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Author: 杨超
Issued Date: 2007-06-05
Degree Grantor: 中国科学院软件研究所
Place of Degree Grantor: 软件研究所
Degree Level: 博士
Keyword: 非规则网格 ; Cauchy-Riemann方程 ; 六边形网格 ; 三向交错网格 ; 成对四点差分格式 ; 六边形有限元 ; 离散谱预条件子
Alternative Title: Discretizations and Fast Solutions for Elliptic PDEs over a type of Irregular Structured Grid
Abstract: 本文着眼于椭圆型偏微分方程的数值求解,重点研究了其在一类非规则结构化网格上的有限差分和有限元离散方法以及相应的快速解法。 经典有限元和有限差分方法在处理二维问题时多采用三角形和四边形网格。然而,平面正则剖分的方式除了三角形和四边形外还有六边形,并且与三角形和四边形相比六边形更接近于圆。六边形也广泛存在于自然界以及诸如材料科学、核工程等很多应用领域之中。这些因素启发我们研究六边形网格。本文研究的六边形网格主要有两类:一是对偶六边形网格,可以看作是平面三向三角形网格(由单一三角形剖分构成的网格)的对偶网格;另一类称作平行六边形网格,是由正六边形网格做仿射变换后得到的。 在本文第一部分,我们构造了对偶六边形网格上Laplace算子的成对四点差分格式,证明并且通过数值实验印证了该格式虽然仅有一阶截断误差,但能够达到二阶整体精度。我们证明的主要思想是利用矩阵变换,可以把对偶六边形网格解耦为两套粗三向三角形网格,而三角形网格上的七点差分格式则具有二阶精度。通过调用粗网格上的快速求解器,也可以把这种解耦思想用于求解四点格式的离散线性方程组。 本文第二部分的工作是研究平行六边形上的有限元。我们简要回顾了三种构建六边形元的思想,并重点研究了基于不完全多项式插值的方法。我们选择的不完全多项式空间是三向坐标意义下的三线性和旋转三线性空间,分别对应于基于边和基于顶点的六自由度六边形元。并且我们还提出对插值条件可以再额外增加单元上平均积分,从而获得两种新的七自由度六边形元。对这四种非协调元,我们都给出了先验误差估计,它们在L2模(或能量模)下都能达到二阶(或一阶)精度。这一误差估计也得到了数值实验的验证。 在第三部分,我们把目光放在典型的一阶椭圆方程组——Cauchy-Riemann 方程上。受传统笛卡尔交错网格的启发,我们提出一类新的非规则交错网格,它由任意三向三角形网格及其对偶六边形网格组成。我们构造了这种网格上的一类三色差分格式,并且证明了这种格式与四点格式类似,都有二阶整体精度,虽然截断误差仅一阶。数值实验验证了差分格式的精度,并且检验了我们利用网格解耦思想提出的快速解法的效果。
English Abstract: This dissertation concerns itself with numerical solutions for elliptic PDEs (Partial Differential Equations). Special attention is paid on discretization and fast solution methods over a type of irregular structured grid. Both finite difference methods (FDMs) and finite element methods (FEMs) are concerned. Classical FDMs/FEMs for two dimensions often treat with triangular or quadrilateral grid. However, besides triangles and quadrilaterals, hexagons also form a regular tessellation ofthe plane. Moreover, Hexagons extensively exist in the nature as well as in some application fields, such as in material sciences and nuclear engineering. This inspires us to consider hexagonal grid. Two types of hexagonal grid are studied in our work. The first one, referred as dual hexagonal grid, can be viewed as the dual grid to the three-directional triangular grid (from triangle tiling). And the second one is consisted by small parallel hexagons that are affine-equivalent to the regular hexagon. In the first part of the dissertation, we focus on the dual hexagonal grid, on which a coupled four-point difference scheme for Laplacian operator is proposed. We prove that the four-point scheme achieves second order global accuracy despite its first order truncation error, which is also verified by our numerical experiments. The main idea in the proof is based on the fact that through matrix transformations, a dual hexagonal grid can be decoupled into two coarse three-directional triangular grid, on which a seven-point difference scheme with second order accuracy is well established. The decoupling techniques also open up ways to utilizing the fast solvers of triangular grid when handling linear systems derived from the coupled four-point scheme over dual hexagonal grid. The second part of our work is finite element methods on parallel hexagons. Three main construction approaches for hexagonal elements are briefly over-viewed. And among them, the incomplete polynomial interpolation methods are further discussed. The incomplete polynomial spaces we choose are the trilinear and rotated trilinear spaces in terms of the so-called three-directional coordinates, subject to edge-oriented and vertex-oriented elements of six degrees of freedom, respectively. By adding an extra degree of freedom over the element face, we construct another two elements. A priori error estimates are given to show that all the four nonconforming elements achieve first order accuracy in the energy norm and second order in the L2 norm. Some numerical results are listed to confirm the theoretical analysis. In the third part, we focus on a typical first order elliptic equations, the Cauchy-Riemann equations. Motivated by the traditional Cartesian staggered grid, we proposed a new irregular staggered grid consisted by an arbitrary three-directional triangular grid and its dual hexagonal grid, on which a three-colored finite difference scheme is constructed. Like the coupled four-point scheme, this scheme has second order accuracy despite first order truncation error. Based on grid decoupling, we give a detailed proof for this phenomenon, as well as a fast solving algorithm. Some numerical tests are also listed to show the accuracy of the scheme and test our fast solver.
Language: 中文
Content Type: 学位论文
Appears in Collections:中科院软件所

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Recommended Citation:
杨超. 一类非规则网格上椭圆问题离散与求解研究[D]. 软件研究所. 中国科学院软件研究所. 2007-06-05.
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