The Gauge-Uzawa finite element method. Part I: the Navier-Stokes equations

Jae-Hong Pyo
Department of Mathematics
Purdue University
West Lafayette , IN 47907, USA

Ricardo H. Nochetto
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA


The Gauge-Uzawa FEM is a new first order fully discrete projection method which combines advantages of both the Gauge and Uzawa methods within a variational framework. A time step consists of a sequence of $d+1$ Poisson problems, $d$ being the space dimension, thereby avoiding both the incompressibility constraint as well as dealing with boundary tangential derivatives as in the Gauge Method. This allows for a simple finite element discretization in space of any order in both 2d and 3d. This first part introduces the method for the Navier-Stokes equations of incompressible fluids and shows unconditional stability and error estimates for both velocity and pressure via a variational approach under realistic regularity assumptions. Several numerical experiments document performance of the Gauge-Uzawa FEM and compare it with other projection methods.