GALERKIN DISCONTINUOUS APPROXIMATION OF THE TRANSPORT-EQUATION AND VISCOELASTIC FLUID-FLOW ON QUADRILATERALS

Numerical simulation of industrial processes involving viscoelastic li quids 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 Galer kin 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 viscoe lastic flow with unknowns u the velocity, p the pressure, and sigma th e 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.