抛物型积分微分方程的对称间断有限体积元方法
发布时间：2019-03-02 14:17
【摘要】：一般情况下,用有限元等方法模拟对称的抛物型积分微分问题得到的刚度矩阵是对称的,因而是一种对称方法,然而用间断有限体积元方法模拟此问题时,我们得到的刚度矩阵是非对称的,因而它是一种非对称的方法,这就造成求解有限元解时方法单一,并且程序运行所占空间大.鉴于此,本文研究对称的间断有限体积元方法.本文首先对如下抛物型积分微分方程的初边值问题(?)提出了一种新的数值模拟方法-—对称间断有限体积元方法.此方法是在间断有限体积元方法的基础上提出的,因此该方法具有间断有限体积元方法的优点,如构造有限元空间时不要求函数在穿越内部单元边界时保持连续,空间构造简单,并且具有高并行性、高精度等优点,同时也具有对称格式的一些优点:计算方法多样且在误差估计时简单明了.文中分别给出了该问题的半离散和全离散的对称间断有限体积元格式,并通过定义该问题的Sobolev投影得出了其对称间断有限体积元解具有L2模和离散的|||·|||1,h的最优阶误差估计;最后,数值实验支持了理论分析结果.
[Abstract]:In general, the stiffness matrix obtained by using finite element method to simulate symmetric parabolic Integro-differential problems is symmetric, so it is a symmetric method. However, when the discontinuous finite volume element method is used to simulate this problem, The stiffness matrix obtained by us is asymmetrical, so it is an asymmetric method, which results in a single method for solving the finite element solution and a large amount of space for the program to run. In view of this, the symmetric discontinuous finite volume element method is studied in this paper. In this paper, we first study the initial boundary value problems of parabolic Integro-differential equations (?) A new numerical simulation method, symmetric discontinuous finite volume element method, is proposed. This method is proposed on the basis of discontinuous finite volume element method, so this method has the advantages of discontinuous finite volume element method, for example, when constructing finite element space, the function is not required to be continuous when crossing the boundary of internal element. The space structure is simple, and it has the advantages of high parallelism, high precision and so on. At the same time, it also has some advantages of symmetric scheme: the calculation method is varied and the error estimation is simple and clear. In this paper, the semi-discrete and fully discrete symmetric discontinuous finite volume element schemes for the problem are given respectively. By defining the Sobolev projection of the problem, it is obtained that the solution of the symmetric discontinuous finite volume element has L2 modes and discrete | 1s. The optimal order error estimate of h; Finally, numerical experiments support the results of theoretical analysis.
【学位授予单位】：山东师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82
本文编号：2433125
[Abstract]:In general, the stiffness matrix obtained by using finite element method to simulate symmetric parabolic Integro-differential problems is symmetric, so it is a symmetric method. However, when the discontinuous finite volume element method is used to simulate this problem, The stiffness matrix obtained by us is asymmetrical, so it is an asymmetric method, which results in a single method for solving the finite element solution and a large amount of space for the program to run. In view of this, the symmetric discontinuous finite volume element method is studied in this paper. In this paper, we first study the initial boundary value problems of parabolic Integro-differential equations (?) A new numerical simulation method, symmetric discontinuous finite volume element method, is proposed. This method is proposed on the basis of discontinuous finite volume element method, so this method has the advantages of discontinuous finite volume element method, for example, when constructing finite element space, the function is not required to be continuous when crossing the boundary of internal element. The space structure is simple, and it has the advantages of high parallelism, high precision and so on. At the same time, it also has some advantages of symmetric scheme: the calculation method is varied and the error estimation is simple and clear. In this paper, the semi-discrete and fully discrete symmetric discontinuous finite volume element schemes for the problem are given respectively. By defining the Sobolev projection of the problem, it is obtained that the solution of the symmetric discontinuous finite volume element has L2 modes and discrete | 1s. The optimal order error estimate of h; Finally, numerical experiments support the results of theoretical analysis.
【学位授予单位】：山东师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82
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