The paper deals with the numerical realization of the 3D Stokes flow subject to threshold slip boundary conditions. The weak velocity-pressure formulation leads to an inequality type problem that is approximated by a mixed finite element method. The resulting algebraic system is non-smooth. Besides the pressure, three additional Lagrange multipliers are introduced: the discrete normal stress releasing the impermeability condition and two discrete shear stresses regularizing the non-smooth slip term. Eliminating the discrete velocity component we obtain the minimization problem for the smooth functional, expressed in terms of the pressure, the normal, and the shear stresses. This problem is solved either by a path following variant of the interior point method or by the semi-smooth Newton method. Numerical scalability is illustrated by computational experiments.

本文讨论了在阈值滑移边界条件下的3D斯托克斯流的数值实现。弱的速度压力公式导致不等式类型问题，该问题通过混合有限元法近似。所得的代数系统是非光滑的。除压力外，还引入了三个附加的拉格朗日乘数：离散的法向应力释放了不可渗透性条件，两个离散的切应力使非光滑滑移项正规化。消除离散速度分量，我们获得了光滑函数的最小化问题，用压力，法向和切应力表示。该问题可以通过内部点方法的变种路径或半光滑牛顿法来解决。