Galerkin discontinuous approximation of the transport equation and viscoelastic fluid flow on quadrilaterals

Numerical simulation of industrial processes involving viscoelastic liquids is often based on finite element methods on quadrilateral meshes. However, numerical analysis of these methods has so far been limited to triangular meshes. In this work, we consider quadrilateral meshes. We first study the approximation of the transport equation by a Galerkin discontinuous method and prove an O(h(k+1/2)) error estimates for the Q(k) finite element. Then we study a differential model for viscoelastic flow with unknowns u the velocity, p the pressure, and sigma the viscoelastic part of the extra-stress tensor. The approximations are ((Q(1))(2) transforms of) Q(k+1) continuous for u, Q(k) discontinuous for sigma, and P-k discontinuous for p, with k greater than or equal to 1. Upwinding for a is obtained by the Galerkin discontinuous method. We show that an error estimate of order O(h(k+1/2)) is valid in the energy norm for the three unknowns. (C) 1998 John Wiley & Sons, Inc.