Adaptive weights for mass conservation in a least-squares finite element method

In this paper, we develop a least-squares (LS) finite element method for the Stokes equations. The LS method uses the L-2-norm of the residuals of the continuity equation multiplied by appropriately adjusted weights. To adjust the weight, we employ an adaptive weight iteration approach based on a residual type a posteriori error estimator for the LS functional. This LS method is applied to flows through a planar channel, lid-driven cavity flows, and flows past a transverse slot. The results demonstrate that mass conservation of the LS method in fluid simulations can be significantly improved by appropriately adjusting weights, and that this can be accomplished using low-order basis functions, without substantial complications in coding. We provide a posteriori error estimates for the linearized velocity-vorticity-pressure first-order system and show numerical results supporting the estimate. Numerical results reveal that the mass conservation constant is problem dependent.