论文标题
PDE近似引起的大型矩阵的光谱分析和快速方法
Spectral analysis and fast methods for large matrices arising from PDE approximation
论文作者
论文摘要
本论文的主要目标是显示在分析PDE近似和设计快速方法来解决相关线性问题的矩阵序列中扮演符号的关键作用。在第一部分中，我们研究由$ \ mathbb {p} _k $ lagrangian有限元元素在dirichlet边界条件下以及操作员为$ \ mathrm {div} \ left（$ a $ a $ a $ a $ a $ c cdot）$ c. mathrm {Div} $ c. $ \叠加ω$，$ω$是$ \ mathbb {r}^d $，$ d \ ge 1 $的开放和有限子集。我们研究了Weyl感中的光谱分布，并简要概述了定位，聚类，极端特征值和渐近条件。我们详细研究了$ω=（0,1）^2 $的恒定系数的情况，并且在可变系数和更通用域的情况下给出了一个简短的帐户。虽然在第二部分中，我们设计了一种快速的多式类型类型的方法，用于解决$ \ mathbb {q} _K $有限元元素近似相同的问题和更高维度中相同问题的有限元近似。分析在一个维度上进行，而黑充血也以较高的尺寸进行$ d \ ge 2 $进行，这表明了对矩阵大小的依赖性和相对于尺寸$ d $的依赖性以及对多项式学位$ k $的稳健性。
The main goal of this thesis is to show the crucial role that plays the symbol in analysing the spectrum the sequence of matrices resulting from PDE approximation and in designing a fast method to solve the associated linear problem. In the first part, we study the spectral properties of the matrices arising from $\mathbb{P}_k$ Lagrangian Finite Elements approximation of second order elliptic differential problem with Dirichlet boundary conditions and where the operator is $\mathrm{div} \left(-a(\mathbf{x}) \nabla\cdot\right)$, with $a$ continuous and positive over $\overline Ω$, $Ω$ being an open and bounded subset of $\mathbb{R}^d$, $d\ge 1$. We investigate the spectral distribution in the Weyl sense, with a concise overview on localization, clustering, extremal eigenvalues, and asymptotic conditioning. We study in detail the case of constant coefficients on $Ω=(0,1)^2$ and we give a brief account in the case of variable coefficients and more general domains. While in the second part, we design a fast method of multigrid type for the resolution of linear systems arising from the $\mathbb{Q}_k$ Finite Elements approximation of the same considered problem in one and higher dimensional. The analysis is performed in one dimension, while the numerics are carried out also in higher dimension $d\ge 2$, demonstrating an optimal behavior in terms of the dependency on the matrix size and a robustness with respect to the dimensionality $d$ and to the polynomial degree $k$.
