Error estimates and superconvergence of mixed covolume approximations for elliptic optimal control problems

In this scholarly article, we analyze the results of error estimates and phenomena of superconvergence associated with the mixed covolume approximation method, which is applied to a particular class of linear elliptic optimal control problems. The control variable is discretized using piecewise constant functions. Additionally, the state and the costate variables are both approximated using the lowest-order Raviart–Thomas ($ RT_0 $) mixed finite element method. First, mixed covolume approximation of optimal control problems is constructed. Second, 'a priori error estimations' for each variable are computed. Third, a superconvergence result is established, and it is proved that there exists a second-order superconvergence relationship between the centroid interpolation of variable $ u $ and its numerical solution. Finally, two carefully designed numerical examples are presented to validate the reliability of the theoretical findings, providing concrete evidence to corroborate the above results and strengthen the coherence of the study's conclusions.