Standard and asymptotic finite element methods for incompressible viscous flows

The effect of the penalty parameter on the solution accuracy is investigated. The existing error estimates have been a posteriori justified for various biquadratic and bicubic velocity elements. New mixed and penalty asymptotic finite element methods are developed. The proposed methods improve the supremum error bounds of the standard mixed and penalty methods by a factor of 0(epsilon/n+1/)(where epsilon is a small characteristic parameter and n is the order of the asymptotic approximation). Numerical examples for contained flows are included and are discussed.