Conforming and divergence-free Stokes elements on general triangular meshes

Guzmán, Johnny; Neilan, Michael

doi:10.1090/S0025-5718-2013-02753-6

Conforming and divergence-free Stokes elements on general triangular meshes
HTML articles powered by AMS MathViewer

by Johnny Guzmán and Michael Neilan PDF

Math. Comp. 83 (2014), 15-36 Request permission

Abstract:

We present a family of conforming finite elements for the Stokes problem on general triangular meshes in two dimensions. The lowest order case consists of enriched piecewise linear polynomials for the velocity and piecewise constant polynomials for the pressure. We show that the elements satisfy the inf-sup condition and converges with order $k$ for both the velocity and pressure. Moreover, the pressure space is exactly the divergence of the corresponding space for the velocity. Therefore the discretely divergence-free functions are divergence-free pointwise. We also show how the proposed elements are related to a class of $C^1$ elements through the use of a discrete de Rham complex.

References

D. N. Arnold, F. Brezzi, and M. Fortin, A stable finite element for the Stokes equations, Calcolo 21 (1984), no. 4, 337–344 (1985). MR 799997, DOI 10.1007/BF02576171
D. N. Arnold and J. Qin, Quadratic velocity/linear pressure Stokes elements, in Advances in Computer Methods for Partial Differential Equations VII, eds. R. Vichnevetsky and R.S. Steplemen, 1992.
Douglas N. Arnold and Ragnar Winther, Mixed finite elements for elasticity, Numer. Math. 92 (2002), no. 3, 401–419. MR 1930384, DOI 10.1007/s002110100348
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 2, 281–354. MR 2594630, DOI 10.1090/S0273-0979-10-01278-4
F. Auricchio, L. Beirão da Veiga, C. Lovadina, and A. Reali, The importance of the exact satisfaction of the incompressibility constraint in nonlinear elasticity: mixed FEMs versus NURBS-based approximations, Comput. Methods Appl. Mech. Engrg. 199 (2010), no. 5-8, 314–323. MR 2576764, DOI 10.1016/j.cma.2008.06.004
Christine Bernardi and Geneviève Raugel, Analysis of some finite elements for the Stokes problem, Math. Comp. 44 (1985), no. 169, 71–79. MR 771031, DOI 10.1090/S0025-5718-1985-0771031-7
Daniele Boffi, Franco Brezzi, Leszek F. Demkowicz, Ricardo G. Durán, Richard S. Falk, and Michel Fortin, Mixed finite elements, compatibility conditions, and applications, Lecture Notes in Mathematics, vol. 1939, Springer-Verlag, Berlin; Fondazione C.I.M.E., Florence, 2008. Lectures given at the C.I.M.E. Summer School held in Cetraro, June 26–July 1, 2006; Edited by Boffi and Lucia Gastaldi. MR 2459075, DOI 10.1007/978-3-540-78319-0
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
A. Buffa, C. de Falco, and G. Sangalli, IsoGeometric Analysis: stable elements for the 2D Stokes equation, Internat. J. Numer. Methods Fluids 65 (2011), no. 11-12, 1407–1422. MR 2808112, DOI 10.1002/fld.2337
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 33–75. MR 343661
M. Fortin and M. Soulie, A nonconforming piecewise quadratic finite element on triangles, Internat. J. Numer. Methods Engrg. 19 (1983), no. 4, 505–520. MR 702056, DOI 10.1002/nme.1620190405
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
Johnny Guzmán and Michael Neilan, A family of nonconforming elements for the Brinkman problem, IMA J. Numer. Anal. 32 (2012), no. 4, 1484–1508. MR 2991835, DOI 10.1093/imanum/drr040
Yunqing Huang and Shangyou Zhang, A lowest order divergence-free finite element on rectangular grids, Front. Math. China 6 (2011), no. 2, 253–270. MR 2780891, DOI 10.1007/s11464-011-0094-0
Alexander Linke, Collision in a cross-shaped domain—a steady 2d Navier-Stokes example demonstrating the importance of mass conservation in CFD, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 41-44, 3278–3286. MR 2571343, DOI 10.1016/j.cma.2009.06.016
Kent Andre Mardal, Xue-Cheng Tai, and Ragnar Winther, A robust finite element method for Darcy-Stokes flow, SIAM J. Numer. Anal. 40 (2002), no. 5, 1605–1631. MR 1950614, DOI 10.1137/S0036142901383910
J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
J.-C. Nédélec, A new family of mixed finite elements in $\textbf {R}^3$, Numer. Math. 50 (1986), no. 1, 57–81. MR 864305, DOI 10.1007/BF01389668
Jinshui Qin and Shangyou Zhang, Stability and approximability of the $\scr P_1$-$\scr P_0$ element for Stokes equations, Internat. J. Numer. Methods Fluids 54 (2007), no. 5, 497–515. MR 2322456, DOI 10.1002/fld.1407
L. R. Scott and M. Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 1, 111–143 (English, with French summary). MR 813691, DOI 10.1051/m2an/1985190101111
L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
Xue-Cheng Tai and Ragnar Winther, A discrete de Rham complex with enhanced smoothness, Calcolo 43 (2006), no. 4, 287–306. MR 2283095, DOI 10.1007/s10092-006-0124-6
C. Taylor and P. Hood, A numerical solution of the Navier-Stokes equations using the finite element technique, Internat. J. Comput. & Fluids 1 (1973), no. 1, 73–100. MR 339677, DOI 10.1016/0045-7930(73)90027-3
Xiaoping Xie, Jinchao Xu, and Guangri Xue, Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models, J. Comput. Math. 26 (2008), no. 3, 437–455. MR 2421892
Shangyou Zhang, On the P1 Powell-Sabin divergence-free finite element for the Stokes equations, J. Comput. Math. 26 (2008), no. 3, 456–470. MR 2421893
Shangyou Zhang, A family of $Q_{k+1,k}\times Q_{k,k+1}$ divergence-free finite elements on rectangular grids, SIAM J. Numer. Anal. 47 (2009), no. 3, 2090–2107. MR 2519595, DOI 10.1137/080728949
Shangyou Zhang, Quadratic divergence-free finite elements on Powell-Sabin tetrahedral grids, Calcolo 48 (2011), no. 3, 211–244. MR 2827006, DOI 10.1007/s10092-010-0035-4
O. C. Zienkiewicz, The finite element method in engineering science, McGraw-Hill, London-New York-Düsseldorf, 1971. The second, expanded and revised, edition of The finite element method in structural and continuum mechanics. MR 0315970