Cubic Lagrange elements satisfying exact incompressibility
The SMAI Journal of computational mathematics, Tome 4 (2018), pp. 345-374.

We prove that an analog of the Scott-Vogelius finite elements are inf-sup stable on certain nondegenerate meshes for piecewise cubic velocity fields. We also characterize the divergence of the velocity space on such meshes. In addition, we show how such a characterization relates to the dimension of C 1 piecewise quartics on the same mesh.

Publié le :
DOI : 10.5802/smai-jcm.38
Classification : 65N30, 65N12, 76D07, 65N85
Guzmán, Johnny 1 ; Scott, L. Ridgway 2

1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
2 Departments of Computer Science and Mathematics, Committee on Computational and Applied Mathematics, University of Chicago, Chicago IL 60637, USA
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     title = {Cubic {Lagrange} elements satisfying exact incompressibility},
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Guzmán, Johnny; Scott, L. Ridgway. Cubic Lagrange elements satisfying exact incompressibility. The SMAI Journal of computational mathematics, Tome 4 (2018), pp. 345-374. doi : 10.5802/smai-jcm.38. http://www.numdam.org/articles/10.5802/smai-jcm.38/

[1] Ahmed, Naveed; Linke, Alexander; Merdon, Christian Towards pressure-robust mixed methods for the incompressible Navier–Stokes equations, International Conference on Finite Volumes for Complex Applications, Springer (2017), pp. 351-359 | Zbl

[2] Alfeld, Peter; Piper, Bruce; Schumaker, Larry L. An explicit basis for C 1 quartic bivariate splines, SIAM Journal on Numerical Analysis, Volume 24 (1987) no. 4, pp. 891-911 | DOI | MR | Zbl

[3] Anbo, Le On the dimension of spaces of pp functions with boundary conditions, Approximation Theory and its Applications, Volume 5 (1989) no. 4, pp. 19-29 | MR | Zbl

[4] Arnold, Douglas N; Qin, Jinshui Quadratic velocity/linear pressure Stokes elements, Advances in computer methods for partial differential equations, Volume 7 (1992), pp. 28-34

[5] Bernardi, Christine; Raugel, Genevieve Analysis of some finite elements for the Stokes problem, Mathematics of Computation (1985), pp. 71-79 | DOI | MR | Zbl

[6] Brenner, Susanne C.; Scott, L. Ridgway The mathematical theory of finite element methods, 15, Springer Science & Business Media, 2008 | MR | Zbl

[7] Chui, C.K.; Schumaker, L.L. On spaces of piecewise polynomials with boundary conditions. II. Type-1 triangulations., Second Edmonton Conference on Approximation Theory (Ditzian, Zeev, ed.) (CMS Conf. Proc., 3), Amer. Math. Soc., Providence, R.I., 1983

[8] Falk, Richard S; Neilan, Michael Stokes complexes and the construction of stable finite elements with pointwise mass conservation, SIAM Journal on Numerical Analysis, Volume 51 (2013) no. 2, pp. 1308-1326 | DOI | MR | Zbl

[9] Guzmán, Johnny; Neilan, Michael Conforming and divergence-free Stokes elements in three dimensions, IMA Journal of Numerical Analysis, Volume 34 (2014) no. 4, pp. 1489-1508 | DOI | MR | Zbl

[10] Guzmán, Johnny; Neilan, Michael Conforming and divergence-free Stokes elements on general triangular meshes, Mathematics of Computation, Volume 83 (2014) no. 285, pp. 15-36 | DOI | MR | Zbl

[11] Guzman, Johnny; Neilan, Michael Inf-sup stable finite elements on barycentric refinements producing divergence–free approximations in arbitrary dimensions, arXiv preprint arXiv:1710.08044 (2017) | Zbl

[12] Guzmán, Johnny; Scott, L. Ridgway The Scott-Vogelius finite elements revisted, Mathematics of Computation, Volume to appear (2017) | Zbl

[13] Harald Christiansen, S.; Hu, K. Generalized Finite Element Systems for smooth differential forms and Stokes problem, ArXiv e-prints (2016) | arXiv

[14] John, Volker; Linke, Alexander; Merdon, Christian; Neilan, Michael; Rebholz, Leo G On the divergence constraint in mixed finite element methods for incompressible flows, SIAM Review (2016) | Zbl

[15] Lai, Ming-Jun; Schumaker, Larry L Spline functions on triangulations, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2007 no. 110 | Zbl

[16] Morgan, J.; Scott, L. R. A nodal basis for C 1 piecewise polynomials of degree n5, Math. Comp., Volume 29 (1975), pp. 736-740 | MR

[17] Morgan, John; Scott, L. R. The Dimension of the Space of C 1 Piecewise–Polynomials (1990) no. 78 (Research Report UH/MD)

[18] Neilan, Michael Discrete and conforming smooth de Rham complexes in three dimensions, Mathematics of Computation, Volume 84 (2015) no. 295, pp. 2059-2081 | DOI | MR | Zbl

[19] Qin, Jinshui On the convergence of some low order mixed finite elements for incompressible fluids, Penn State (1994) (Ph. D. Thesis) | MR

[20] Qin, Jinshui; Zhang, Shangyou Stability and approximability of the P1–P0 element for Stokes equations, International journal for numerical methods in fluids, Volume 54 (2007) no. 5, pp. 497-515 | MR | Zbl

[21] Scott, L. Ridgway; Vogelius, Micheal Conforming Finite Element Methods for Incompressible and Nearly Incompressible Continua., Large Scale Computations in Fluid Mechanics, B. E. Engquist, et al., eds., Volume 22 (Part 2), Providence: AMS (1985), pp. 221-244 | Zbl

[22] Scott, LR; Vogelius, Michael Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials, RAIRO-Modélisation mathématique et analyse numérique, Volume 19 (1985) no. 1, pp. 111-143 | Numdam | MR | Zbl

[23] Strang, Gilbert Piecewise polynomials and the finite element method, Bulletin of the American Mathematical Society, Volume 79 (1973) no. 6, pp. 1128-1137 | DOI | MR | Zbl

[24] Vogelius, Michael A right-inverse for the divergence operator in spaces of piecewise polynomials, Numerische Mathematik, Volume 41 (1983) no. 1, pp. 19-37 | DOI | MR | Zbl

[25] Zhang, Shangyou A new family of stable mixed finite elements for the 3d Stokes equations, Mathematics of computation, Volume 74 (2005) no. 250, pp. 543-554 | DOI | MR | Zbl

[26] Zhang, Shangyou Divergence-free finite elements on tetrahedral grids for k6, Mathematics of Computation, Volume 80 (2011) no. 274, pp. 669-695 | DOI | MR | Zbl

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