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.

This paper describes numerical solutions of incompressible Navier-Stokes equations with a new boundary condition. To solve this problem, we use the discretization by mixed finite element method. We use a vector extrapolation method for computing numerical solutions of the steady-state Navier-Stokes equations. In addition, two types of a posteriori error indicator are introduced and are shown to give global error estimates that are equivalent to the true error. A numerical experiment on the driven cavity flow is given to demonstrate the effectiveness of the vector extrapolation method. We compare the result with the solution from commercial code like ADINA system as well as with values from other simulations.

In this work, we introduce the steady Stokes equations with a new boundary condition, generalizes the Dirichlet and the Neumann conditions. Then we derive an adequate variational formulation of Stokes equations. It includes algorithms for discretization by mixed finite element methods. We use a block diagonal preconditioners for Stokes problem. We obtain a faster convergence when applying the preconditioned MINRES method. Two types of a posteriori error indicator are introduced and are shown to give global error estimates that are equivalent to the true discretization error. In order to evaluate the performance of the method, the numerical results are compared with some previously published works or with others coming from commercial code like Adina system.