Title:         Finite element approximation on quadrilateral meshes

Authors:       Douglas N. Arnold, Daniele Boffi, Richard S. Falk, and Lucia Gastaldi

Source:        Communications in Numerical Methods in Engineering 17 (2001), pp. 805-812

Status:        Published

Abstract:  Quadrilateral finite elements are generally constructed by
starting from a given finite dimensional space of polynomials V^
on the unit reference square K^. The elements of V^ are
then transformed by using the bilinear isomorphisms F_K which map
K^ to each convex quadrilateral element K. It has been recently
proven that a necessary and sufficient condition for approximation of
order r+1 in L^2 and r in H^1 is that V^ contains the
space Q_r of all polynomial functions of degree r separately in
each variable. In this paper several numerical experiments are
presented which confirm the theory. The tests are taken from various
examples of applications: the Laplace operator, the Stokes problem and
an eigenvalue problem arising in fluid-structure interaction modeling.

Keywords:      quadrilateral, finite element, approximation, serendipity, mixed finite element

Subj. class.:  65N30, 41A10, 41A25, 41A27, 41A63

URL:           http://ima.umn.edu/~arnold/papers/quadmeshes.pdf