Least-Squares Methods for Incompressible Newtonian Fluid Flow: Linear Stationary Problems

This paper develops and analyzes two least-squares methods for the numerical solution of linear, stationary incompressible Newtonian fluid flow in two and three dimensions. Both approaches use the L norm to define least-squares functionals. One is based on the stress-velocity formulation (see section 3.2), and it applies to general boundary conditions. The other is based on an equivalent formulation for the pseudo-stress and velocity (see section 4.2), and applies to pure velocity Dirichlet boundary conditions. The velocity gradient and vorticity can be obtained algebraically from this new tensor variable. It is shown that the homogeneous least-squares functionals are elliptic and continuous in the H(div; Ω) × H(Ω) norm. This immediately implies optimal error estimates for conforming finite element approximations. As well, it admits optimal multigrid solution methods if Raviart-Thomas finite element spaces are used to approximate the stress or the pseudo-stress tensor.