Homogenization And Numerical Solutions Of Two Kinds Of Flow Equations

In the field of composite materials, physics, mechanics, engineering, crystal and molec-ular structures, nuclear design, the initial and boundoury condtion problems are usually con-sidered on the periodic structures. Further, the scale of the period is comparatively small tothe scale of the whole media, which leads to the obstacle of convergence analysis and numer-ical calculations. However, the methods of homogenization overcome the difficulties above,the main idea of this method is to solve the homogenized equation instead of the originalone. The methods for homogenization mainly include multi-scale asymptotic expansions,energy method, two-scale convergence and stochastic method. In this research, two kindsof flow equations are taken into considerations: the Stokes equation and the cometary flowequation.As the first goal of this research, numerical procedures for solving the Stokes equationin porous media are shown: firstly the homogenized equation is given by homogenizationmethod, then the numerical method of solving cellar Stokes equation is given by mixed finiteelement method, and lastly the numerical results are given by Uzawa conjugate-gradientmethod.As the second goal of this research, the homogenization results of linear cometary flowequation with strong external field are shown. First, the homogenization result of the linearcometary flow equation with strong external magnetic field is given. Then the result ofthe linear cometary flow equation with strong external magnetic and electric field is given.Further, the homogenization result of the one dimensional linear cometary flow equationwith strong external electric field is given. Last, numerical results of one dimensionalcometary flow equation are calculated by splitting method and PWENO method.