Convergence Analysis of Fractional Time-Stepping Techniques for Incompressible Fluids with Microstructure

Abstract

We present and analyze fully discrete fractional time stepping techniques for the solution of the micropolar Navier Stokes equations, which is a system of equations that describes the evolution of an incompressible fluid whose material particles possess both translational and rotational degrees of freedom. The proposed schemes uncouple the computation of the linear and angular velocity and the pressure. We develop a first order scheme which is unconditionally stable and delivers optimal convergence rates, and an almost unconditionally stable second order scheme.

A decoupling finite element method with different time steps for the micropolar fluid model

“…However, Algorithm 1 with r = 5 can save much computational time. Moreover, it is found that the results of our algorithm are similar to those in [15,33,42]. Therefore, Algorithm 1 captures this model well.…”

“…As expected, Algorithm 1 with r = 5 requires less CPU-time than Algorithm 1 with r = 1 to achieve nearly the same relative error. In this example, we consider an important practical problem, the stirring of a passive scalar, which has been tested in [33]. In order to model the passive scalar, we combine the micropolar fluid model Firstly, consider Ω = (−1, 1) 2 .…”

“…Nochetto et al [26] have developed a semi-implicit fully discrete scheme which, at each time step, decouples the computation of the linear and angular velocities but requires the solution of a saddle-point problem for the determination of the linear velocity and the pressure. In order to remedy this shortcoming, by applying a fractional time stepping technique, Salgado [33] has developed an unconditionally stable first-order scheme and an almost unconditionally stable second-order scheme. In [27], a fully discrete penalty finite element method has been proposed and analyzed, and suboptimal error estimates have been proved.…”

“…The three nondimensional numbers appearing in (2.1) are the Newtonian kinematic viscosity µ, the dynamic micro-rotation viscosity κ, and the angular viscosity γ [24]. Furthermore, f = (f 1 , f 2 , 0) and g = (0, 0, g 3 ) represent smooth, externally applied forces and moments [33], respectively.…”

The equations of ferrohydrodynamics: Modeling and numerical methods

“…[29,23]) is LBB stable for ≥ 2 under minor restrictions of the mesh T h . Note also in (66), that we are using a continuous finite element space P for the pressure, which is something we might have to change (the velocity space too) if we want to consider convergence of a numerical scheme under minimal regularity assumptions. The use of discontinuous pressures will be considered later in §5 for a slightly different model.…”