LOCALLY STABILIZED FINITE ELEMENT METHOD FOR STOKES PROBLEM WITH NONLINEAR SLIP BOUNDARY CONDITIONS

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.

AParallel Pressure Projection Stabilized Finite Element Method for Stokes Equation with Nonlinear Slip Boundary Conditions

For the low-order finite element pair P1􀀀P1,based on full domain partition technique,a parallel pressure projection stabilized finite element algorithm for the Stokes equation with nonlinear slip boundary conditions is designed and analyzed.From the definition of the subdifferential,the variational formulation of this equation is the variational inequality problem of the second kind.Each subproblem is a global problem on the composite grid,which is easy to program and implement.The optimal error estimates of the approximate solutions are obtained by theoretical analysis since the appropriate stabilization parameter is chosen.Finally,some numerical results are given to demonstrate the hight efficiency of the parallel stabilized finite element algorithm.

Uzawa Iteration Method for Stokes Type Variational Inequality of the Second Kind

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.

Two-Level Newton Iteration Methods for Navier-Stokes Type Variational Inequality Problem

This paper deals with the two-level Newton iteration method based on the pressure projection stabilized finite element approximation to solve the numerical solution of the Navier-Stokes type variational inequality problem.We solve a small Navier-Stokes problem on the coarse mesh with mesh size H and solve a large linearized Navier-Stokes problem on the fine mesh with mesh size h.The error estimates derived show that if we choose h=O(|logh|^(1/2)H^(3)),then the two-level method we provide has the same H1 and L^(2) convergence orders of the velocity and the pressure as the one-level stabilized method.However,the L^(2) convergence order of the velocity is not consistent with that of one-level stabilized method.Finally,we give the numerical results to support the theoretical analysis.