Based on the low-order conforming finite element subspace （Vh, Mh） such as the P1-P0 triangle element or the Q1-P0 quadrilateral element, the locally stabilized finite element method for the Stokes problem with nonlinear slip boundary conditions is investigated in this paper. For this class of nonlinear slip boundary conditions including the subdifferential property, the weak variational formulation associated with the Stokes problem is an variational inequality. Since （Vh, Mh） does not satisfy the discrete inf-sup conditions, a macroelement condition is introduced for constructing the locally stabilized formulation such that the stability of （Vh, Mh） is established. Under these conditions, we obtain the H1 and L2 error estimates for the numerical solutions.
In this paper, the Uzawa iteration algorithm is applied to the Stokes problem with nonlinear slip boundary conditions whose variational formulation is the variational inequality of the second kind. Firstly, the multiplier in a convex set is introduced such that the variational inequality is equivalent to the variational identity. Moreover, the solution of the variational identity satisfies the saddle-point problem of the Lagrangian functional ζ. Subsequently, the Uzawa algorithm is proposed to solve the solution of the saddle-point problem. We show the convergence of the algorithm and obtain the convergence rate. Finally, we give the numerical results to verify the feasibility of the Uzawa algorithm.
This paper deals with the boundary integral method to study the Navier-Stokes equations around a rotating obstacle. The detail of this method is that the exterior domain is truncated into a bounded domain and a new exterior domain by introducing some open ball BR, and the nonlinear problem in the bounded domain and the linearized problem in the new exterior domain are considered and the approximation coupled problem is obtained. We show that the error between the solution u of Navier-Stokes equations around a rotating obstacle and the solution ue of the approximation coupled problem is O（R-1/4） in the Hl-seminorm when Iwl does not exceed some constant.
In this work, a new numerical scheme is proposed for thermal/isothermal incompressible viscous flows based on operator splitting. Unique solvability and stability analysis are presented. Some numerical result are given, which show that the proposed scheme is highly efficient for the thermal/isothermal incompressible viscous flows.
