Adaptive Mesh Method For Optimal Control Problem Of Stokes Equation

All the fluids in nature have certain viscosity.Because viscosity affects the shape and properties of fluid flow,the existence of viscosity brings great difficulties to the mathematical description and treatment of fluid flow.When studying the problem of water flow phenomenon,the change of density can be ignored,therefore liquid is regarded as incompressible fluid.The solution of many hydrodynamics problems is essentially to solve PDE problems,the complexity of the PDE problems also makes the finite element approach not reach the ideal precision.The adaptive mesh method uses the intelligent advantage as human to adjust the mesh by increasing,deleting or moving grid nodes,making the calculation more flexible.In this paper,the problems related to viscous incompressible Stokes flow arc considered.The r method(moving mesh method)in adaptive mesh method is adopted to solve the problems related to the viscous incompressible Stokes flow.The main research contents are as follows:1.For Stokes problem,the model,discrete form and inf-sup conditions used to guarantee the uniqueness of the solution are provided.The residual a posterior error estimation of Stokes equation is derived,the upper and lower bounds are proved to ensure its reliability and effectiveness.The residual a posterior error estimation is used to guide the process of grid change.A moving mesh algorithm based on residual a posterior error estimation is proposed.Two numerical examples are given to verify the efficiency of the algorithm,and the accuracy of the algorithm is improved while the number of the mesh nodes is kept unchanged.2.For the optimal control problem of Stokes equation,the model of the optimal control problem and the mixed finite element approximation are given.Secondly,the state equation of Stokes equation is analyzed,the sensitivity analysis results of the objective functional and the optimality condition are obtained by introducing Lagrange function,and the residual a posterior error estimation of the optimal control problem of Stokes equation is derived.A moving mesh algorithm-based on the results of sensitivity analysis of objective functional and error estimation-are proposed.The efficiency of the algorithm is verified by two benchmark examples,the results are also verified by comparing with the given objective state.