Penalty finite element method for the Navier-Stokes equations

An analysis of a penalty formulation of the stationary Navier-Stokes equations for an incompressible fluid is presented. Subject to restrictions on the viscosity and prescribed body force, it is shown that there exists a unique solution to this penalty problem. The solution to the penalty problem is shown to converge to the solution of the Navier-Stokes problem as O(epsilon) where epsilon approaches zero is the penalty parameter. Existence, uniqueness and stability properties for the approximate problem are then developed and estimates for finite element approximation of the penalized Navier-Stokes problem are derived. Numerical studies are conducted to examine rates of convergence and sample numerical results presented for test cases.