Optimal Mixed $H-P$ Finite Element Methods for Stokes and Non-Newtonian Flow

Abstract

Based upon a new mixed variational formulation for the three-field Stokes equations and linearized Non-Newtonian flow, an $h-p$ finite element method is presented with or without a stabilization. As to the variational formulation without stabilization, optimal error bounds in $h$ as well as in $p$ are obtained. As with stabilization, optimal error bounds are obtained which is optimal in $h$ and one order deterioration in $p$ for the pressure, that is consistent with numerical results in [9,12] and therefore solved the problem therein. Moreover, we proposed a stabilized formulation which is optimal in both $h$ and $p$.