The gradient discretisation method for optimal control problems, with super-convergence for non-conforming finite elements and mixed-hybrid mimetic finite differences

TL;DR

This paper develops and analyzes the gradient discretisation method for optimal control problems governed by diffusion equations, demonstrating super-convergence for non-conforming finite elements and mimetic finite differences through theoretical error estimates and numerical experiments.

Contribution

It introduces a unified gradient discretisation framework for optimal control problems and proves super-convergence results for non-conforming finite elements and mimetic finite differences.

Findings

Super-convergence achieved for non-conforming $ ext{P}_1$ finite elements.

Super-convergence demonstrated for mixed-hybrid mimetic finite differences.

Numerical experiments confirm theoretical error estimates.

Abstract

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.