# Numerical analysis for the pure Neumann control problem using the gradient discretisation method

## TL;DR

This paper presents a unified analysis of the gradient discretisation method for Neumann boundary control problems, deriving optimal error estimates and demonstrating super-convergence through numerical experiments across various numerical schemes.

## Contribution

It introduces a general framework that applies to multiple numerical schemes for Neumann control problems, including new super-convergence results for post-processed control variables.

## Key findings

- Optimal order error estimates for state, adjoint, and control variables.
- Super-convergence of post-processed control variables.
- Numerical experiments confirm theoretical convergence rates.

## Abstract

The article discusses the gradient discretisation method (GDM) for distributed optimal control problems governed by diffusion equation with pure Neumann boundary condition. Using the GDM framework enables to develop an analysis that directly applies to a wide range of numerical schemes, from conforming and non-conforming finite elements, to mixed finite elements, to finite volumes and mimetic finite differences methods. Optimal order error estimates for state, adjoint and control variables for low order schemes are derived under standard regularity assumptions. A novel projection relation between the optimal control and the adjoint variable allows the proof of a super-convergence result for post-processed control. Numerical experiments performed using a modified active set strategy algorithm for conforming, nonconforming and mimetic finite difference methods confirm the theoretical rates of convergence.