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 FK 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 L2 and r in H1 is that V̂ contains the space Qr 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 and Sons, Ltd.