不完全双二次有限体积元法

发布时间：2018-10-24 14:17

【摘要】：有限体积(元)法自上世纪七十年代末被李荣华教授以广义差分法的名称提出以来,研究成果层出不穷.该方法涉及到两套网格剖分和与之对应的两个函数空间:原始网格剖分上的试探函数空间,对偶网格剖分上的分片常数或分片低次多项式空间,即检验函数空间.本文取不完全双二次有限元空间作为试探函数空间.所谓的不完全双二次元是指:限制在一个原始单元上的每个型值点处的型函数是一个不完全双二次多项式.它的型值定义在四边形单元的四个顶点和四边中点上.本文研究了不完全双二次元,构造了新的数值方法——不完全双二次有限体积元法.试探函数空间取等参的不完全双二次有限元空间,检验函数空间取定义在对偶单元上的分片常数函数空间.构造了四种不同的对偶网格剖分,前两种是容易想到的非退化的对偶网格剖分,后两种是退化的对偶网格剖分.针对四种不同的网格分别建立了相应的有限体积格式,并给出了稳定性分析和收敛性分析.当对偶网格非退化时,给出了格式稳定的网格比范围;当对偶网格退化时,分析格式的稳定性条件,发现此时双线性形式限制在一个单元上对应的矩阵的最小特征值接近于0甚至小于0,说明这种格式的双线性形式在一个单元上不正定.进而证明了基于非退化格式的不完全双二次有限体积元法按H~1模度量为2阶收敛.最后本文使用所构造的格式求解Poisson方程的Dirichlet问题.数值结果表明,前两种格式的数值解按H~1模度量达到了最佳的2阶收敛;后两种格式,其数值解按H~1模度量为1阶收敛.这些结果进一步验证了理论分析的正确性。
[Abstract]:Since the finite volume (element) method was put forward by Professor Li Ronghua in the late 1970s as the name of the generalized difference method, the research results have been produced one after another. This method involves two sets of mesh generation and two corresponding function spaces: the heuristic function space on the original mesh generation, the piecewise constant or the piecewise low degree polynomial space on the dual mesh generation, that is, the test function space. In this paper, the incomplete biquadratic finite element space is taken as the heuristic function space. The so-called incomplete double quadratic element means that the type function at every type value point on a primitive unit is an incomplete biquadratic polynomial. Its type value is defined on the four vertices and midpoints of the quadrilateral element. In this paper, the incomplete double quadratic element is studied, and a new numerical method, the incomplete double quadratic finite volume element method, is constructed. The test function space is the piecewise constant function space defined on the dual element, and the test function space takes the incomplete biquadratic finite element space with isoparametric function space, and the test function space takes the piecewise constant function space defined on the dual element. Four different dual meshes are constructed. The first two are non-degenerate dual meshes which are easy to think of, and the latter two are degenerate dual meshes. The finite volume schemes are established for four different meshes, and the stability analysis and convergence analysis are given. When the dual mesh is nondegenerate, the stable mesh ratio range is given, and the stability condition of the scheme is analyzed when the dual grid is degenerate. It is found that the minimum eigenvalue of the matrix of bilinear form is close to 0 or less than 0, which indicates that the bilinear form of bilinear form is not positive definite on a unit. Furthermore, it is proved that the incomplete biquadratic finite volume element method based on nondegenerate scheme is of second order convergence according to the metric of H1 norm. Finally, we use the constructed scheme to solve the Dirichlet problem of Poisson equation. The numerical results show that the numerical solutions of the first two schemes have the best convergence of order 2 according to the metric of the first two schemes, and the numerical solutions of the latter two schemes have the convergence of the first order according to the metric of the first norm. These results further verify the correctness of the theoretical analysis.
【学位授予单位】：吉林大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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