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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
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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.

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Additional Information

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

Daniele Boffi
Affiliation: Dipartimento di Matematica, Università di Pavia, 27100 Pavia, Italy

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

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