Self-adaptive Alternating Direction Multiplier Method For Stokes Problems With Nonlinear Slip Boundary Conditions

The Stokes problem has a wide range of applications in fluid mechanics,and the nonlinear Stokes problem is usually transformed into a variational form in many research.In this paper,the alternating direction multiplier method is applied to Stokes problems with nonlinear slip boundary conditions.To improve the performance of method,a self-adaptive alternating direction multiplier method is proposed with automatically selects the penalty parameter.In the two-dimensional case,the Stokes problem under nonlinear slip boundary conditions is equivalently transformed into a variational equation problem.An auxiliary variable on the boundary is introduced to derive saddle point problem based on an augmented Lagrangian function,which can be solved by the alternating direction multiplier method.So get a three-step iterative method,this method mainly solves a linear sub-problem,and at the same time explicitly solves auxiliary variables and Lagrange multipliers.Since the convergence speed of the alternating direction multiplier method is significantly dependent on the penalty parameter,and it is difficult to determine the appropriate parameter in advance.In this paper,the self-alternating direction multiplier method is combined with an adaptive rule that uses an iterative function to automatically select appropriate penalty parameters.Thus,the self-adaptive alternating direction multiplier method is obtained.This solves the problem that the algorithm convergence speed obviously depends on the penalty parameter,and the penalty parameter can be selected automatically and effectively.Then the convergence of the self-adaptive alternating direction multiplier method for the Stokes problem under nonlinear slip boundary conditions is analyzed theoretically,and the numerical results verify the feasibility and effectiveness of the proposed method.Finally,this paper summarizes the research work of the self-adaptive alternating direction multiplier method for the Stokes problem with nonlinear slip boundary conditions,and prospects the subsequent research in the future.