The Gradient Discretization Method for Optimal Control Problems, with Superconvergence for Nonconforming Finite Elements and Mixed-Hybrid Mimetic Finite Differences

摘要

In this paper, optimal control problems governed by diffusion equations withDirichlet and Neumann boundary conditions are investigated in the framework ofthe gradient discretisation method. Gradient schemes are defined for theoptimality system of the control problem. Error estimates for state, adjointand control variables are derived. Superconvergence results for gradientschemes under realistic regularity assumptions on the exact solution isdiscussed. These super-convergence results are shown to apply to non-conforming$mathbb{P}_1$ finite elements, and to the mixed/hybrid mimetic finitedifferences. Results of numerical experiments are demonstrated for theconforming, nonconforming and mixed/hybrid mimetic finite difference schemes.