A nonconforming finite element method for fourth order curl equations in ℝ³

Zheng, Bin; Hu, Qiya; Xu, Jinchao

doi:10.1090/s0025-5718-2011-02480-4

Cited by 58 publications

(73 citation statements)

References 31 publications

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“…In the experiments, we find that there exist some superconvergence phenomena, which will be given a detailed proof in our future work. Besides, we are also interested in the H 2 (curl)-conforming finite elements in 3-D. 13,14,15,16,17,18,19,20 in the left and 1, 2, 3, 16,17,18,19,20,21 in the right are the node DOFs. Similarly, we can obtain the standard basis functions in the triangleK which is denoted as φ i = (φ 1 i , φ 2 i ) (i = 1, 2, · · · , 24).…”

Section: Resultsmentioning

confidence: 99%

H(curl$^2$)-Conforming Finite Elements in 2 Dimensions and Applications to the Quad-Curl Problem

Zhang¹,

Wang²,

Zhang³

2019

SIAM J. Sci. Comput.

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In this paper, we first construct the H 2 (curl)-conforming finite elements both on a rectangle and a triangle. They possess some fascinating properties which have been proven by a rigorous theoretical analysis. Then we apply the elements to construct a finite element space for discretizing quad-curl problems. Convergence orders O(h k ) in the H(curl) norm and O(h k−1 ) in the H 2 (curl) norm are established. Numerical experiments are provided to confirm our theoretical results. 2000 Mathematics Subject Classification. 65N30 and 35Q60 and 65N15 and 35B45.

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Section: Resultsmentioning

confidence: 99%

H(curl$^2$)-Conforming Finite Elements in 2 Dimensions and Applications to the Quad-Curl Problem

Zhang¹,

Wang²,

Zhang³

2019

SIAM J. Sci. Comput.

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“…One of the sequences connects the C 1 finite element space of Alfeld [1] to the inf-sup stable Stokes pair of Zhang [21] globally. This is done by introducing a new H 1 (curl)conforming finite element space that may be useful for fourth order curl problems [22]. Finally, we show that the complexes are exact on contractible domains.…”

Section: Introductionmentioning

confidence: 90%

Exact smooth piecewise polynomial sequences on Alfeld splits

Guzmán

Neilan

2020

Math. Comp.

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We develop exact piecewise polynomial sequences on Alfeld splits in any spatial dimension and any polynomial degree. An Alfeld split of a simplex is obtained by connecting the vertices of an n-simplex with its barycenter. We show that, on these triangulations, the kernel of the exterior derivative has enhanced smoothness. Byproducts of this theory include characterizations of discrete divergence-free subspaces for the Stokes problem, commutative projections, and simple formulas for the dimensions of smooth polynomial spaces.Hence, κω ∈ P r+1 Λ k−1 (T ).Suppose that ω = σ a σ dλ σ ∈ P r Λ k (T ) and suppose that f = [τ (0), τ (1), τ (2), . . . , τ (s)] is an ssimplex of T . Then the trace of ω on f is given bywhere tr f a σ := a σ | f is simply the restriction of a σ to f . We say that σ ⊂ τ if {σ(1), . . . , σ(k)} ⊂ {τ (0), τ (1), . . . , τ (s)}.We define the spaceThe following result is contained in [3, Theorem 4.8].We also need a result in the case r = 0. To do this, we first state a result from Arnold et al. [3, Lemma 4.6].Proposition 2.2. Let ω ∈ P r Λ k (T ). Suppose that tr Fi ω = 0, for 1 ≤ i ≤ n and T ω ∧ η, for all η ∈ P r−n+k Λ n−k (T ).Then, ω = 0. In particular, if ω ∈ P 0 Λ k (T ) with k ≤ n − 1 satisfies tr Fi ω = 0 for 1 ≤ i ≤ n, then ω = 0.Lemma 2.3. Define the set of k-simplices that have x 0 as a vertex:

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“…This model problem arises in many different applications, such as in the resistive magnetohydrodynamics (MHD) and in the inverse electromagnetic scattering theory. The resistive MHD system reads ( [17,34]): find the velocity u, the pressure p and the magnetic induction field B such that…”

Section: Introductionmentioning

confidence: 99%