Finite element approximation on quadrilateral meshes

SUMMARY Quadrilateral 7nite elements are generally constructed by starting from a given 7nite dimensional space of polynomials ˆ V on the unit reference square ˆ K. The elements of ˆ V are then transformed by using the bilinear isomorphisms FK which map ˆ K to each convex quadrilateral element K. It has been recently proven that a necessary and su2cient condition for approximation of order r +1 inL 2 and r in H 1 is that ˆ V contains the space Qr of all polynomial functions of degree r separately in each variable. In this paper several numerical experiments are presented which con7rm the theory. The tests are taken from various examples of applications: the Laplace operator, the Stokes problem and an eigenvalue problem arising in ?uid-structure interaction modelling. Copyright ? 2001 John Wiley & Sons, Ltd.

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