Variational and Virtual Discretizations of Optimal Control Problems Governed by Diffusion Problems

摘要

In this paper, we aim to develop virtual element methods for approximating the optimal control problem governed by the diffusion equation with distributed control. Variational and virtual element discretization approaches are used for the approximation of the control variable, which depends on the state and costate variables that appeared in the formulation. In variational discretization, the control variable is not discretized explicitly, whereas, for virtual element discretizations, the approximation of control is sought in a discrete virtual space consisting of polynomial and non-polynomial functions. Moreover, the discretize-then-optimize approach is used for the computation of optimal control when the virtual element method is employed for the control. For seeking a numerical solution of state and costate equations, a virtual element method is used, and optimal a priori error estimates are established in suitable norms for the state, adjoint, and control variables. Numerical experiments are conducted over various convex and concave polygonal meshes to verify the theoretical findings and robustness of the method under mesh distortion.