Abstract
The linear nonconforming finite element, combined with constant finite element for pressure, is stable for the Stokes problem. But it does not satisfy the discrete Korn inequality. The linear conforming finite element satisfies the discrete Korn inequality, but is not stable for the Stokes problem and fails for the nearly incompressible elasticity problems. We enrich the linear conforming finite element by some nonconforming P1 bubbles, i.e., select a subspace of the linear nonconforming finite element space, so that the resulting linear nonconforming element is both stable and conforming enough to satisfy the Korn inequality, on HTC-type triangular and tetrahedral grids. Numerical tests in 2D and 3D are presented, confirming the analysis.
Acknowledgment
The author thanks an anonymous reviewer who suggested using the Nedelec edge element basis to prove Lemma 3.1.
References
[1] D. N. Arnold and J. Qin, Quadratic velocity/linear pressure Stokes elements. In: Advances in Computer Methods for Partial Differential Equations, Vol. VII, (Eds. R. Vichnevetsky and R. S. Steplemen), 1992, pp. 28–34.Search in Google Scholar
[2] C. Bacuta, P. Vassilevski, and S. Zhang, A new approach for solving Stokes systems arising from a distributive relaxation method, Numer. Methods Partial Differ. Equ., 27 (2011), No. 4, 898–914.10.1002/num.20560Search in Google Scholar
[3] S. C. Brenner, K. Wang, J. Zhao, Poincaré–Friedrichs inequalities for piecewise H2 functions, Numer. Funct. Anal. Optim., 25 (2004), No. 5-6, 463–478.10.1081/NFA-200042165Search in Google Scholar
[4] S. C. Brenner, Korn’s inequalities for piecewise H1 vector fields, Math. Comp., 73 (2004), No. 247, 1067–1087.10.1090/S0025-5718-03-01579-5Search in Google Scholar
[5] M. Crouzeix and P. A. Raviart, Conforming and nonconforming finite elements for solving the stationary Stokes equations, I. In: Rev. Francaise Automat. Informat. Recherche Operationnelle Ser. Rouge, 7 (1973), 33–75.10.1051/m2an/197307R300331Search in Google Scholar
[6] M. Fabien, J. Guzmán, M. Neilan, and A. Zytoon, Low-order divergence-free approximations for the Stokes problem on Worsey– Farin and Powell–Sabin splits, Comput. Methods Appl. Mech. Engrg., 390 (2022), Paper 114-444.10.1016/j.cma.2021.114444Search in Google Scholar
[7] R. S. Falk, Nonconforming finite element methods for the equations of linear elasticity, Math. Comp., 57 (1991), No. 196, 529–550.10.1090/S0025-5718-1991-1094947-6Search in Google Scholar
[8] R. S. Falk and M. Neilan, Stokes complexes and the construction of stable finite elements with pointwise mass conservation, SIAM J. Numer. Anal., 51 (2013), No. 2, 1308–1326.10.1137/120888132Search in Google Scholar
[9] G. Fu, J. Guzmán, and M. Neilan, Exact smooth piecewise polynomial sequences on Alfeld splits, Math. Comp., 89 (2020), No. 323, 1059–1091.10.1090/mcom/3520Search in Google Scholar
[10] J. Guzmán and M. Neilan, Conforming and divergence-free Stokes elements on general triangular meshes, Math. Comp., 83 (2014), No. 285, 15–36.10.1090/S0025-5718-2013-02753-6Search in Google Scholar
[11] J. Guzmán and M. Neilan, Conforming and divergence-free Stokes elements in three dimensions, IMA J. Numer. Anal., 34 (2014), No. 4, 1489–1508.10.1093/imanum/drt053Search in Google Scholar
[12] J. Guzmán and M. Neilan, inf-sup stable finite elements on barycentric refinements producing divergence-free approximations in arbitrary dimensions, SIAM J. Numer. Anal., 56 (2018), No. 5, 2826–2844.10.1137/17M1153467Search in Google Scholar
[13] J. Guzmán, A. Lischke, and M. Neilan, Exact sequences on Powell–Sabin splits, Calcolo 57 (2020), No. 2, Paper 13.10.1007/s10092-020-00361-xSearch in Google Scholar
[14] J. Hu and M. Schedensack, Two low-order nonconforming finite element methods for the Stokes flow in three dimensions, IMA J. Numer. Anal., 39 (2019), No. 3, 1447–1470.10.1093/imanum/dry021Search in Google Scholar
[15] Y. Huang and S. Zhang, A lowest order divergence-free finite element on rectangular grids, Front. Math. China, 6 (2011), No. 2, 253–270.10.1007/s11464-011-0094-0Search in Google Scholar
[16] R. Kouhia and R. Stenberg, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow, Comput. Methods Appl. Mech. Engrg., 124 (1995), No. 3, 195–212.10.1016/0045-7825(95)00829-PSearch in Google Scholar
[17] M. Neilan, Discrete and conforming smooth de Rham complexes in three dimensions, Math. Comp., 84 (2015), No. 295, 2059–2081.10.1090/S0025-5718-2015-02958-5Search in Google Scholar
[18] J. Qin, On the convergence of some low order mixed finite elements for incompressible fluids, Thesis, Pennsylvania State University, 1994.Search in Google Scholar
[19] L. R. Scott and M. Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials, RAIRO, Modelisation Math. Anal. Numer., 19 (1985), 111–143.10.1051/m2an/1985190101111Search in Google Scholar
[20] L. R. Scott and M. Vogelius, Conforming finite element methods for incompressible and nearly incompressible continua, Lectures in Applied Mathematics, Vol. 22, 1985, pp. 221–244.Search in Google Scholar
[21] L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), No. 190, 483–493.10.1090/S0025-5718-1990-1011446-7Search in Google Scholar
[22] X. Xu and S. Zhang, A new divergence-free interpolation operator with applications to the Darcy–Stokes–Brinkman equations, SIAM J. Sci. Comput., 32 (2010), No. 2, 855–874.10.1137/090751049Search in Google Scholar
[23] M. Zhang and S. Zhang, A 3D conforming-nonconforming mixed finite element for solving symmetric stress Stokes equations, Int. J. Numer. Anal. Model., 14 (2017), No. 4-5, 730–743.Search in Google Scholar
[24] S. Zhang, A new family of stable mixed finite elements for 3D Stokes equations, Math. Comp., 74 (2005), No. 250, 543–554.10.1090/S0025-5718-04-01711-9Search in Google Scholar
[25] S. Zhang, On the P1 Powell–Sabin divergence-free finite element for the Stokes equations, J. Comp. Math., 26 (2008), 456–470.Search in Google Scholar
[26] S. Zhang, A family of Qk+1,k × Qk,k+1 divergence-free finite elements on rectangular grids, SIAM J. Numer. Anal., 47 (2009), No. 3, 2090– 2107.10.1137/080728949Search in Google Scholar
[27] S. Zhang, Quadratic divergence-free finite elements on Powell–Sabin tetrahedral grids, Calcolo, 48 (2011), No. 3, 211–244.10.1007/s10092-010-0035-4Search in Google Scholar
[28] S. Zhang, Divergence-free finite elements on tetrahedral grids for k ⩾ 6, Math. Comp., 80 (2011), No. 274, 669–695.10.1090/S0025-5718-2010-02412-3Search in Google Scholar
[29] S. Zhang, Coefficient jump-independent approximation of the conforming and nonconforming finite element solutions, Adv. Appl. Math. Mech., 8 (2016), No. 5, 722–736.10.4208/aamm.2015.m931Search in Google Scholar
[30] S. Zhang, A P4 bubble enriched P3 divergence-free finite element on triangular grids, Comput. Math. Appl., 74 (2017), No. 11, 2710–2722.10.1016/j.camwa.2017.06.036Search in Google Scholar
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