Recovery A Posteriori Error Estimates For General Convex Optimal Control Problems

Superconvergence and recovery a posteriori error estimates of the finite element approximation for general convex optimal control problems are investigated in this paper. The control variable is approximated by piecewise constant functions, and both the state y and the co-state p by piecewise linear finite element functions. We prove the superconvergence error estimate in L~2-norm between the approximated solution and the L~2-projection of the control, and superconvergence error estimates in H~1-norm between the approximated solutions and the elliptic projections of the state and co-state. Then, we use the recovery operator mentioned in [30] to postprocess the approximated solutions. By using the superconvergence results, we get recovery a posteriori error estimates which are asymptotically exact under some regularity conditions. Because the regularity of the control variable and the state is very different, and the singularity of them is often in the different position in the area, so in our experiment, different finite element spaces on different meshes are used for the approximation of the singularity of the control and the state. Then we must look for a suitable preconditioning for the projection algorithm. A new model is mentioned in our experiment, while the preconditioning used in [30] is inefficient for the new model. We adopt an interpolation function as a preconditioning instead, which is efficient for both our model and quadratic convex optimal control problems. Numerical examples are provided to verify the theoretical results.