A comparative study of mixed least-squares FEMs for the incompressible Navier-Stokes equations

In the present contribution we compare different mixed least-squares finite element formula-tions (LSFEMs) with respect to computational costs and accuracy. In detail, we consider an approach for Newtonian fluid flow, which is described by the incompressible Navier-Stokes equa-tions. Starting from the residual forms of the equilibrium equation and the continuity condition, various first-order systems are derived. From these systems least-squares functionals are con-structed by means of L2-norms, which are the basis for the associated minimization problems. The first formulation under consideration is a div-grad first-order system resulting in a three-field formulation with stresses, velocities, and pressure as unknowns. This S-V-P formulation is approximated in H(div) × H1 × L2 on triangles and for comparison also in H1 ×H1 × L2 on quadrilaterals. The second formulation is the well-known div-curl-grad first-order velocity-vorticity-pressure (V-V-P) formulation. Here all unknowns are approximated in H1 on quadri-laterals. Besides some numerical advantages, as e.g. an inherent symmetric structure of the system of equations and a directly available error estimator, it is known that least-squares methods have also a drawback concerning mass conservation, especially when lower-order ele