Numerical analysis for the pure Neumann control problem using the gradient discretisation method
Provides a general framework for analyzing a wide class of numerical schemes for Neumann control problems, but the contribution is incremental as it extends existing GDM analysis to a specific boundary condition.
The paper develops a unified analysis for distributed optimal control problems with pure Neumann boundary conditions using the gradient discretisation method, deriving optimal error estimates and a super-convergence result for post-processed control, confirmed by numerical experiments with various schemes.
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.