Optimal control for Darcy's equation in a heterogeneous porous media

In this paper, we investigate optimal control problems in heterogeneous porous media. The optimal control problem is governed by the Darcy's flow equation; where the pressure is the state variable and the source/sink is the control variable. Then we introduce the reduced optimal control problem which contains only the state variable by replacing the control variable with a dependent quantity of the state variable based on the Darcy's equation. Here we employ $C^0$ interior penalty finite element methods for the spatial discretization to solve the reduced optimal control problem resulting in a fourth-order variational inequality. We use $P_2$ Lagrange finite elements for $C^0$ interior penalty methods, which require fewer degrees of freedom than $C^1$ finite element methods. We provide a priori error estimates and stability analyses by considering a heterogeneous permeability coefficient. Several numerical examples validate the given theories and illustrate the capabilities of the proposed algorithm.