The aim of this paper is to study a stationary Brinkman problem in an anisotropic porous medium by using a mini-element method with a general boundary condition. One of the important aspects of the $P1-Bubble/P1$ method is satisfying the inf-sup condition, which allows us the existence and the uniqueness of the weak solution to our problem. To go further in this theoretical study, an a priori error estimate is established. To see the importance of this method in reality, we applied this method to a real problem. The numerical simulation studies support our results and demonstrate the effectiveness of this method.

This paper considers numerical methods for approximating and simulating the Stokes–Darcy problem, with a new boundary condition. We study a robust stabilized fully mixed discretization technique. This method ensures stability of the finite element scheme and does not use any Lagrange multipliers to introduce a stabilization term in the temporal Stokes–Darcy problem discretization. A correct finite element scheme is obtained and its convergence analysis is done. Finally, the efficiency and accuracy of these numerical methods are illustrated by different numerical tests.