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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 two and three dimensions. 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.

The Gauge--Uzawa Finite Element Method. Part I: The Navier--Stokes Equations