Weak imposition of boundary conditions for the gauge formulation of the incompressible Navier–Stokes equations

Abstract

The projection method was first introduced by Chorin [Bull. AMS 73 (1967), pp. 928–931] and Temam [Arch. Rat. Mech. Anal. 33 (1969), pp. 377–385] as a computationally efficient numerical method to solve the incompressible Navier–Stokes equations. Despite its success in decoupling the computations of velocity and pressure, it suffers from inaccurate numerical boundary layers. As an effort to resolve this inaccuracy, E and Liu [Int. J. Numer. Meth. Fluids 34 (2000), pp. 701–710] proposed the gauge method, which is a reformulation of the Navier–Stokes equations in terms of an auxiliary vector field and a gauge variable. This method utilizes the freedom of choosing a boundary condition for the gauge variable to reduce the numerical coupling between the considered variables. Nevertheless, the computational implementation of the boundary conditions for the auxiliary vector field is difficult in the context of finite elements since they involve either the normal or tangential derivative of the gauge variable. In order to circumvent this issue, we propose a weak formulation of the boundary conditions based on the symmetric Nitsche method. Computational results are presented to illustrate the accuracy of the proposed method.

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