Cubic Lagrange elements satisfying exact incompressibility
The SMAI journal of computational mathematics, Volume 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.

Published online: 2018-11-19
Classification: 65N30,  65N12,  76D07,  65N85
     author = {Johnny Guzm\'an and L. Ridgway Scott},
     title = {Cubic Lagrange elements satisfying exact incompressibility},
     journal = {The SMAI journal of computational mathematics},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {4},
     year = {2018},
     pages = {345-374},
     doi = {10.5802/smai-jcm.38},
     language = {en},
Guzmán, Johnny; Scott, L. Ridgway. Cubic Lagrange elements satisfying exact incompressibility. The SMAI journal of computational mathematics, Volume 4 (2018) , pp. 345-374. doi : 10.5802/smai-jcm.38.

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

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

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

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

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

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

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

[8] Richard S Falk; Michael Neilan 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

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

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

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

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

[13] S. Harald Christiansen; K. Hu Generalized Finite Element Systems for smooth differential forms and Stokes problem, ArXiv e-prints (2016) | arXiv:1605.08657

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

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

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

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

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

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

[20] Jinshui Qin; Shangyou Zhang 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

[21] L. Ridgway Scott; Micheal Vogelius 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) (1985), pp. 221-244

[22] LR Scott; Michael Vogelius 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

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

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

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

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