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

doi:10.1007/s42967-025-00476-4

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (3): 851-867.doi: 10.1007/s42967-025-00476-4

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

Hongying Man¹, Shangyou Zhang²

1. School of Mathematics, Beijing Institute of Technology, Beijing, 100081, China;
2. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

收稿日期:2024-02-21 修回日期:2024-12-30 出版日期:2026-06-20 发布日期:2026-05-29
通讯作者: Hongying Man, Email: manhy@bit.edu.cn E-mail:manhy@bit.edu.cn
作者简介:Shangyou Zhang, Email: szhang@udel.edu
基金资助:
Not applicable.

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

Hongying Man¹, Shangyou Zhang²

1. School of Mathematics, Beijing Institute of Technology, Beijing, 100081, China;
2. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

Received:2024-02-21 Revised:2024-12-30 Online:2026-06-20 Published:2026-05-29
Contact: Hongying Man, Email: manhy@bit.edu.cn E-mail:manhy@bit.edu.cn
Supported by:
This work was subsidized by the National Natural Science Foundation of China (No. 12361078) and the Guangxi Natural Science Foundation of China (Nos. 2018JJB110062, 2019AC20062, 2021JJB110006, and 2021AC19147).

摘要/Abstract

摘要： 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.

关键词: Mixed finite element, Linear elasticity, Conforming finite element, Superconvergence, 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.

Key words: Mixed finite element, Linear elasticity, Conforming finite element, Superconvergence, Square grids

中图分类号:

Hongying Man, Shangyou Zhang. On the Superconvergence of a Conforming Mixed Finite Element for Linear Elasticity on Uniform n-Square Grids[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 851-867.

参考文献

[1] Adams, S., Cockburn, B.: A mixed finite element method for elasticity in three dimensions. J. Sci. Comput. 25, 515-521 (2005)
[2] Amara, M., Thomas, J.M.: Equilibrium finite elements for the linear elastic problem. Numer. Math. 33, 367-383 (1979)
[3] Arnold, D.N., Awanou, G.: Rectangular mixed finite elements for elasticity. Math. Models Methods Appl. Sci. 15, 1417-1429 (2005)
[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, D.N., Douglas, J., Jr., Gupta, C.P.: A family of higher order mixed finite element methods for plane elasticity. Numer. Math. 45, 1-22 (1984)
[7] Arnold, D.N., Falk, R., Winther, R.: Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comp. 76, 1699-1723 (2007)
[8] Arnold, D.N., Winther, R.: Mixed finite element for elasticity. Numer. Math. 92, 401-419 (2002)
[9] Arnold, D.N., Winther, R.: Nonconforming mixed elements for elasticity. Math. Models. Methods Appl. Sci. 13, 295-307 (2003)
[10] Awanou, G.: Two remarks on rectangular mixed finite elements for elasticity. J. Sci. Comput. 50, 91-102 (2012)
[11] Boffi, D., Brezzi, F., Fortin, M.: Reduced symmetry elements in linear elasticity. Commun. Pure Appl. Anal. 8, 95-121 (2009)
[12] Brandts, J.H.: Superconvergence and a posteriori error estimation for triangular mixed finite elements. Numer. Math. 68, 311-324 (1994)
[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 8(2), 129-151 (1974)
[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. 200, 2903-2915 (2011)
[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, S.C., Wang, Y.N.: Conforming rectangular mixed finite elements for elasticity. J. Sci. Comput. 47, 93-108 (2011)
[18] Clément, P.: Approximation by finite element functions using local regularization. RAIRO Anal. Numer. 2, 77-84 (1975)
[19] Cockburn, B., Gopalakrishnan, J., Guzmán, J.: A new elasticity element made for enforcing weak stress symmetry. Math. Comp. 79, 1331-1349 (2010)
[20] Douglas, J., Wang, J.: Superconvergence of mixed finite element methods on rectangular domains. Calcolo 26, 121-134 (1989)
[21] Duran, R.: Superconvergence for rectangular mixed finite elements. Numer. Math. 58, 2-15 (1990)
[22] Ewing, R.E., Lazarov, R.D., Wang, J.: Superconvergence of the velocity along the Gauss lines in mixed finite element methods. SIAM J. Numer. Anal. 28, 1015-1029 (1991)
[23] Gopalakrishnan, J., Guzmán, J.: Symmetric nonconforming mixed finite elements for linear elasticity. SIAM J. Numer. Anal. 49, 1504-1520 (2011)
[24] Gopalakrishnan, J., Guzmán, J.: A second elasticity element using the matrix bubble. IMA J. Numer. Anal. 32, 352-372 (2012)
[25] Guzmán, J.: A unified analysis of several mixed methods for elasticity with weak stress symmetry. J. Sci. Comput. 44, 156-169 (2010)
[26] Hu, J.: Finite element approximations of symmetric tensors on simplicial grids in Rⁿ: 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. 53, 1438-1463 (2015)
[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. 35(2), 716-732 (2019)
[30] Hu, J., Ma, R.: Partial relaxation of C⁰ 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-rectangular grids. Comput. Math. Appl. 71(7), 1317-1336 (2016)
[32] Hu, J., Man, H.Y., Zhang, S.: A simple conforming mixed finite element for linear elasticity on rectangular grids in any space dimension. J. Sci. Comput. 58(2), 367-379 (2014)
[33] Hu, J., Shi, Z.C.: Lower order rectangular nonconforming mixed elements for plane elasticity. SIAM J. Numer. Anal. 46, 88-102 (2007)
[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. 58(2), 297-307 (2015)
[37] Hu, J., Zhang, S.: Finite element approximations of symmetric tensors on simplicial grids in Rⁿ: 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. 30, 103-116 (1978)
[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, H.Y., Hu, J., Shi, Z.-C.: Lower order rectangular nonconforming mixed finite element for the three-dimensional elasticity problem. Math. Models Methods Appl. Sci. 19, 51-65 (2009)
[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. 54, 483-493 (1990)
[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. 48, 447-462 (1986)
[47] Stenberg, R.: Two low-order mixed methods for the elasticity problem. In: Whiteman, J.R. (ed.) The Mathematics of Finite Elements and Applications, VI, pp. 271-280. Academic Press, London (1988)
[48] Stenberg, R.: A family of mixed finite elements for the elasticity problem. Numer. Math. 53, 513-538 (1988)
[49] Xie, X.P., Xu, J.C.: New mixed finite elements for plane elasticity and Stokes equations. Sci. China Math. 54, 1499-1519 (2011)
[50] Wang, K.: Superconvergence and extrapolation for mixed finite element methods on rectangular domains. Math. Comp. 56, 477-503 (1991)
[51] Ye, X., Zhang, S.: A P_k+2 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. 14(3), 367-386 (1998)
[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)

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

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

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