The gradient discretisation method for optimal control problems, with super-convergence for non-conforming finite elements and mixed-hybrid mimetic finite differences
It provides a unified theoretical framework and practical super-convergence guarantees for a class of numerical schemes in optimal control, benefiting researchers in numerical analysis and computational optimal control.
The paper develops error estimates and super-convergence results for optimal control problems governed by diffusion equations using the gradient discretisation method, achieving super-convergence for non-conforming P1 finite elements and mixed-hybrid mimetic finite differences under realistic regularity assumptions.
In this paper, optimal control problems governed by diffusion equations with Dirichlet and Neumann boundary conditions are investigated in the framework of the gradient discretisation method. Gradient schemes are defined for the optimality system of the control problem. Error estimates for state, adjoint and control variables are derived. Superconvergence results for gradient schemes under realistic regularity assumptions on the exact solution is discussed. These super-convergence results are shown to apply to non-conforming $\mathbb{P}_1$ finite elements, and to the mixed/hybrid mimetic finite differences. Results of numerical experiments are demonstrated for the conforming, nonconforming and mixed/hybrid mimetic finite difference schemes.