Stokes Complexes and the Construction of Stable Finite Elements with Pointwise Mass Conservation

Falk, Richard S.; Neilan, Michael

doi:10.1137/120888132

Cited by 163 publications

(133 citation statements)

References 24 publications

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“…Here, the authors showed that the pair P k − P dc k−1 is stable in two dimensions on simplicial triangulations provided the polynomial degree satisfies k ≥ 4 and if the triangulation does not contain singular vertices. These results have since been expanded in [13,16]. Similar to the simplicial case, the construction of Stokes pairs yielding divergence-free approximations on Cartesian meshes is mostly limited to the two dimensional case [4,17,26].…”

Section: 4)mentioning

confidence: 99%

“…As pointed out in [13], imposing boundary conditions of finite element spaces while preserving the surjectivity of the divergence is a non-trivial issue. For example, if v is a globally continuous function on Ω and vanishes on ∂Ω, then the derivatives of v vanish at corners (n = 2) and edges (n = 3) of ∂Ω.…”

Section: Global Finite Element Spaces With Imposed Boundary Conditionsmentioning

confidence: 99%

“…The construction of our finite element pairs is motivated by a smooth de Rham complex (or Stokes complex [13]). In two dimensions, this complex is given by the sequence…”

Section: Introductionmentioning

confidence: 99%

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Stokes elements on cubic meshes yielding divergence-free approximations

Neilan

Sap

2015

Calcolo

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Conforming piecewise polynomial spaces with respect to cubic meshes are constructed for the Stokes problem in arbitrary dimensions yielding exactly divergence-free velocity approximations. The derivation of the finite element pair is motivated by a smooth de Rham complex that is well-suited for the Stokes problem. We derive the stability and convergence properties of the new elements as well as the construction of reduced elements with less global unknowns.

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Section: 4)mentioning

confidence: 99%

Section: Global Finite Element Spaces With Imposed Boundary Conditionsmentioning

confidence: 99%

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Stokes elements on cubic meshes yielding divergence-free approximations

Neilan

Sap

2015

Calcolo

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“…Due to the work of, e.g., Arnold, Qin, Zhang, Neilan, Guzman, and Falk [1][2][3][4][5], several choices such as Scott-Vogelius elements are now available that are LBB stable (sometimes requiring a mesh condition), are sufficiently low degree so as to be practical, provide ∥∇ · u h ∥ = 0 for the discrete velocity solution, and also deliver optimal asymptotic accuracy of the velocity in the energy norm. We refer to such mixed finite elements as divergence-free elements.…”

Section: Introductionmentioning

confidence: 99%

A note on the importance of mass conservation in long-time stability of Navier–Stokes simulations using finite elements

Belenli

Rebholz

Tone

2015

Applied Mathematics Letters

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a b s t r a c tWe prove a long-time stability result for the finite element in space, linear extrapolated Crank-Nicolson in time discretization of the Navier-Stokes equations (NSE). From this result and a numerical experiment, we show the importance of discrete mass conservation in long-time simulations of the NSE. That is, we show that using elements that strongly enforce mass conservation can provide significantly more accurate solutions over long times, compared to those that enforce it weakly.

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“…There are several other such divergence-free finite elements, cf. [2,14,16,17,19,20,[29][30][31]33].…”

mentioning

confidence: 99%

Supercloseness of the Divergence-Free Finite Element Solutions on Rectangular Grids

Huang

Zhang

2013

Commun. Math. Stat.

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By the standard theory, the stable Q k+1,k − Q k,k+1 /Q dc k divergence-free element converges with the optimal order of approximation for the Stokes equations, but only order k for the velocity in H 1 -norm and the pressure in L 2 -norm. This is due to one polynomial degree less in y direction for the first component of velocity, which is a Q k+1,k polynomial of x and y. In this manuscript, we will show by supercloseness of the divergence free element that the order of convergence is truly k + 1, for both velocity and pressure. For special solutions (if the interpolation is also divergence-free), a two-order supercloseness is shown to exist. Numerical tests are provided confirming the accuracy of the theory.

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Stokes Complexes and the Construction of Stable Finite Elements with Pointwise Mass Conservation

Cited by 163 publications

References 24 publications

Stokes elements on cubic meshes yielding divergence-free approximations

Stokes elements on cubic meshes yielding divergence-free approximations

A note on the importance of mass conservation in long-time stability of Navier–Stokes simulations using finite elements

Supercloseness of the Divergence-Free Finite Element Solutions on Rectangular Grids

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