Several Finite Element Iterative Methods For The Incompressible Conduction-Convection Equations

The incompressible conduction-convection problem constitute an important system of equations in fluid dynamics, which is the coupled nonlinear dynamic sys-tem of viscous incompressible flow and temperature field. Because of the nonlinear, strong-coupling of velocity and pressure characteristics of the conduction-convection problem. And it is not very easy to realize the essence of the nonlinear phenomena, the numerical methods have played an important role to investigate the conduction-convection system. However, numerical simulation of conduction-convection equa-tions has a great difficulty, i.e., the contradiction between huge problem size and the limited computing ability. Hence, constructing and studying an algorithm with high efficiency and precision are very important. In this thesis, several finite ele-ment iterative methods for the two-dimensional steady and unsteady incompressible conduction-convection equations are studied as follows:1、First, we propose a two-level defect-correction method for the two-dimensional stationary conduction-convection equations. The method combine the defect-correction method, the two-level strategy and the stabilized techniques based on the local Gauss integration, which main idea is:solve a linearized defect-correction problem in coarse grid by Oseen iteration and correct the solution in the fine grid by Newton itera-tion. The stability and convergence of the proposed method are deduced. Finally, numerical examples show that the proposed algorithm is highly efficient and reliable for the stationary conduction-convection equations at high Reynolds number.2、Secondly, the two-level nonconforming stabilized finite element method is proposed for the two-dimensional stationary conduction-convection equations. The scheme combine the lowest equal order nonconforming finite element pair (Pinc-P1-P1) and two-level method. The method includes three different corrections:Stokes correction, Oseen correction and Newton correction. Moreover, the stability and convergence of the proposed method are deduced. Numerical results demonstrate that the proposed method saves a lot of CPU time and shows higher accuracy than the conforming one.3、Then, a Crank-Nicolson extrapolation scheme is proposed for the two-dimensional time-dependent conduction-convection equations. Mixed finite element method is applied for the spatial approximation of the velocity, pressure and tem-perature. The time discretization is based on the Crank-Nicolson scheme for the linear term and semi-implicit scheme for the nonlinear term. Moreover, the sec-ond order convergence of error estimations are derived. Finally, numerical tests show that the suggested method can quickly and effectively deal with the unsteady conduction-convection problem.4、Finally, a rigorous analysis of the defect-correction finite element method is investigated for the two-dimensional time-dependent conduction-convection prob-lem which based on the Crank-Nicolson scheme. The method mainly consists of two steps:solve a nonlinear problem with an added artificial viscosity term on a finite element grid and correct the solutions on the same grid using a linearized defect-correction technique. The stability and error estimate of the fully discrete scheme are derived. As a consequence, the effectiveness of the method to deal with time-dependent conduction-convection equations at high Reynolds number is illustrated in several numerical experiments.