Stabilization Algorithm For Optimal Control Problem Governed By Stokes Equation

In this paper,the grad-div stabilized finite element method for control con-strained optimal control problem governed by steady Stokes equation and unsteady Stokes equation with small viscosity coefficient is investigated.A priori error anal-ysis is given for state,adjoint state and control variables.Firstly,we consider steady Stokes optimal control problem:finding(y,p,u)?V × Q × Uad such thatsubject toHere yd is a target velocity field,y(x)denotes the velocity,and p(x)stands for the pressure.? = Re-1 denotes the viscosity coefficient with Re being the Reynolds number.?>0 is a given regularization(or control cost)parameter.Uad is the admissible set of control space.We derive the continuous first order optimality condition by using the La-grangian multiplier method.The regularity of the control problem is analyzed.Grad-div stabilization technique is used to approximate the state equation,while the control variable is discretized by variational discretization method.The grad-div stabilization discrete scheme for the control problem is built up.Discrete first order optimality conditions for the control problem are derived by using the La-grangian multiplier method.A priori error estimates for the state variable,adjoint variable and control variable are derived.Numerical example is presented to verify the theoretical findings.Then,we consider unsteady Stokes optimal control problem:finding(y,p,u)?V × Q × Uad such thatsubject towhere yd ?L2(0,T;L2(?))is a target velocity field?y(x,t)denotes the velocity and p(x,t)denotes the pressure,?=Re-1 is the viscosity coefficient with Re being is the Reynolds number??>0 is a given regularization(or control cost)parameter and ?>0,Uad is the admissible set of control space.The continuous first order optimality condition is derived by using the La-grangian multiplier method.The backward-Euler grad-div stabilized finite elemnt discrete scheme for the control problem is constructed by continuous piecewise fi-nite element discretization of state variable and piecewise constant discretization of control variable.Discrete first order optimality conditions for the control problem are derived by using the Lagrangian multiplier method.A priori error estimates for the state variable,adjoint variable and control variable are derived.