Abstract
The lowest order nonconforming rectangular element in three dimensions involves 54 degrees of freedom for the stress and 12 degrees of freedom for the displacement. With a modest increase in the number of degrees of freedom (24 for the stress), we obtain a conforming rectangular element for linear elasticity in three dimensions. Moreover, unlike the conforming plane rectangular or simplicial elements, this element does not involve any vertex degrees of freedom. Second, we remark that further low order elements can be constructed by approximating the displacement with rigid body motions. This results in a pair of conforming elements with 72 degrees of freedom for the stress and 6 degrees of freedom for the displacement.
Similar content being viewed by others
References
Adams, S., Cockburn, B.: A mixed finite element method for elasticity in three dimensions. J. Sci. Comput. 25(3), 515–521 (2005)
Amara, M., Thomas, J.M.: Equilibrium finite elements for the linear elastic problem. Numer. Math. 33(4), 367–383 (1979)
Arnold, D.N., Awanou, G.: Rectangular mixed finite elements for elasticity. Math. Models Methods Appl. Sci. 15(9), 1417–1429 (2005)
Arnold, D.N., Awanou, G., Winther, R.: Finite elements for symmetric tensors in three dimensions. Math. Comput. 77(263), 1229–1251 (2008)
Arnold, D.N., Brezzi, F., Douglas, J. Jr.: PEERS: a new mixed finite element for plane elasticity. Jpn. J. Appl. Math. 1(2), 347–367 (1984)
Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)
Arnold, D.N., Falk, R.S., Winther, R.: Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comput. 76(260), 1699–1723 (2007) (electronic)
Arnold, D.N., Winther, R.: Mixed finite elements for elasticity. Numer. Math. 92(3), 401–419 (2002)
Arnold, D.N., Winther, R.: Nonconforming mixed elements for elasticity. Math. Models Methods Appl. Sci. 13(3), 295–307 (2003). Dedicated to Jim Douglas, Jr. on the occasion of his 75th birthday
Awanou, G.: Rectangular mixed elements for elasticity with weakly imposed symmetry condition. Preprint (2010). arXiv:1012.1906
Awanou, G.: A rotated nonconforming rectangular mixed element for elasticity. Calcolo 46(1), 49–60 (2009)
Awanou, G.: Symmetric matrix fields in the finite element method. Symmetry 2, 1375–1389 (2010)
Bernardi, C., Girault, V.: A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35(5), 1893–1916 (1998) (electronic)
Boffi, D., Brezzi, F., Demkowicz, L.F., Durán, R.G., Falk, R.S., Fortin, M.: In: Mixed Finite Elements, Compatibility Conditions, and Applications. Lecture Notes in Mathematics, vol. 1939. Springer, Berlin (2008). Lectures given at the C.I.M.E. Summer School held in Cetraro, June 26–July 1, 2006, edited by Boffi and L. Gastaldi
Chen, S.-C., Yang, Y.-N.: Conforming rectangular mixed finite elements for elasticity. J. Sci. Comput. (2010, to appear). doi:10.1007/s10915-010-9422-x
Cockburn, B., Gopalakrishnan, J., Guzmán, J.: A new elasticity element made for enforcing weak stress symmetry. Math. Comput. 79(271), 1331–1349 (2010)
Gopalakrishnan, J., Guzmán, J.: A second elasticity element using the matrix bubble (2010, submitted)
Gopalakrishnan, J., Guzmán, J.: Symmetric non-conforming mixed finite elements for linear elasticity (2010, submitted)
Guzmán, J.: A unified analysis of several mixed methods for elasticity with weak stress symmetry. J. Sci. Comput. 44(2), 156–169 (2010)
Hu, J., Shi, Z.-C.: Lower order rectangular nonconforming mixed finite elements for plane elasticity. SIAM J. Numer. Anal. 46(1), 88–102 (2007/08)
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(1), 51–65 (2009)
Morley, M.E.: A family of mixed finite elements for linear elasticity. Numer. Math. 55(6), 633–666 (1989)
Nicaise, S., Witowski, K., Wohlmuth, B.I.: An a posteriori error estimator for the Lamé equation based on equilibrated fluxes. IMA J. Numer. Anal. 28(2), 331–353 (2008)
Stenberg, R.: On the construction of optimal mixed finite element methods for the linear elasticity problem. Numer. Math. 48(4), 447–462 (1986)
Stenberg, R.: A family of mixed finite elements for the elasticity problem. Numer. Math. 53(5), 513–538 (1988)
Stenberg, R.: Two low-order mixed methods for the elasticity problem. In: The Mathematics of Finite Elements and Applications, VI, Uxbridge, 1987, pp. 271–280. Academic Press, London (1988)
Yi, S.-Y.: Nonconforming mixed finite element methods for linear elasticity using rectangular elements in two and three dimensions. Calcolo 42(2), 115–133 (2005)
Yi, S.-Y.: A new nonconforming mixed finite element method for linear elasticity. Math. Models Methods Appl. Sci. 16(7), 979–999 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Awanou, G. Two Remarks on Rectangular Mixed Finite Elements for Elasticity. J Sci Comput 50, 91–102 (2012). https://doi.org/10.1007/s10915-011-9474-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-011-9474-6