Hydrodynamic design optimization using non stationary porous media model

In this paper, we focus on the penalty finite element method for the non stationary porous media model. We begin by showing the existence and uniqueness of the solution for the initial problem. Error estimates for the velocity and the pressure are obtained via the energy method. We introduce a time discretization by the use of a backward Euler scheme combined with fully discrete finite element method to approximate the penalized problem and establish an error estimate for the velocity and the pressure which will be used to show the convergence of the approximate solution to the solution of the initial problem. The shape optimization problem is to find the shape which is optimal in that it minimizes a cost functional related to a comfort fish population. We derive the adjoint system associated to the penalized problem. We compute the gradient in terms of state and adjoint variables. The optimization procedure is implemented using the continuous adjoint method and the finite element method. Numerical simulations are presented to show the efficiency and the robustness of the proposed method. •We are interested in the existence and uniqueness of the solution for the initial porous media problem.•We derive an interesting error estimates for the velocity and the pressure for penalized problem via the energy method.•We demonstrate the convergence of the finite element approximation solution to the solution of the initial problem.•We provide an intrinsic characterization of the gradient in terms of state and adjoint variables.