Two-level Variational Multiscale Finite Element Methods For Two Class Of Equations In Fluid Dynamics

It knows that Navier-Stokes equation is a mathematical physics equation and is used to describe the law of ?uid ?ow, which has been widely used in many ?elds. While the numerical solution of large Reynolds number problem is an important problem in numerical simulation of incompressible ?ow problems. When we make the numerical simulation directly, the required mesh scale is small enough. Even now,the computer technology is highly developed, the direct numerical simulation of high Reynolds number problem is still a big challenge. In this paper, based on the variational multiscale method, we study twolevel ?nite element algorithms for steady Navier-Stokes equations and Smagorinsky model. The speci?c contents are as follows:1. Based on the variational multiscale method, we study two-level iteration penalty ?nite element algorithm for steady Navier-Stokes equations. The two-level mesh algorithms are as follows: on the coarse mesh,we apply the iteration penalty method and the variational multiscale method for solving the nonlinear Navier-Stokes equation; on the ?ne mesh, based on Newton linearization method, we solve the linearized Navier-Stokes equations. We get the H1 errors for velocity and L2 errors for pressure in theory. Finally, we give numerical examples to verify the results of theoretical analysis by appropriately select the mesh scale and the penalty parameters.2. Based on the the variational multiscale method, we study twolevel ?nite element algorithm for steady Smagorinsky model. The twolevel mesh algorithms are as follows: on the coarse mesh, we apply the the variational multiscale method for solving the nonlinear Smagorinsky model; on the ?ne mesh, based on Newton linearization method, we solve the linearized Smagorinsky model. Similarly, we get the H1 errors for velocity and L2 errors for pressure in theory. Finally, we also give numerical examples to verify the results of theoretical analysis.