论文标题
用于Helmholtz问题的两级移动的Laplace预处理问题：价值分析和独立于波数的收敛性
A two-level shifted Laplace preconditioner for Helmholtz problems: Field-of-values analysis and wavenumber-independent convergence
论文作者
论文摘要
求解由Helmholtz方程式离散化引起的线性系统的主要工具之一是移动的Laplace预处理，这是由于扰动的Helmholtz问题$-ΔU-（K^2 + I \ i \ i \ VAREPSILON）U = F $ = F $ wery $ 0 \ neq \ neq \ varepsilon $ bersor n oster and and \ ins there \ ins \ ins the \ ins the \ ins there \ ins they。在这项工作中，我们重新审视了将移位的拉普拉斯预处理与两级放气相结合的想法，并将其应用于通过线性有限元元素离散的Helmholtz问题。 We use the convergence theory of GMRES based on the field of values to prove that GMRES applied to the two-level preconditioned system with a shift parameter $\varepsilon \sim k^2$ converges in a number of iterations independent of the wavenumber $k$,provided that the coarse mesh size $H$ satisfies a condition of the form $Hk^{2} \leq C$ for some constant $C$ depending on该域，但独立于波数$ k $。这种行为与独立移动的laplacian截然不同，仅在$ \ varepsilon \ sim k $的条件下，才建立了与波数单无关的gmres收敛。甘德（I.G.）格雷厄姆和E.A. Spence，Numer。 Math。，131（2015），567-614]。最后，我们提供了数值证据，表明gmres的波数非依赖性收敛也适用于无污染的网格，其中粗网格尺寸满足$ hk^{3/2} \ leq c $，并且不彻底的粗网格求解。
One of the main tools for solving linear systems arising from the discretization of the Helmholtz equation is the shifted Laplace preconditioner, which results from the discretization of a perturbed Helmholtz problem $-Δu - (k^2 + i \varepsilon )u = f$ where $0 \neq \varepsilon \in \mathbb{R}$ is an absorption parameter. In this work we revisit the idea of combining the shifted Laplace preconditioner with two-level deflation and apply it to Helmholtz problems discretized with linear finite elements. We use the convergence theory of GMRES based on the field of values to prove that GMRES applied to the two-level preconditioned system with a shift parameter $\varepsilon \sim k^2$ converges in a number of iterations independent of the wavenumber $k$,provided that the coarse mesh size $H$ satisfies a condition of the form $Hk^{2} \leq C$ for some constant $C$ depending on the domain but independent of the wavenumber $k$. This behaviour is sharply different to the standalone shifted Laplacian, for which wavenumber-independent GMRES convergence has been established only under the condition that $\varepsilon \sim k$ by [M.J. Gander, I.G. Graham and E.A. Spence, Numer. Math., 131 (2015), 567-614]. Finally, we present numerical evidence that wavenumber-independent convergence of GMRES also holds for pollution-free meshes, where the coarse mesh size satisfies $Hk^{3/2} \leq C $, and inexact coarse grid solves.
