Stability of the finite elements 9/(4c+1) and 9/5c for stationary Stokes equations

Qin, Jinshui; Zhang, Shangyou

doi:10.1016/j.compstruc.2005.07.002

Cited by 11 publications

(7 citation statements)

References 12 publications

(29 reference statements)

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“…This idea was suggested in [12,13,20] for the lowest order Hood-Taylor element, where numerical evidence of the improvement was given (see also [9]). The proof of the stability of the enhanced lowest order Hood-Taylor scheme for triangular and rectangular meshes, can be found in [14,15,19]. In this paper we consider the enhancement of the generalized Hood-Taylor schemes (for triangular and tetrahedral meshes in two and three space dimensions, respectively), obtained by adding piecewise constant functions to the space of continuous piecewise polynomials of degree k (k ≥ 1 in 2D and k ≥ 2 in 3D) and prove that the inf-sup condition holds also in this case under the restriction that each element has at least one internal node.…”

Section: Introductionmentioning

confidence: 99%

Local Mass Conservation of Stokes Finite Elements

et al. 2011

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In this paper we discuss the stability of some Stokes finite elements. In particular, we consider a modification of Hood-Taylor and Bercovier-Pironneau schemes which consists in adding piecewise constant functions to the pressure space. This enhancement, which had been already used in the literature, is driven by the goal of achieving an improved mass conservation at element level. The main result consists in proving the inf-sup condition for the enhanced spaces in a general setting and to present some numerical tests which confirm the stability properties. The improvement in the local mass conservation is shown in a forthcoming paper (Boffi et al. In: Papadrakakis, M., Onate, E., Schrefler, B. (eds.) Coupled Problems 2011. Computational Methods for Coupled Problems in Science and Engineering IV, Cimne, 2011) where the presented schemes are used for the solution of a fluid-structure interaction problem.

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Section: Introductionmentioning

confidence: 99%

Local Mass Conservation of Stokes Finite Elements

et al. 2011

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“…Finally, we make a brief comparison between the new divergence-free element and the most popular Q 1 /P 0 element, cf. the references in [19]. The convergence data for the Q 1 /P 0 element are listed in Table 3.…”

Section: Numerical Testsmentioning

confidence: 99%

A lowest order divergence-free finite element on rectangular grids

Huang

Zhang

2011

Front. Math. China

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It is shown that the conforming Q 2,1;1,2 -Q 1 mixed element is stable, and provides optimal order of approximation for the Stokes equations on rectangular grids. Here, Q denotes the space of continuous piecewise-polynomials of degree 2 or less in the x direction but of degree 1 in the y direction. Q 1 is the space of discontinuous bilinear polynomials, with spurious modes filtered. To be precise, Q 1 is the divergence of the discrete velocity space Q 2,1;1,2 . Therefore, the resulting finite element solution for the velocity is divergence-free pointwise, when solving the Stokes equations. This element is the lowest order one in a family of divergence-free element, similar to the families of the Bernardi-Raugel element and the RaviartThomas element.

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“…As a result, mass conservation is guaranteed at the element level in the sense that the average of the divergence of the velocity over each element is zero. This enriched Taylor-Hood finite element approximation was first used by [10,11] on linear Stokes and Navier-Stokes problems and its stability was proven by [23,25,29] on triangular and rectangular meshes. The stability for the enriched Taylor-Hood element of even higher order accuracy on general meshes is proven in [6].…”

Section: Introductionmentioning

confidence: 98%