Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Approximation by quadrilateral finite elements

Authors: Douglas N. Arnold, Daniele Boffi and Richard S. Falk
Journal: Math. Comp. 71 (2002), 909-922
MSC (2000): Primary 65N30, 41A10, 41A25, 41A27, 41A63
Published electronically: March 22, 2002
MathSciNet review: 1898739
Full-text PDF Free Access
View in AMS MathViewer New

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element, which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element. It is known that for affine isomorphisms, a necessary and sufficient condition for approximation of order $r+1$ in $L^p$ and order $r$ in $W^1_p$ is that the given space of functions on the reference element contain all polynomial functions of total degree at most $r$. In the case of bilinear isomorphisms, it is known that the same estimates hold if the function space contains all polynomial functions of separate degree $r$. We show, by means of a counterexample, that this latter condition is also necessary. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements.

References [Enhancements On Off] (What's this?)

  • Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174
  • B. Nagy, On cosine operator functions in Banach spaces, Acta Sci. Math. (Szeged) 36 (1974), 281–289. MR 374995
  • Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
  • Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383
  • F. Kikuchi, M. Okabe, and H. Fujio, Modification of the 8-node serendipity element, Comp. Methods Appl. Mech. Engrg. 179 (1999), 91–109.
  • R. H. McNeal and R. L. Harder, Eight nodes or nine?, Int. J. Numer. Methods Engrg. 33 (1992), 1049–1058.
  • R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element, Numer. Methods Partial Differential Equations 8 (1992), no. 2, 97–111. MR 1148797, DOI
  • P. Sharpov and Y. Iordanov, Numerical solution of Stokes equations with pressure and filtration boundary conditions, J. Comp. Phys. 112 (1994), 12–23.
  • G. Strang and G. Fix, A Fourier analysis of the finite element variational method, Constructive Aspects of Functional Analysis (G. Geymonat, ed.), C.I.M.E. II Ciclo, 1971, pp. 793–840.
  • Jing Zhang and Fumio Kikuchi, Interpolation error estimates of a modified 8-node serendipity finite element, Numer. Math. 85 (2000), no. 3, 503–524. MR 1760933, DOI
  • O. C. Zienkiewicz and R. L. Taylor, The finite element method, fourth edition, volume 1: Basic formulation and linear problems, McGraw-Hill, London, 1989.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N30, 41A10, 41A25, 41A27, 41A63

Retrieve articles in all journals with MSC (2000): 65N30, 41A10, 41A25, 41A27, 41A63

Additional Information

Douglas N. Arnold
Affiliation: Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 55455
MR Author ID: 27240

Daniele Boffi
Affiliation: Dipartimento di Matematica, Università di Pavia, 27100 Pavia, Italy
MR Author ID: 348743

Richard S. Falk
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854

Keywords: Quadrilateral, finite element, approximation, serendipity, mixed finite element
Received by editor(s): March 10, 2000
Published electronically: March 22, 2002
Article copyright: © Copyright 2002 American Mathematical Society