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

Abstract

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.

Fully H(gradcurl)-nonconforming Finite Element Method for The Singularly Perturbed Quad-curl Problem on Cubical Meshes

“…Then estimate (5.10) follows from [21,Theorem 3.3]. From integration by parts and the fact that p = 0, we get…”

“…Therefore, we can glue together all p K for K ∈ T h to get a function p h ∈ S r h (T h ). The exactness at Q h (T h ) follows from the exactness at the same position of smooth Stokes complex and the commutativity of interpolations and div h (Lemma 4.1) [21].…”

“…Hence, Huang and Zhang in [10] developed two families of H(grad curl)-nonconforming but H(curl)-conforming finite elements on tetrahedral meshes, which can solve problem (1.1) when tends to zero, see also [19]. On cubical meshes, one of the authors and her collaborators proposed a family of nonconforming finite elements with at least 48 DOFs in [21]. This family is also H(curl)-conforming, and hence can solve problem (1.1) as vanishes.…”