A subspace of linear nonconforming finite element for nearly incompressible elasticity and Stokes flow

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.Suche 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.20560Suche 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-200042165Suche 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/S0025-5718-03-01579-5Suche 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/197307R300331Suche 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.2021.114444Suche 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/S0025-5718-1991-1094947-6Suche 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, 1308–1326.10.1137/120888132Suche 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/3520Suche 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/S0025-5718-2013-02753-6Suche 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/drt053Suche 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, 2826–2844.10.1137/17M1153467Suche 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-xSuche 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/dry021Suche 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/s11464-011-0094-0Suche 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-PSuche 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/S0025-5718-2015-02958-5Suche in Google Scholar

[18] J. Qin, On the convergence of some low order mixed finite elements for incompressible fluids, Thesis, Pennsylvania State University, 1994.Suche 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/1985190101111Suche 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.Suche 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/S0025-5718-1990-1011446-7Suche 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, 855–874.10.1137/090751049Suche 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.Suche 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/S0025-5718-04-01711-9Suche 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.Suche in Google Scholar

[26] S. Zhang, A family of Q_k+1,k × Q_k,k+1 divergence-free finite elements on rectangular grids, SIAM J. Numer. Anal., 47 (2009), No. 3, 2090– 2107.10.1137/080728949Suche 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/s10092-010-0035-4Suche 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/S0025-5718-2010-02412-3Suche 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.m931Suche 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.2017.06.036Suche in Google Scholar