Prediction-Correction Method For Solvingstokes Problems With Leak Boundaryconditions

The Stokes problems is an important elliptic boundary value problem,which appears in many engineering applications.Although the Stokes equation is linear and easy to discretize,some complicated boundary conditions lead to great difficulties in mathematical modeling,theoretical analysis and numerical solution.In this work,the numerical method of Stokes problem under leak boundary conditions is considered.Firstly,the Uzawa iterative method is extended to the Stokes problem with leak boundary conditions,then a prediction-correction method is proposed to deal with the difficulty of parameter selection.In Section 3,Uzawa iterative method is considered for the Stokes problem with leak boundary conditions.To solve the problem numerically,a multiplier in a convex set is introduced such that the variational inequality is equivalent to the variational identity,and the solution of the variational identity satisfies the saddle point problem of Lagrangian function.The Uzawa iterative method is proposed to solve the saddle point problem.The convergence of the algorithm is proved and the convergence rate is obtained.Numerical results verify the feasibility of Uzawa iterative method,and show the leak and no-leak phenomena of the problem.Section 4 focuses on the prediction-correction method for the Stokes problem with leak boundary conditions.Since the Uzawa iterative method converges only for the small and positive penalty parameter,and it is difficult to choose the proper penalty parameter.In order to improve the Uzawa iterative method,a prediction-correction method is proposed to adjust parameter automatically in correction step.The main advantage of this method is that each iterative step consists of two linear problems and the method is convergent for any initial parameter.The convergence of the method is analyzed,and the numerical results show the feasibility and effectiveness of the prediction-correction method.