Superconvergence Of Triangular Mixed Finite Elements Solutions For Optimal Control Problems

Efficient numerical method is the key for us to investigate the practical problemsusing the optimal control. Although there are lots of numerical methods had been used, the finite element method is the most important method, which has perfect theory and common program. Presently, there are few works have done to analysis the theory about the optimal control problems with the mixed finite element method.In this paper, to obtain a high accuracy about the gradient for the optimalcontrol problems, which absorb the gradient of the state variables in the objective function, we use triangular mixed finite element methods to investigatethe quadratic convex optimal control problems. We employ two different finite element spaces to approximate the control and state variables, either the order k = 1 Raviart-Thomas mixed finite element spaces to approximate the state and co-state, or piecewise constant functions to approximate the control variable. Firstly, we get the optimal conditions with the variational principle, which translatesthe minimized question to a system about the state, co-state and variational inequality. Then, we define the interpolation u_I for the control, and we replace the approximation of control with the approximation of co-state variable with the variational inequality. We introduce state function and co-state function for the elements in the control set, so, we can approximate the error with some sub-errors. We prove the superconvergence error estimate of h² between the approximated solution and the interpolation of the exact control. We give a postprocessing operatorfor the low-regularity of control. Obviously, the postprocessing operator is different for different control set, because the postprocessing operator matchescontrol set. In this paper, the control set is U_ad = {u∈L²(Ω):∫_Ωu(x)dx≥0},so, we define the postprocessing operator is u^* = g(h) = (max(0,(?)_h)-z_h)/v. Moreover, by postprocessing technique, we find that the projection of the discrete co-state is superclose (in order h²) to the exact control. And we present two numerical examples to test the superconvergence theory.