We formulate a projection-based stabilization finite element technique for solving steadystate natural convection problems. In particular, we consider heat transport through combined solid and fluid media. This stabilization does not act on the large flow structures. Based on the projection stabilization idea, finite element error analysis of the problem is investigated and optimal errors for the velocity, temperature and pressure are established. We also present some numerical tests which both verify the theoretical predictions and demonstrate the method's promise.

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The present work is devoted to introduce the backward Euler based modular time filter method for MHD flow. The proposed method improves the accuracy of the solution without a significant change in the complexity of the system. Since time filters for fluid variables are added as separate post processing steps, the method can be easily incorporated into an existing backward Euler code. We investigate the conservation and long time stability properties of the improved scheme. Stability and second order convergence of the method are also proven. The influences of introduced time filter method on several numerical experiments are given, which both verify the theoretical findings and illustrate its usefulness on practical problems.Here, u, P := p + s 2 |B| 2 , p and B denote the unknown velocity, modified pressure, pressure and magnetic field, respectively. The body forces f and ∇ × g are forcing on the velocity and magnetic field, respectively. Also, Re is the Reynolds number, Re m is the magnetic Reynolds number, and s is the coupling number. The Lagrange multiplier (dummy variable) λ corresponds to the solenoidal constraint on the magnetic field. In the continuous case, provided the initial condition B 0 is solenoidal, then the use of λ is unnecessary, see [5]. However, when discretizing with the finite element method, this solenoidal constraint is needed to be enforced explicitly and thus the additional variable is required. We also assume that the system (1.1)-(1.4) is equipped with homogeneous Dirichlet boundary conditions for velocity and the magnetic field.Due to the coupling of the equations of the velocity and the magnetic field, developing efficient, accurate numerical methods for solving MHD system (1.1)-(1.4) remains a great challenge in computational fluid dynamics community. It is well known that time filter methods combined with leapfrog scheme are commonly used in geophysical fluid dynamics to reduce spurious oscillations to improve predictions, see e.g. [3], but these methods degrade the numerical accuracy and over damps the physical mode. A successfully tuned model was developed by Williams [29] reducing undesired numerical damping of [3] with higher order accuracy, see [2,22,24,30] and references therein. On the other hand, in practice, the use of the backward Euler method is often preferred to extend a code for the steady state problem and this yields stable but inefficient time accurate transient solutions, see [10]. To improve this behavior, time filters are used to stabilize the backward Euler discretizations in [14] for the classical numerical ODE theory.The present work extends the method of [7] tailored to MHD flows for constant time step. As it is mentioned in this study, the constant time step method is equivalent to a general second order, two step and A-stable method given in [9] and [19]. The scheme we consider is the time filtered backward Euler method, which is efficient, O(∆t 2 ) and amenable to implementation in existing legacy codes. In addition, we also consider the numerical conse...

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