Prove the sharper theoretical state-error convergence rate

Establish theoretically that the state error for the monolithic discontinuous Galerkin discretization of Darcy optimal control with Radon-measure tracking and pointwise control constraints converges at the sharper rate O(h^{2\alpha}), rather than only at the established rate O(h^{\min\{\theta,1-\epsilon\}}), where \theta=\min\{\alpha,2\alpha-1\}.

Background

Theorem establishes an L^2 error bound for the discrete state at rate O(h^{\min{\theta,1-\epsilon}}), with \theta=\min{\alpha,2\alpha-1} determined by the elliptic regularity index and the reduced regularity of the measure-driven adjoint. The numerical experiments nevertheless suggest that the state error itself converges at the sharper rate O(h^{2\alpha}). The unresolved issue is to close this theoretical gap by proving the sharper state-error estimate for the SIPG discretization and coupled pointwise-control optimality system analyzed in the paper.