Efficient stress-velocity least-squares finite element formulations for the incompressible Navier-Stokes equations

In this contribution three different mixed least-squares finite element methods (LSFEMs) are investigated with respect to accuracy and efficiency with simultaneous consideration of regular and adaptive meshing strategies. The deliberations are made for the incompressible Navier-Stokes equations. The generally known first-order div-grad system in terms of the (total) stress, velocity and pressure (SVP) formulation is the basis for two further div-grad least-squares formulations in terms of (total) stress and velocity (SV), whereby both formulations are derived from different fountainheads. The extended SV formulation is an enrichment of the known stress-velocity formulation proposed by Cai et al. (2004). The second SV formulation is based on the substitution of the pressure in the SVP formulation, thus it is based on a discontinuous pressure interpolation. Advantage of the SV formulations is a smaller system matrix size due to the reduction of the relevant degrees of freedom. Nevertheless, the approximation quality has to last the demand of the established least-squares formulations. The drawback of a poor mass conservation is well-investigated for the LSFEM and known to overcome by using high-order interpolations for all solution variables. Therefore, the main attention of this contribution is focused on accuracy, while aiming for high efficiency, which is intrinsic for the fundamental idea in here. In the light of efficient LSFEM solutions the advantage of the inherent a posteriori error estimator of the method is used for deliberations on different marking strategies in an h-type adaptive mesh refinement.