A collocation penalty least-squares finite element formulation for incompressible flows

Abstract

We present a penalty least-squares finite element formulation for incompressible flows. The pressure degree of freedom is eliminated using the classical penalty approach and the least-squares model is formed in terms of velocity, vorticity, and dilatation. An iterative penalization is implemented, which allows the use of low penalty parameters and retains a manageable conditioning number of the global coefficient matrix. The h-convergence is verified using the exact solution of Kovasznay’s flow. Numerical results are presented for a number of benchmark problems, e.g. steady flow past a large circular cylinder in a channel, transient flow over a backward facing step, unsteady flow past a circular cylinder, and buoyancy driven natural convection in a square cavity. For all numerical examples, the effect of penalty parameter on the accuracy is investigated thoroughly and it is concluded that the present model produces accurate results for low penalty parameters (10–100). These problems are also solved on coarse meshes to show that good mass conservation is achieved.