论文标题
二阶差分运算符的广义频谱
Generalized spectrum of second order differential operators
论文作者
论文摘要
我们分析了操作员$δ^{ - 1} [\ nabla \ cdot（k \ nabla u）] $的频谱，其中$δ$表示laplacian，$ k = k = k（x，y）$是对称的张量。我们的主要结果表明，该光谱可以从频谱分解$ k =qλq^t $中得出，其中$ q = q = q（x，y）$是正交矩阵，$λ=λ（x，y）$是对角矩阵。更确切地说，只要$ k $是连续的，频谱等于$λ$的对角线功能条目的范围。假定所涉及的域是有界和Lipschitz的，并且考虑了均匀的Dirichlet和均匀的Neumann边界条件。我们研究了在无限尺寸Sobolev空间上定义的操作员。我们的理论研究通过离散的问题通过数值实验来照亮。 本文介绍的结果扩展了以前的分析，这些分析已经解决了具有标量系数函数的椭圆差分运算符。我们的调查是由预处理问题（有效的数值计算）和进一步发展二阶PDES频谱理论（核心分析）的需要的动机。
We analyze the spectrum of the operator $Δ^{-1} [\nabla \cdot (K\nabla u)]$, where $Δ$ denotes the Laplacian and $K=K(x,y)$ is a symmetric tensor. Our main result shows that this spectrum can be derived from the spectral decomposition $K=Q ΛQ^T$, where $Q=Q(x,y)$ is an orthogonal matrix and $Λ=Λ(x,y)$ is a diagonal matrix. More precisely, provided that $K$ is continuous, the spectrum equals the convex hull of the ranges of the diagonal function entries of $Λ$. The involved domain is assumed to be bounded and Lipschitz, and both homogeneous Dirichlet and homogeneous Neumann boundary conditions are considered. We study operators defined on infinite dimensional Sobolev spaces. Our theoretical investigations are illuminated by numerical experiments, using discretized problems. The results presented in this paper extend previous analyses which have addressed elliptic differential operators with scalar coefficient functions. Our investigation is motivated by both preconditioning issues (efficient numerical computations) and the need to further develop the spectral theory of second order PDEs (core analysis).
