Some Finite Element Algorithms Based On Domain Decomposition For Incompressible Flow Problems

The Navier-Stokes equations and its coupling equation depict a series of im-portant physical models in Computational Fluid Dynamics.They describe basic movement of some fluids in our life.By analyzing the model,it can help us to better understand the essence of fluid motion.However,due to its strong coupling and nonlinearity,the exact solutions of these equations are very difficult to obtain.Therefore,constructing stable and efficient numerical algorithms to solve the Navie-Stokes equations or its coupling equations are very important.A lot of computation time and storage space are needed when solving the equa-tions.Therefore,in order to improve computational efficiency and reduce computing resources,the following works are done in this paper:1.Propose a parallel two-step algorithm based on the finite element method and domain decomposition technique to solve the Navier-Stokes equation.The algo-rithm mainly combines a finite element method with domain decomposition method to improve the computational efficiency.The domain decomposition technique can achieve parallel computation,so the computing time will be greatly reduced.An-other advantage is that the combination of the two step method and the domain decomposition makes the error smaller than the original two-step method.That is to say,we can get more accurate results with less degrees of freedom.2.Devise a simplified two-grid algorithm to solve stationary incompressible natural convection problem.The efficient two-grid algorithm consists of solving one initial problem based on a lower order finite element pair P₁b-P₁-P₁on a coarse mesh,then solving a linearized system based on a higher order finite element pair P₂-P₁-P₂on a fine mesh.The stability and the convergence of the simplified two-grid algorithm are deduced,which demonstrate the convergence rates of the method are the same as the P₂-P₁-P₂finite element method for solving natural convection equations.Next,a parallel two-grid algorithm is proposed based on the simplified two-grid algorithm and fully overlapping domain decomposition technique.The parallel algorithm is more efficient.