Optimal Error Estimate of the Penalty Finite Element Method for the Micropolar Fluid Equations

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

“…Despite the fact that, as mentioned in the previous paragraph, system (1.1) has a great deal of practical applications, to the best of our knowledge only two references deal with its discretization: [27] proposes and analyzes a penalty projection-method and suboptimal error estimates are proved. Reference [26], develops a semi-implicit fully discrete scheme which, at each time step, decoupled the computation of the linear and angular velocities but required the solution of a saddle-point problem for the determination of the linear velocity and pressure.…”

Pressure‐correction projection methods for the time‐dependent micropolar fluids

“…Chen 4 studied a numerical method based on the projection method and space finite difference method to solve unsteady incompressible micropolar problem. Ortega‐Torres and Rojas‐Medar 5 proved the optimal error estimates of the velocity, pressure and angular velocity, are proved for the fully discrete penalty finite‐element method of the MNSE problem. Nochetto 6 proposed a first‐order semi‐implicit fully discrete finite element method of MNSE, which decouple the linear velocity and angular velocity of each time step.…”

“…A popular strategy to overcome this difficulty is to relax the incompressibility constraint in an appropriate way, resulting in a class of projection methods, 8‐12 among which are rotation pressure‐correction method, 13 the penalty method, 5,14,15 the velocity correction method, 16 and the Gauge‐Uzawa projection method 17 and so on. More importantly and appealing, using projection methods, one only needs to solve a sequence of decoupled elliptic equations for the velocity and the pressure at each time step, making it very efficient for large scale numerical simulations.…”

Efficient and Unconditional Energy Stable Schemes for the Micropolar Navier-Stokes Equations

“…From a numerical point of view, there are some attempts in designing numerical schemes for the micropolar Navier-Stokes equations. For instance, a penalty projection method is proposed and suboptimal error estimates are proved in [17]; a semi-implicit fully discrete scheme is presented in [15] which requires to solve a saddle point problem for velocity and pressure at each time step; in a related work [21], a fractional time stepping technique is proposed to decouple the computation of pressure and velocity. The nonlinear terms in these work are treated either implicitly or semi-implicitly, so that a coupled linear or nonlinear system with variable coefficients has to be solved at each time step.…”