An alternative least-squares formulation of the Navier–Stokes equations with improved mass conservation

The focus of this paper is on incompressible flows in three dimensions modeled by least-squares finite element methods (LSFEM) and using a novel reformulation of the Navier–Stokes equations. LSFEM are attractive because the resulting discrete equations yield symmetric, positive definite systems of algebraic equations and the functional provides both a local and global error measure. On the other hand, it has been documented for existing reformulations that certain types of boundary conditions and high-aspect ratio domains can yield very poor mass conservation. It has also been documented that improved mass conservation with LSFEM can be achieved by strengthening the coupling between the pressure and velocity. The new reformulation presented here is demonstrated to provide both improved multigrid convergence rates because it is differentially diagonally dominant and improved mass conservation over existing methods because it increases the pressure–velocity coupling along the inflow and outflow boundaries.