A Legendre Galerkin Spectral Method For Optimal Control Problem Governed By Stokes Equations

Flow control problem,discussed in this paper,has become a very active and successful research area,and has been wildly used in petroleum,chemical and aeronautical engineering.In fact,it had a great deal of effect upon society and economical profits.As other optimal controls,efficient numerical methods are very important to successful applications of flow control.Nowadays,the finite element method is undoubtedly the most wildly used numerical method in computing flow control problems and other control problems. Also,finite element approximation can achieve higher accuracy when the solutions have lower regularities.In recent years,adaptive finite element method and mixed finite element method have got great development in computing optimal control problems,and spectral method has been used in computing unconstrained control problems.Spectral method is a kind of both old and new numerical method for solving partial differential equations,which has been wildly used in numerical computation of meteorology,physics,mechanics and so on for the last four decades.At the same time,its numerical analysis theory has become more and more perfected. Nowadays,spectral method becomes the third numerical method to solve partial differential equations after the finite difference method and the finite element method.Spectral method has the great superiority of infinite convergence rate with smooth solutions to the above two methods.With Smooth solutions,spectral method needs only few work on computation,and can get higher accuracy.In this paper,we adopt the Legendre-Galerkin method,used in[46],to solve the distributed optimal control problem governed by stokes equations.The discretized control problem satisfies the well-known Babu(?)ka-Brezzi conditions by choosing an appropriate pair of discretization spaces for the velocity and the pressure, Constructing suitable base functions of discretization spaces leads to systems with sparse matrixes.We first derive a priori error estimates of the Legendre-Galerkin approximation of the unconstrained control problems,and the convergence order for the velocity is O(N^1-m),and O(N^3/2-m) for the pressure when the state is in H^m(Ω).Then,thanks to the higher regularity of control,both the priori and the posteriori error estimates of the Legendre-Galerkin approximation are obtained for the control problem with integral constrained set by adopting appropriate auxiliary system where the convergence order of priori error estimate is like above.At last,two numerical examples are implied to illustrate the error estimates.