间断、组合多尺度有限元方法的分析与计算

发布时间：2018-10-21 20:07

【摘要】：关于多尺度模型问题(包括带奇性多尺度问题)的研究,在科学和工程上有着非常广泛的应用.本文针对多尺度模型问题分别提出了多尺度间断Galerkin方法(包括多尺度间断有限元方法和多尺度间断Petrov-Galerkin方法)；传统有限元和超样本多尺度Petrov-Galerkin方法相结合的组合方法；多尺度间断有限体积元方法.在第二章中,我们研究了多尺度间断Galerkin方法(MsDGM),包括多尺度间断有限元和多尺度间断Petrov-Galerkin方法.DG方法在处理曲边问题和非一致,非结构网格等情况时更有优势,而且DG格式具有局部守恒性质.这些优点在多尺度问题中有很多应用MsDGM是多尺度方法和DG方法的耦合,其主要思想是在超样本多尺度有限元空间中利用DG格式进行多尺度数值模拟.本章针对多尺度问题分别采用了间断有限元和间断Petrov-Galerkin两种数值求解方法.在DG的格式下,共振误差消失了.另外,Petrov-Galerkin方法可以降低计算复杂性.我们给出了误差分析并进行了数值模拟.数值试验显示数值方法是有效的.在第三章中,我们提出了组合有限元和超样本多尺度Petrov-Galerkin方法(FE-OMsPGM)用于求解奇性多尺度椭圆问题.例如,地下水流模拟中的通道问题或井-区域附近的奇性问题.FE-OMsPGM的基本思想是：先将计算区域分为问题区域和普通区域,其中问题区域是奇性所在的区域.然后在问题区域上利用细网格上的传统有限元方法,在普通区域上利用超样本多尺度Petrov-Galerkin方法,两种区域的交界面上的衔接问题利用加罚技术来处理FE-OMsPGM融合了FEM和OMsPGM的优点,自由度比传统FEM的少,处理奇性问题的效果比OMsPGM的效果好.我们给出了误差分析和相应的数值试验.数值结果显示了FE-OMsPGM的正确性和有效性.在第四章中,我们研究了间断有限体积元方法(DFVEM).有限体积元方法是一种质量守恒格式,在计算流体动力学中有很广泛的应用.间断有限体积元方法融合了DG和FVEM二者的优点.我们构造了一种新的DFVEM,较之前的DFVEM,不同之处在于控制体的选取.本章研究的目的是为了提出求解多尺度模型问题的多尺度间断有限体积元方法.误差分析和相应的数值试验验证了方法的正确性.在第五章中,我们提出了多尺度间断有限体积元方法(MsDFVEM)求解多尺度问题MsDFVEM是多尺度方法与间断有限体积元方法的一种耦合,其基本思想是在超样本多尺度有限元空间中利用间断有限体积元方法逼近多尺度解,不仅可以准确的抓住小尺度的信息,同时也可以获得粗网格上的质量守恒.MsDFVEM可以看作是MsDPGM的一个小的扰动,因此在MsDPGM的基础上,我们只需对扰动项进行分析,进而给出MsDFVEM的误差分析.
[Abstract]:The study of multi-scale model problems (including singularity multi-scale problems) has been widely used in science and engineering. In this paper, the multi-scale discontinuous Galerkin method (including the multi-scale discontinuous finite element method and the multi-scale discontinuous Petrov-Galerkin method) and the combination of the traditional finite element method and the super-sample multi-scale Petrov-Galerkin method are proposed to solve the multi-scale model problems, respectively. Multiscale discontinuous finite volume element method. In the second chapter, we study the multiscale discontinuous Galerkin method (MsDGM), which includes the multiscale discontinuous finite element method and the multiscale discontinuous Petrov-Galerkin method. The DG method has advantages in dealing with curved edge problems and non-uniform, unstructured meshes, etc. Moreover, the DG scheme has the property of local conservation. Many of these advantages are coupled by MsDGM and DG methods in multi-scale problems. The main idea is to use DG scheme to carry out multi-scale numerical simulation in super-sample multi-scale finite element space. In this chapter, two numerical methods, discontinuous finite element method and discontinuous Petrov-Galerkin method, are used to solve the multi-scale problem. In the DG scheme, the resonance error disappears. In addition, the Petrov-Galerkin method can reduce computational complexity. Error analysis and numerical simulation are given. Numerical experiments show that the numerical method is effective. In chapter 3, we propose a combined finite element method and a super-sample multi-scale Petrov-Galerkin method (FE-OMsPGM) for solving singular multi-scale elliptic problems. For example, the passage problem in groundwater flow simulation or the singularity problem near the well- region. The basic idea of FE-OMsPGM is that the computational region is divided into the problem region and the ordinary region, where the problem region is the region where the singularity is located. Then the traditional finite element method on fine mesh is used in the problem area, and the hyper-sample multi-scale Petrov-Galerkin method is used in the ordinary area. The problem of connection at the interface between the two regions is solved by adding penalty technique to deal with the advantages of FEM and OMsPGM. The degree of freedom is less than that of traditional FEM, and the effect of dealing with singularity is better than that of OMsPGM. Error analysis and corresponding numerical experiments are given. Numerical results show the correctness and validity of FE-OMsPGM. In chapter 4, we study the discontinuous finite volume element method (DFVEM).) Finite volume element method (FVM) is a mass conservation scheme, which is widely used in computational fluid dynamics (CFD). The discontinuous finite volume element method combines the advantages of DG and FVEM. We construct a new DFVEM, which is different from the previous DFVEM, in the selection of control bodies. The purpose of this chapter is to propose a multiscale discontinuous finite volume element method for solving multiscale model problems. The correctness of the method is verified by error analysis and corresponding numerical experiments. In chapter 5, we propose a multiscale discontinuous finite volume element method (MsDFVEM) for solving multiscale problem MsDFVEM is a coupling of the multiscale method and the discontinuous finite volume element method. Its basic idea is to use the discontinuous finite volume element method to approach the multi-scale solution in the hyper-sample multi-scale finite element space, which can not only accurately grasp the small scale information. At the same time, we can obtain the mass conservation on rough meshes. MsDFVEM can be regarded as a small disturbance of MsDPGM. Therefore, on the basis of MsDPGM, we only need to analyze the perturbation term and then give the error analysis of MsDFVEM.
【学位授予单位】：南京大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82

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