Stable finite element pair for Stokes problem and discrete Stokes complex on quadrilateral grids

Zhang, Shuo

doi:10.1007/s00211-015-0749-y

Cited by 31 publications

(18 citation statements)

References 54 publications

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“…We instead figure out the exact connection between the finite element functions designed in this paper and the element in [9], and construct the approximation estimate in an indirect way. The exact connection can be viewed as a specialization of the one given in [17]. Moreover, this relation indicates an analogue correspondence with the exactness relation between the Crouzeix-Raviart element and the Morley element on triangular grids.…”

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confidence: 78%

Minimal consistent finite element space for the biharmonic equation on quadrilateral grids

Zhang

2019

IMA Journal of Numerical Analysis

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In this paper, a finite element space is presented on quadrilateral grids which can provide consistent discretization for the biharmonic equations. The space consists of piecewise quadratic polynomials and is of minimal degree for the variational problem. introductionIn the study of qualitative and numerical analysis of partial differential equations and, in general, of approximation theory, we are often interested in the approximation of functions in Sobolev spaces by piecewise polynomials defined on a partition of the domain. In order for simpler interior structure, lower degree polynomials are often expected to be used. It is of theoretical and practical interests whether and how a minimal-degree finite element space can be constructed for certain problems, namely, particularly, whether and how a consistent finite element space can be constructed for H m elliptic problem with m-th degree polynomials.When the subdivision consists of simplexes, a systematic family minimal-degree family of nonconforming finite elements has been proposed by Wang and Xu [14] for 2m-th order elliptic partial differential equations in R n for any n m with polynomials with degree m. Known as Wang-Xu or Morley-Wang-Xu family, the elements have been playing bigger and bigger role in numerical analysis. The elements are constructed based on the perfect matching between the dimension of m-th degree polynomials and the dimension of (n −k)-subsimplexes with 1 k m. The generalization to the cases n < m is attracting more and more research interests, c.f., e.g., [16].When the subdivision consists of geometrical shapes other than simplex, the problem is more complicated. The minimal conforming element spaces have been constructed for H m problem on R n rectangle grids by Hu and Zhang [7], where Q k polynomials are used for 2k-th order problems. Some low-degree rectangle elements have been designed, such as the rectangular Morley element and incomplete P 3 element for biharmonic equation. It remains open if the degrees can be further reduced for consistent nonconforming element spaces with minimal degree. Generally, the cells of 2000 Mathematics Subject Classification. Primary 65L60, 65M60, 65N30, 31A30.

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confidence: 78%

Minimal consistent finite element space for the biharmonic equation on quadrilateral grids

Zhang

2019

IMA Journal of Numerical Analysis

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“…The estimate falls into the category of standard techniques; c.f., e.g., [55,65,72]. Actually, since ũ ∈ (H 2 (Ω)) 2 and p ∈ H 1 (Ω), we have…”

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confidence: 99%

“…This way, the condition SC 3 is desirable in many applications. Besides, for pairs that satisfy SC 1 and SC 3, a discrete Stokes complex can usually be expected, the importance of which has been more and more clear; see [37,72] and the references therein for some examples. For linear elasticity problem, SC 3 can help avoid the reduced integration in the divergence part of the bilinear form.…”

Section: Introductionmentioning

confidence: 99%

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A conservative stable finite element method for Stokes flow and nearly incompressible linear elasticity on rectangular grid

Chen

Zhang

2017

Journal of Computational and Applied Mathematics

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In this paper, we discuss finite element methods for the incompressible Stokes problem and the nearly incompressible linear elasticity problem. Specifically, we present a finite element pair for the incompressible Stokes problem, which satisfies the discrete inf-sup condition and the discrete Korn's inequality, and moreover, which is element-wise conservative. The pair provides a lockingfree method for the nearly incompressible linear elasticity problem without reduced integration.2000 Mathematics Subject Classification. 65M60, 76M10.

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“…Just a very few works are concerned over general quadrilaterals. Zhang proposed a lowest order quadrilateral divergence‐free Stokes element with an H 1 ‐nonconforming structure. A conforming element was recently introduced by Neilan and Sap to achieve a second‐order approximation, where the spline element method (also known as macro or composite element method) was utilized.…”

Section: Introductionmentioning

confidence: 99%

A note on a lowest order divergence‐free Stokes element on quadrilaterals

Zhou

Meng

Fan

et al. 2019

Math Methods in App Sciences

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This short note reports a lowest order divergence‐free Stokes element on quadrilateral meshes. The velocity space is based on a P1 spline element over the crisscross partition of a quadrilateral, and the pressure is approximated by piecewise constant. For a given quadrilateral mesh, this element is stable if and only if the well‐known Q1‐P0 element is also stable. Although this method is a subspace method of Qin‐Zhang's P1‐P0 element, their velocity solutions are precisely equal. Moreover, an explicit basis representation is also provided. These theoretical findings are verified by numerical tests.

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Stable finite element pair for Stokes problem and discrete Stokes complex on quadrilateral grids

Cited by 31 publications

References 54 publications

Minimal consistent finite element space for the biharmonic equation on quadrilateral grids

Minimal consistent finite element space for the biharmonic equation on quadrilateral grids

A conservative stable finite element method for Stokes flow and nearly incompressible linear elasticity on rectangular grid

A note on a lowest order divergence‐free Stokes element on quadrilaterals

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