Research Of Two-level Brezzi-pitkaranta Stabilized Finite Methods For Navier-stokes Equations

Navier-Stokes equations is very important in the field of science andengineering and is a typical problem. Because it is difficult to find a exactsolutions of Navier-Stokes equations, people can understand theproperties of solutions to Navier-Stokes equations from numericalsolutions.This thesis mainly studies the numerical solutions of Navier-Stokesequations and Navier-Stokes type variational inequality problems. It isorganized as follows.In Chapter2, we study the two-level stabilized methods forNavier-Stokes equations. We present a new stabilized finite elementmethod for incompressible flows based on Brezzi-Pitkaranta stabilizedmethod. For classical one-level method, error estimates of numericalsolutions in some norms are derived. Combining the techniques oftwo-level methods, we propose two-level Oseen/Stokes/Newton iteration methods corresponding to three different linearized methods, andshow the error estimates of three methods. We also propose a Newtoncorrection scheme based on the above two-level iteration methods.Finally, some numerical experiments are given to support the theoreticalresults and to verify the efficiency of these two-level iteration methods.In Chapter3, we study the two-level stabilized methods forNavier-Stokes type variational inequality problems. Similarily, we presentthe classical one-level method and two-level Newton method forNavier-Stokes type variational inequality problems based on Brezzi-Pitkaranta stabilized method. The error estimates for finite elementsolution are derived. Finally, the numerical experiments are given tosupport the theoretical results and to verify the efficiency of thesetwo-level Newton methods.