Driss Yakoubi

Aziz Takhirov, Driss Yakoubi

We develop a novel iterative approach for solving the incompressible magnetohydrodynamics problem. The main idea is to split the velocity-momentum and magnetic induction equations with respect to the diffusive terms. As a result, we get a smaller system that is iteration-level-dependent, along with two Stokes systems that need to be assembled only once. We also extended the scheme to the Els{ä}sser variables reformulation of the equations. For both schemes, we established boundedness and convergence. Several numerical experiments are presented to show the effectiveness of the schemes.

Aziz Takhirov, Driss Yakoubi

We develop a novel and efficient iterative scheme for solving incompressible steady Navier-Stokes equations. The method is an adaptation of the Incremental Viscosity Splitting approximation for unsteady flows to steady equations. At each nonlinear iteration, the scheme requires solving an elliptic PDE for the velocity variable and a system with an SPD matrix for the pressure variable, which remains the same across all nonlinear iterations. The method can also be interpreted as an algebraic splitting approach. We prove boundedness and geometric convergence. Numerical tests illustrate the efficiency of the proposed algorithm.

Sarra Maarouf, Driss Yakoubi

This paper presents the unsteady Darcy's equations coupled with two nonlinear reaction-diffusion equations, namely this system describes the mass concentration and heat transfer in porous media. The existence and uniqueness of the solution are established for both the variational formulation problem and for its discrete one obtained using spectral discretization. Optimal a priori estimates are given using the Brezzi-Rappaz-Raviart theorem. We conclude by some numerical tests which are in agreement with our theoretical results.