Optimal control model governed by Stokes equations has important applications in fluid mechanics and aerospace.In recent years,virtual element method has become one of the effective method to solve the partial differential equations due to its high flexibility of mesh generation.In this thesis we apply virtual element method to solve the optimal control problem governed by Stokes equations.We construct the virtual element discrete scheme and deduce the a priori as well as a posteriori error estimates.Consider the following optimal control problem:find(y,p,u)∈V × Q × Uad satisfying(?)J(y,u)=1/2‖y-yd‖L22(Ω)+γ/2‖u‖L22(Ω)subject to where J(y,u)is the objective functional,yd is the desired state,γ>0 is the regularization parameter,and Ω is a bounded domain in R2 with the boundary Γ.The spaces V and Q are defined as V:=H01(Ω),Q:L02(Ω)={q∈L2(Ω)s.t ∫ΩqdΩ=0}.The admissible control set Uad is defined by Uad={u∈L2(Ω):ua≤u(x)≤ub a.e.in Ω}.The quantities ua,ub∈R2 are constant vectors and the inequality ua≤u(x)≤ub is understood componentwise.For above optimal control problem,firstly,based on the virtual element approximation of the Stokes equations and variational discretization of the control variable u,we construct the virtual element discrete scheme for optimal control problem.Then a priori error estimates for state,adjoint state and control variable in L2 and H1 norms are derived.Further,based on the a posteriori error estimates of virtual element method for Stokes equations and the equivalence between the error of the optimal control problem and the error of the auxiliary problems,the upper and lower bounds for the a posteriori error estimates of the optimal control problem are derived.Finally,by using of projected gradient algorithm and adaptive virtual element algorithm,numerical experiments are given to verify the correctness of the theoretical analysis of the a priori error estimate and a posteriori error estimate. |