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.

在本文中，我们将重点放在非平稳多孔介质模型的惩罚有限元方法上。我们首先说明最初问题的解决方案的存在性和唯一性。速度和压力的误差估计是通过能量方法获得的。

我们通过使用反向Euler方案与完全离散有限元方法相结合来引入时间离散化来近似受罚问题，并建立速度和压力的误差估计，这将用于显示近似解的收敛性。解决最初的问题。

形状优化问题是找到最佳形状，因为它使与舒适鱼种群有关的成本函数最小化。我们推导与惩罚问题相关的伴随系统。我们根据状态和伴随变量来计算梯度。优化过程是使用连续伴随方法和有限元方法实现的。

数值仿真表明了该方法的有效性和鲁棒性。