Error estimates for semi-discrete gauge methods for the Navier-Stokes equations

Abstract

The gauge formulation of the Navier-Stokes equations for incompressible fluids is a new projection method. It splits the velocity u = a + ∇ ϕ \mathbf {u}=\mathbf {a}+\nabla \phi in terms of auxiliary (nonphysical) variables a \mathbf {a} and ϕ \phi and replaces the momentum equation by a heat-like equation for a \mathbf {a} and the incompressibility constraint by a diffusion equation for ϕ \phi . This paper studies two time-discrete algorithms based on this splitting and the backward Euler method for a \mathbf {a} with explicit boundary conditions and shows their stability and rates of convergence for both velocity and pressure. The analyses are variational and hinge on realistic regularity requirements on the exact solution and data. Both Neumann and Dirichlet boundary conditions are, in principle, admissible for ϕ \phi but a compatibility restriction for the latter is uncovered which limits its applicability.