Title: Approximation by quadrilateral finite elements
Authors: Douglas N. Arnold, Daniele Boffi, and Richard S. Falk
Source: Math. Comp., 71(239):909-922, 2002
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 L2 and order r in H1 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.
Keywords: quadrilateral, finite element, approximation, serendipity,
mixed finite element
URL: http://www.math.rutgers.edu/~falk/papers/quadapprox.pdf