Finite element approximation on quadrilateral meshes

Quadrilateral finite elements are generally constructed by starting from a given finite dimensional space of polynomials V̂ on the unit reference square K̂ . The elements of V̂ are then transformed by using the bilinear isomorphisms F K which map K̂ 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 V̂ 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 modelling. Copyright © 2001 John Wiley & Sons, Ltd.