On the Superconvergence of a Conforming Mixed Finite Element for Linear Elasticity on Uniform n-Square Grids

Abstract

This paper is to prove one-order superconvergence of both stress and displacement of a conforming symmetric mixed finite element on uniform n-square grids, for the linear elasticity equation in the Hellinger-Reissner variational formulation. Numerical examples on 2D and 3D uniform square grids are computed, verifying the theory.

Keywords

Mixed finite element / Linear elasticity / Conforming finite element / Superconvergence / Square grids / 65N15 / 65N30 / 73C02

References

[1] Adams S, Cockburn B. A mixed finite element method for elasticity in three dimensions. J. Sci. Comput., 2005, 25: 515-521

[2] Amara M, Thomas JM. Equilibrium finite elements for the linear elastic problem. Numer. Math., 1979, 33: 367-383

[3] Arnold DN, Awanou G. Rectangular mixed finite elements for elasticity. Math. Models Methods Appl. Sci., 2005, 15: 1417-1429

[4] Arnold, D.N., Awanou, G., Winther, R.: Finite elements for symmetric tensors in three dimensions. Math. Comp. 77, 1229–1251 (2008)

[5] Arnold, D.N., Brezzi, F., Douglas, J., Jr.: PEERS: a new mixed finite element for plane elasticity. Jpn. J. Appl. Math. 1, 347–367 (1984)

[6] Arnold DN, Douglas JJr, Gupta CP. A family of higher order mixed finite element methods for plane elasticity. Numer. Math., 1984, 45: 1-22

[7] Arnold DN, Falk R, Winther R. Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comp., 2007, 76: 1699-1723

[8] Arnold DN, Winther R. Mixed finite element for elasticity. Numer. Math., 2002, 92: 401-419

[9] Arnold DN, Winther R. Nonconforming mixed elements for elasticity. Math. Models. Methods Appl. Sci., 2003, 13: 295-307

[10] Awanou G. Two remarks on rectangular mixed finite elements for elasticity. J. Sci. Comput., 2012, 50: 91-102

[11] Boffi D, Brezzi F, Fortin M. Reduced symmetry elements in linear elasticity. Commun. Pure Appl. Anal., 2009, 8: 95-121

[12] Brandts JH. Superconvergence and a posteriori error estimation for triangular mixed finite elements. Numer. Math., 1994, 68: 311-324

[13] Brezzi F. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Francaise Automat. Informat. Recherche Operationnelle Ser. Rouge, 1974, 8(2): 129-151

[14] Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991)

[15] Carstensen C, Eigel M, Gedicke J. Computational competition of symmetric mixed FEM in linear elasticity. Comput. Methods Appl. Mech. Engrg., 2011, 200: 2903-2915

[16] Carstensen, C., Günther, D., Reininghaus, J., Thiele, J.: The Arnold-Winther mixed FEM in linear elasticity. Part I: implementation and numerical verification. Comput. Methods Appl. Mech. Engrg. 197, 3014–3023 (2008)

[17] Chen SC, Wang YN. Conforming rectangular mixed finite elements for elasticity. J. Sci. Comput., 2011, 47: 93-108

[18] Clément P. Approximation by finite element functions using local regularization. RAIRO Anal. Numer., 1975, 2: 77-84

[19] Cockburn B, Gopalakrishnan J, Guzmán J. A new elasticity element made for enforcing weak stress symmetry. Math. Comp., 2010, 79: 1331-1349

[20] Douglas J, Wang J. Superconvergence of mixed finite element methods on rectangular domains. Calcolo, 1989, 26: 121-134

[21] Duran R. Superconvergence for rectangular mixed finite elements. Numer. Math., 1990, 58: 2-15

[22] Ewing RE, Lazarov RD, Wang J. Superconvergence of the velocity along the Gauss lines in mixed finite element methods. SIAM J. Numer. Anal., 1991, 28: 1015-1029

[23] Gopalakrishnan J, Guzmán J. Symmetric nonconforming mixed finite elements for linear elasticity. SIAM J. Numer. Anal., 2011, 49: 1504-1520

[24] Gopalakrishnan J, Guzmán J. A second elasticity element using the matrix bubble. IMA J. Numer. Anal., 2012, 32: 352-372

[25] Guzmán J. A unified analysis of several mixed methods for elasticity with weak stress symmetry. J. Sci. Comput., 2010, 44: 156-169

[26] Hu, J.: Finite element approximations of symmetric tensors on simplicial grids in Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb{R}}^n$$\end{document}: the higher order case. J. Comp. Math. 33, 283–296 (2015)

[27] Hu J. A new family of efficient conforming mixed finite elements on both rectangular and cuboid meshes for linear elasticity in the symmetric formulation. SIAM J. Numer. Anal., 2015, 53: 1438-1463

[28] Hu, J., Ma, R.: Conforming mixed triangular prism elements for the linear elasticity problem. Int. J. Numer. Anal. Model. 15(1/2), 228–242 (2018)

[29] Hu J, Ma R. Nonconforming mixed finite elements for linear elasticity on simplicial grids. Numer. Methods Part. Differ. Equ., 2019, 35(2): 716-732

[30] Hu, J., Ma, R.: Partial relaxation of C0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^0$$\end{document} vertex continuity of stresses of conforming mixed finite elements for the elasticity problem. Comput. Methods Appl. Math. 21(1), 89–108 (2021)

[31] Hu, J., Man, H., Wang, J., Zhang, S.: The simplest nonconforming mixed finite element method for linear elasticity in the symmetric formulation on n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n$$\end{document}-rectangular grids. Comput. Math. Appl. 71(7), 1317–1336 (2016)

[32] Hu J, Man HY, Zhang S. A simple conforming mixed finite element for linear elasticity on rectangular grids in any space dimension. J. Sci. Comput., 2014, 58(2): 367-379

[33] Hu J, Shi ZC. Lower order rectangular nonconforming mixed elements for plane elasticity. SIAM J. Numer. Anal., 2007, 46: 88-102

[34] Hu, J., Zhang, S.: Superconvergence of simple conforming mixed finite elements for linear elasticity on rectangular grids in any space dimension. arXiv:1407.4190 (2014)

[35] Hu, J., Zhang, S.: A family of conforming mixed finite elements for linear elasticity on triangular grids. arXiv:1406.7457 (2014)

[36] Hu J, Zhang S. A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids. Sci. China Math., 2015, 58(2): 297-307

[37] Hu, J., Zhang, S.: Finite element approximations of symmetric tensors on simplicial grids in Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb{R}}^n$$\end{document}: the lower order case. Math. Models Methods Appl. Sci. 26(9), 1649–1669 (2016)

[38] Johnson C, Mercier B. Some equilibrium finite element methods for two-dimensional elasticity problems. Numer. Math., 1978, 30: 103-116

[39] Lin, Q., Lin, J.F.: Finite Element Methods: Accuracy and Improvement. Science Press, Beijing (2006)

[40] Lin, Q., Yan, N.N.: The Construction and Analysis of High Efficiency Finite Element Methods. Hebei University Press, Baoding (1996). ((in Chinese))

[41] Lin, Y., Xu, X., Zhang, S.: Superconvergent P1 honeycomb virtual elements and lifted P3 solutions. Calcolo 61(4), 67 (2024)

[42] Man HY, Hu J, Shi Z-C. Lower order rectangular nonconforming mixed finite element for the three-dimensional elasticity problem. Math. Models Methods Appl. Sci., 2009, 19: 51-65

[43] Morley, M.: A family of mixed finite elements for linear elasticity. Numer. Math. 55, 633–666 (1989)

[44] Scott L, Zhang S. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp., 1990, 54: 483-493

[45] Shi, D.Y., Li, M.H.: Superconvergence analysis of a stable conforming rectangular mixed finite elements for the linear elasticity problem. J. Comput. Math. 32(2), 205–214 (2014). https://doi.org/10.4208/jcm.1401-m3837

[46] Stenberg R. On the construction of optimal mixed finite element methods for the linear elasticity problem. Numer. Math., 1986, 48: 447-462

[47] Stenberg RWhiteman JR. Two low-order mixed methods for the elasticity problem. The Mathematics of Finite Elements and Applications, VI, 1988LondonAcademic Press271-280

[48] Stenberg R. A family of mixed finite elements for the elasticity problem. Numer. Math., 1988, 53: 513-538

[49] Xie XP, Xu JC. New mixed finite elements for plane elasticity and Stokes equations. Sci. China Math., 2011, 54: 1499-1519

[50] Wang K. Superconvergence and extrapolation for mixed finite element methods on rectangular domains. Math. Comp., 1991, 56: 477-503

[51] Ye, X., Zhang, S.: A Pk+2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P_{k+2}$$\end{document} polynomial lifting operator on polygons and polyhedrons. Appl. Math. Lett. 116, 107033 (2021)

[52] Ye, X., Zhang, S.: Four-order superconvergent CDG finite elements for the biharmonic equation on triangular meshes, J. Comput. Appl. Math. 440, 115516 (2024)

[53] Yi, S.Y.: Nonconforming mixed finite element methods for linear elasticity using rectangular elements in two and three dimensions. Calcolo 42, 115–133 (2005)

[54] Yi, S.Y.: A new nonconforming mixed finite element method for linear elasticity. Math. Models. Methods Appl. Sci. 16, 979–999 (2006)

[55] Zhang Z. Superconvergence in the projected-shear plate-bending finite element method. Numer. Meth. Part. Differ. Equ., 1998, 14(3): 367-386

[56] Zhang, Z., Zhang, S.: Derivative superconvergence of rectangular finite elements for the Reissner-Mindlin plate. Comput. Methods Appl. Mech. Engrg. 134(1/2), 1–16 (1996)

RIGHTS & PERMISSIONS

Shanghai University