Finite element approximation on quadrilateral meshes

Quadrilateral finite elements are generally constructed by starting from a given finite dimensional space of polynomials (V) over cap on the unit refe rence square (K) over cap. The elements of (V) over cap are then transforme d by using the bilinear isomorphisms F-K which map (K) over cap to each con vex quadrilateral element K. It has been recently proven that a necessary a nd sufficient condition for approximation of order r + 1 in L-2 and r in H- 1 is that (V) over cap contains the space Q(r) of all polynomial functions of degree r separately in each variable. In this paper several numerical ex periments 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 modelling. Copyright (C) 2001 John Wiley & Sons, Ltd.