热传导—对流问题的混合有限元法的一些研究

【作者】 田向军；

【导师】 罗振东； 谢正辉；

【作者基本信息】 首都师范大学 ， 应用数学， 2003， 硕士

【摘要】 本文分别给出了非定常的热传导-对流问题的Crank-Nicolson混合元法时间二阶精度全离散格式，非线性Galerkin混合元法时间二阶精度全离散格式以及定常的热传导-对流问题回溯二重水平法。讨论了时间上的Crank-Nicolson离散方法应用于非定常的热传导-对流问题的空间离散的Galerkin混合元近似。在对方程的解的正则性进行适宜假设的条件下证明了时间二阶精度的误差估计。给出了热传导-对流问题非线性Galerkin全离散混合元解的存在性和收敛性。在某些已有结论的基础之上，我们证明了这种格式对于时间离散上的二阶精度。提出了一种解决定常的热传导-对流问题的有限元近似中出现的非线性问题的两层方法。这种二层方法解一个小的，非线性的粗网格系统，一个细网格上的Oseen问题以及一个粗网格上的线性校正问题。同时，给出了这种近似解的存在性和收敛性。更多还原

【Abstract】 In this paper, a Crank-Nicolson mixed element method , a nonlinear Galerkin mixed element method for the non stationary conduction-convection problems time second order accuracy fully discrete formats and a two-level mixed element method with backtracing for the stationary conduction-convection problems are presented and analyed,respectively, an error analysis are provided for the Crank-Nicolson method of time discretization applied to spatially discrete Galerkin mixed element approximations of the nonstationary conduction-convection problems. Second order estimates are proven in time under realistic assumptions about the regularity of the solution. The existence and the convergence of the fully discrete format of nonlinear Galerkin mixed element method with time second order accuracy for the non stationary conduction-convection problems are showen. On the basis of some conclusion, we have proved that the schemes have second-order convergence accuracy for the time discretization, a two-level method for resolving the nonlinearity in finite element approximation of the stationary conduction-convection problems is presented. The two-level method involves solving one small, nonlinear coarse mesh system, one Oseen problem on the fine mesh and one linear correction problem on the coarse mesh. Meanwhile, the existence and the convergence of the approximate solutions are shown.更多还原