二阶椭圆问题自适应有限体积元方法的多重网格算法
发布时间：2018-07-29 11:25
【摘要】：有限体积元方法又称为控制体积法,盒式方法,广义差分法,是在有限差分方法和有限元方法的基础上发展起来的求解偏微分方程的重要数值方法.有限体积元方法既保持了有限差分方法便于计算的优点,又保持了有限元方法的计算精度,同时又具有局部守恒性质,这使得该方法在流体力学和热传导方程等诸多领域上获得广泛的关注.本文着重研究了二阶椭圆问题的自适应有限体积元方法的多重网格算法,即在网格加密过程中不再进行一致加密,只在后验误差较大的区域利用最新顶点二分法进行局部加密,从而使得问题的自由度大幅降低,减少问题的计算量,提高计算的精度.在处理问题时,我们可以将有限体积元方法看作是有限元方法的一个扰动,然后依据自适应有限元方法多重网格算法的收敛性结果,建立自适应有限体方法的相关收敛性理论.文章首先简单介绍了自适应有限体积元方法的国内外研究现,和本文选题的研究意,其次叙述了最新顶点二分法的步骤,有限体积元方法的构造思想和多重网格的相关理论.最后文章证明了问题的收敛性分析结果,并给出数值算例来验证理论的结果.
[Abstract]:The finite volume element method, also known as the control volume method, the box method and the generalized difference method, is an important numerical method to solve the partial differential equation based on the finite difference method and the finite element method. The finite volume element method not only maintains the advantages of the finite difference method, but also keeps the calculation precision of the finite element method. In this paper, the multigrid algorithm of adaptive finite volume element method for two order elliptic problems is studied in this paper, that is, no uniform encryption is carried out in the process of grid encryption, only the posteriori error is larger. The region uses the latest vertex dichotomy for local encryption so that the degree of freedom of the problem is greatly reduced, the computational complexity of the problem is reduced, and the accuracy of the calculation is improved. The convergence theory of the adaptive finite body method is established. First, the paper briefly introduces the research at home and abroad of the adaptive finite volume element method and the research meaning of this topic. Secondly, it describes the steps of the latest vertex dichotomy, the construction of the finite volume element method and the related theory of the multigrid. Finally, the theory of the finite volume element method and the related theory of the multigrid are also described. The convergence of the problem is proved and a numerical example is given to verify the theoretical results.
【学位授予单位】：烟台大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82
本文编号：2152530
[Abstract]:The finite volume element method, also known as the control volume method, the box method and the generalized difference method, is an important numerical method to solve the partial differential equation based on the finite difference method and the finite element method. The finite volume element method not only maintains the advantages of the finite difference method, but also keeps the calculation precision of the finite element method. In this paper, the multigrid algorithm of adaptive finite volume element method for two order elliptic problems is studied in this paper, that is, no uniform encryption is carried out in the process of grid encryption, only the posteriori error is larger. The region uses the latest vertex dichotomy for local encryption so that the degree of freedom of the problem is greatly reduced, the computational complexity of the problem is reduced, and the accuracy of the calculation is improved. The convergence theory of the adaptive finite body method is established. First, the paper briefly introduces the research at home and abroad of the adaptive finite volume element method and the research meaning of this topic. Secondly, it describes the steps of the latest vertex dichotomy, the construction of the finite volume element method and the related theory of the multigrid. Finally, the theory of the finite volume element method and the related theory of the multigrid are also described. The convergence of the problem is proved and a numerical example is given to verify the theoretical results.
【学位授予单位】：烟台大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82
【参考文献】
相关期刊论文 前2条
1 ;A COARSENING ALGORITHM ON ADAPTIVE GRIDS BY NEWEST VERTEX BISECTION AND ITS APPLICATIONS[J];Journal of Computational Mathematics;2010年06期
2 ;Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems[J];Science in China(Series A:Mathematics);2006年10期
，本文编号：2152530
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