A least-squares finite element formulation for unsteady incompressible flows with improved velocity–pressure coupling

In the weak form Galerkin formulation for incompressible flows, the pressure has a well-understood role. At all times, it may be interpreted as a Lagrange multiplier that enforces the divergence-free constraint on the velocity field. This is not the case in least-squares formulations for incompressible flows, where the divergence-free constraint is enforced in a least-squares sense in a variational setting of residual minimization. Thus, the role of the pressure in a least-squares formulation is rather vague. We find that this lack of velocity–pressure coupling in least-squares formulations may induce spurious temporal pressure oscillations when using the non-stationary form of the equations. We present a least-squares formulation with improved velocity–pressure coupling, based on the use of a regularized divergence-free constraint. A first-order system least-squares (FOSLS) approach based on velocity, pressure and vorticity is used to allow the use of practical C0 element expansions in the finite element model. We use high-order spectral element expansions in space and second- and third-order time stepping schemes. Excellent conservation of mass and accuracy of computed pressure metrics are demonstrated in the numerical results.