Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T19:50:54.281Z Has data issue: false hasContentIssue false
  • English
  • Français

A class of nonparametric DSSY nonconforming quadrilateralelements∗

Published online by Cambridge University Press:  07 October 2013

Youngmok Jeon ,
Hyun Nam ,
Dongwoo Sheen  and
Kwangshin Shim
Youngmok Jeon
Affiliation:
Department of Mathematics, Ajou University, Suwon 443–749, Korea.. [email protected]
Hyun Nam
Affiliation:
Department of Mathematics, Seoul National University, Seoul 151–747, Korea.; [email protected]
Dongwoo Sheen
Affiliation:
Department of Mathematics, and Interdisciplinary Program in Computational Science and Technology, Seoul National University, Seoul 151–747, Korea.; [email protected]
Kwangshin Shim
Affiliation:
Department of Mathematics, Seoul National University, Seoul 151–747, Korea.; [email protected]
Get access

Abstract

A new class of nonparametric nonconforming quadrilateral finite elements is introducedwhich has the midpoint continuity and the mean value continuity at the interfaces ofelements simultaneously as the rectangular DSSY element [J. Douglas, Jr., J.E. Santos, D.Sheen and X. Ye, ESAIM: M2AN 33 (1999) 747–770.] Theparametric DSSY element for general quadrilaterals requires five degrees of freedom tohave an optimal order of convergence [Z. Cai, J. Douglas, Jr., J.E. Santos, D. Sheen andX. Ye, Calcolo 37 (2000) 253–254.], while the newnonparametric DSSY elements require only four degrees of freedom. The design of newelements is based on the decomposition of a bilinear transform into a simple bilinear mapfollowed by a suitable affine map. Numerical results are presented to compare the newelements with the parametric DSSY element.

Keywords

Nonconformingfinite elementquadrilateral
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 6 , November 2013 , pp. 1783 - 1796
Copyright
© EDP Sciences, SMAI 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, D.N., Boffi, D. and Falk, R.S., Approximation by quadrilateral finite elements. Math. Comput. 71 (2002) 909–922. Google Scholar
Brenner, S.C. and Sung, L.Y., Linear finite element methods for planar elasticity. Math. Comput. 59 (1992) 321–338. Google Scholar
Cai, Z., Douglas, J. Jr., Santos, J.E., Sheen, D. and Ye, X., Nonconforming quadrilateral finite elements: A correction. Calcolo 37 (2000) 253–254. Google Scholar
Cai, Z., Douglas, J. Jr. and Ye, X., A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier-Stokes equations. Calcolo 36 (1999) 215–232. Google Scholar
Chen, Z., Projection finite element methods for semiconductor device equations. Computers Math. Appl. 25 (1993) 81–88. Google Scholar
Crouzeix, M. and Raviart, P.-A., Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO – Math. Model. Numer. Anal. 7 (1973) 33–75. Google Scholar
Douglas, J. Jr., Santos, J.E., Sheen, D. and Ye, X., Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems. ESAIM: M2AN 33 (1999) 747–770. Google Scholar
Han, H., Nonconforming elements in the mixed finite element method. J. Comput. Math. 2 (1984) 223–233. Google Scholar
Hu, J. and Shi, Z.-C., Constrained quadrilateral nonconforming rotated Q 1-element. J. Comput. Math. 23 (2005) 561–586. Google Scholar
Y. Jeon, H. Nam, D. Sheen and K. Shim, A nonparametric DSSY nonconforming quadrilateral element with maximal inf-sup constant (2013). In preparation.
Köster, M., Ouazzi, A., Schieweck, F., Turek, S. and Zajac, P., New robust nonconforming finite elements of higher order. Appl. Numer. Math. 62 (2012) 166–184. Google Scholar
Kouhia, R. and Stenberg, R., A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Engrg. 124 (1995) 195–212. Google Scholar
Lee, C.-O., Lee, J. and Sheen, D., A locking-free nonconforming finite element method for planar elasticity. Adv. Comput. Math. 19 (2003) 277–291. Google Scholar
Ming, P. and Shi, Z.-C., Nonconforming rotated Q 1 element for Reissner-Mindlin plate. Math. Models Methods Appl. Sci. 11 (2001) 1311–1342. Google Scholar
Ming, P. and Shi, Z.-C., Two nonconforming quadrilateral elements for the Reissner–Mindlin plate. Math. Models Methods Appl. Sci. 15 (2005) 1503–1517. Google Scholar
Nam, H., Choi, H.J., Park, C. and Sheen, D., A cheapest nonconforming rectangular finite element for the Stokes problem. Comput. Methods Appl. Mech. Engrg. 257 (2013) 77–86. Google Scholar
Park, C. and Sheen, D., P 1-nonconforming quadrilateral finite element methods for second-order elliptic problems. SIAM J. Numer. Anal. 41 (2003) 624–640. Google Scholar
Park, C. and Sheen, D., A quadrilateral Morley element for biharmonic equations. Numer. Math. 124 (2013) 395–413. Google Scholar
Rannacher, R. and Turek, S., Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equ. 8 (1992) 97–111. Google Scholar
Shi, Z.-C., An explicit analysis of Stummel’s patch test examples. Int. J. Numer. Meth. Engrg. 20 (1984) 1233–1246. Google Scholar
Shi, Z.C., The FEM test for convergence of nonconforming finite elements. Math. Comput. 49 (1987) 391–405. Google Scholar
S. Turek, Efficient solvers for incompressible flow problems, vol. 6. Lecture Notes in Comput. Sci. Engrg. Springer, Berlin (1999).
Wang, M., On the necessity and sufficiency of the patch test for convergence of nonconforming finite elements. SIAM J. Numer. Anal. 39 (2001) 363–384. Google Scholar
Zhang, Z., Analysis of some quadrilateral nonconforming elements for incompressible elasticity. SIAM J. Numer. Anal. 34 (1997) 640–663. Google Scholar