 
Summary: MATHEMATICS OF COMPUTATION
Volume 71, Number 239, Pages 909922
S 00255718(02)014394
Article electronically published on March 22, 2002
APPROXIMATION BY QUADRILATERAL FINITE ELEMENTS
DOUGLAS N. ARNOLD, DANIELE BOFFI, AND RICHARD S. FALK
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 ele
ment, which is then transformed to a space of functions on each convex quadri
lateral 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 Lp 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
