds on each e ∈ E(K), (5.2)…”

Section: Lemma 41 [36] There Exists An Operator πmentioning

confidence: 99%

“…In the mean time, quad-curl operator also plays an important role in approximating the Maxwell transmission eigenvalue problem [32,10]. Recently, the designing of the approximations for quadcurl problems gain quite a few attentions from the finite element community ( [26,24,39,42,43,23,44,45]). The structures of the quad-curl problem are unique as the operator has a much bigger kernel than the one in the curl-curl problem.…”

mentioning

confidence: 99%

“…In [13,26], a novel way of further simplifying the structure of the quad-curl problem is proposed. The quad-curl problem is decoupled into three sub-problems, two curl-curl equations, and one Stokes equation, all of which have mature finite element approximation theories (e.g., [17,33,28,27]).…”

mentioning

confidence: 99%

“…Conforming finite element spaces for H 2 0 (curl, Ω) has been recently constructed in [42,24] for two dimensional dimensions and [43,23] in three dimensions. Nonconforming and low order finite element spaces can be found in [45,26].…”

mentioning

confidence: 99%

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Error analysis of a decoupled finite element method for quad-curl problems

Cao,

Chen,

Huang

2021

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Finite element approximation to a decoupled formulation for the quad-curl problem is studied in this paper. The difficulty of constructing elements with certain conformity to the quad-curl problems has been greatly reduced. For convex domains, where the regularity assumption holds for Stokes equation, the approximation to the curl of the true solution has quadratic order of convergence and first order for the energy norm. If the solution shows singularity, a posterior error estimator is developed and a separate marking adaptive finite element procedure is proposed, together with its convergence proved. Both a priori and a posteriori error analysis are supported by the numerical examples.

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“…Discrete complexes in three dimensions. First recall a nonconforming discretization of the following Stokes complex in three dimensions [26] (5.1) 0GGGAH 1 0 (Ω)…”

Section: Lemma 41 [36] There Exists An Operator πmentioning

confidence: 99%

Section: Lemma 41 [36] There Exists An Operator πmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Error analysis of a decoupled finite element method for quad-curl problems

Cao,

Chen,

Huang

2021

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An arbitrary-order fully discrete Stokes complex on general polyhedral meshes

Hanot¹

2021

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In this paper we present an arbitrary-order fully discrete Stokes complex on general polygonal meshes. Based upon the recent construction of the de Rham fully discrete complex [14] we extend it using the same principle. We complete it with other polynomial spaces related to vector calculus operators and to the Koszul complex required to accommodate the increased smoothness of the Stokes complex. This complex is especially well suited for problem involving Jacobian, divergence and curl, like e.g. the Stokes system or magnetohydrodynamics. We show a complete set of results on the novelties of this complex complementing those of [14]: exactness properties, uniform Poincaré inequalities and primal and adjoint consistency. We use our new complex on the Stokes system and validate the expected convergence rates with various numerical tests.

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Nonconforming finite elements for the Brinkman and $-\text{curl}Δ\text{curl}$ problems on cubical meshes

Zhang¹,

Zhang²,

Zhang³

2022

Preprint

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We propose two families of nonconforming elements on cubical meshes: one for the − curl ∆ curl problem and the other for the Brinkman problem. The element for the − curl ∆ curl problem is the first nonconforming element on cubical meshes. The element for the Brinkman problem can yield a uniformly stable finite element method with respect to the parameter ν. The lowest-order elements for the − curl ∆ curl and the Brinkman problems have 48 and 30 degrees of freedom, respectively. The two families of elements are subspaces of H(curl; Ω) and H(div; Ω), and they, as nonconforming approximation to H(grad curl; Ω) and [H 1 (Ω)] 3 , can form a discrete Stokes complex together with the Lagrange element and the L 2 element.2000 Mathematics Subject Classification. 65N30 and 35Q60 and 65N15 and 35B45. Key words and phrases. nonconforming elements, Brinkman problem, − curl ∆ curl problem, finite element de Rham complex, Stokes complex.

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Nonconforming finite element Stokes complexes in three dimensions

Cited by 8 publications

References 35 publications

Error analysis of a decoupled finite element method for quad-curl problems

Error analysis of a decoupled finite element method for quad-curl problems

An arbitrary-order fully discrete Stokes complex on general polyhedral meshes

Nonconforming finite elements for the Brinkman and $-\text{curl}Δ\text{curl}$ problems on cubical meshes

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