多尺度间断有限体积元方法
发布时间：2018-09-12 17:00
【摘要】：有限体积元方法(FVEM)是求解偏微分方程的一类重要的数值方法.它可以被视为广义的有限差分方法,也可以看作一种广义的Petrov-Galerkin方法.该方法的关键之处在于控制体单元以及测试函数空间、检验函数空间的选取.间断有限体积元方法(DFVEM)可以被视为有限体积元方法与间断Galerkin方法的一种耦合,方法不需要逼近函数在单元边界处连续.本文考虑求解带有振荡系数的多尺度椭圆问题.由于系数的振荡性质,间断有限体积元方法无法准确地求解多尺度问题.超样本多尺度基函数可以有效地抓住解的多尺度信息.本文考虑基于超样本多尺度基函数的间断有限体积元方法求解多尺度问题,称之为多尺度间断有限体积元方法(MsDFVEM).本文给出的多尺度间断有限体积元方法,可以被视为多尺度间断Petrov-Galerkin方法(MsDPGM)的扰动.结合已有MsDPGM的结论,以及对扰动项的估计,在周期系数情形下,我们给出了 MsDFVEM严格的理论分析.最后,我们通过数值实验验证了方法的准确性和有效性.
[Abstract]:Finite volume element method (FVEM) is an important numerical method for solving partial differential equations. It can be regarded as a generalized finite difference method or a generalized Petrov-Galerkin method. The key points of this method are the control unit, the test function space and the selection of the function space. The discontinuous finite volume element method (DFVEM) can be regarded as a coupling between the finite volume element method and the discontinuous Galerkin method. In this paper, we consider solving multi-scale elliptic problems with oscillatory coefficients. Because of the oscillation property of the coefficients, the discontinuous finite volume element method can not solve the multi-scale problem accurately. The hypersample multiscale basis function can effectively capture the multi-scale information of the solution. In this paper, the discontinuous finite volume element method based on the supersample multiscale basis function is considered to solve the multi-scale problem, which is called the multi-scale discontinuous finite volume element method (MsDFVEM). The multiscale discontinuous finite volume element method presented in this paper can be regarded as the perturbation of the (MsDPGM) method of the multiscale discontinuous Petrov-Galerkin method. Combined with the existing conclusions of MsDPGM and the estimation of the perturbation term, we give a rigorous theoretical analysis of MsDFVEM in the case of periodic coefficients. Finally, the accuracy and validity of the method are verified by numerical experiments.
【学位授予单位】：南京大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82
，
本文编号：2239644
[Abstract]:Finite volume element method (FVEM) is an important numerical method for solving partial differential equations. It can be regarded as a generalized finite difference method or a generalized Petrov-Galerkin method. The key points of this method are the control unit, the test function space and the selection of the function space. The discontinuous finite volume element method (DFVEM) can be regarded as a coupling between the finite volume element method and the discontinuous Galerkin method. In this paper, we consider solving multi-scale elliptic problems with oscillatory coefficients. Because of the oscillation property of the coefficients, the discontinuous finite volume element method can not solve the multi-scale problem accurately. The hypersample multiscale basis function can effectively capture the multi-scale information of the solution. In this paper, the discontinuous finite volume element method based on the supersample multiscale basis function is considered to solve the multi-scale problem, which is called the multi-scale discontinuous finite volume element method (MsDFVEM). The multiscale discontinuous finite volume element method presented in this paper can be regarded as the perturbation of the (MsDPGM) method of the multiscale discontinuous Petrov-Galerkin method. Combined with the existing conclusions of MsDPGM and the estimation of the perturbation term, we give a rigorous theoretical analysis of MsDFVEM in the case of periodic coefficients. Finally, the accuracy and validity of the method are verified by numerical experiments.
【学位授予单位】：南京大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82
，
本文编号：2239644
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