论文标题
分析与线性自动化差异操作员离散化方案相关的光谱符号
Analysis of the spectral symbol associated to discretization schemes of linear self-adjoint differential operators
论文作者
论文摘要
给定许多数值应用中，在许多数值应用中，有线性的自动接合差异操作员$ \ MATHCAL {l} $以及离散方案（例如有限差异，有限元素，Galerkin Isoogementic分析等），在许多数值应用中，重要的是了解（相对）的总体近似值， $ \ MATHCAL {L}^{（\ MathBf {n}）} $是与连续运算符$ \ Mathcal {l} $的光谱相比。广义本地toeplitz序列的理论允许计算与离散矩阵$ \ Mathcal {l}^{（\ Mathbf {n}）} $相关的光谱符号函数$ω$。 我们证明符号$ω$可以渐近地测量最大光谱相对错误$ \ MATHCAL {E} \ geq 0 $。它衡量了该方案的远非$ \ MATHCAL {L} $的良好相对近似值，并且表明一个合适的（可能是不均匀的）网格，以至于如果耦合到该方案准确性的越来越多，请保证$ \ nathCal {e} = 0 $。
Given a linear self-adjoint differential operator $\mathcal{L}$ along with a discretization scheme (like Finite Differences, Finite Elements, Galerkin Isogeometric Analysis, etc.), in many numerical applications it is crucial to understand how good the (relative) approximation of the whole spectrum of the discretized operator $\mathcal{L}^{(\mathbf{n})}$ is, compared to the spectrum of the continuous operator $\mathcal{L}$. The theory of Generalized Locally Toeplitz sequences allows to compute the spectral symbol function $ω$ associated to the discrete matrix $\mathcal{L}^{(\mathbf{n})}$. We prove that the symbol $ω$ can measure, asymptotically, the maximum spectral relative error $\mathcal{E}\geq 0$. It measures how the scheme is far from a good relative approximation of the whole spectrum of $\mathcal{L}$, and it suggests a suitable (possibly non-uniform) grid such that, if coupled to an increasing refinement of the order of accuracy of the scheme, guarantees $\mathcal{E}=0$.
