Uniform convergence of a multigrid method for elliptic equations with anisotropic coefficients. (23rd June 2018)

• Record Type: Journal Article
• Authors: Yun, Xiaofan; Qin, Chenxiang; Wu, Jinbiao; Zheng, Hui
• Is Part Of: IMA journal of numerical analysis. Volume 39:Number 2(2019)
• Page Start: 1058
• Page End: 1084
• Publication Date: 2018-06-23
• Subjects: multigrid method -- anisotropic elliptic equation -- Xu–Zikatanov identity -- line Gauss–Seidel iteration -- uniform convergence; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/dry019
• Languages: English
• ISSNs: 0272-4979

Abstract

Abstract: We prove the uniform convergence of the V-cycle multigrid method with line Gauss–Seidel iteration as its smoother for anisotropic elliptic equations. We define a rectified projection operator to decompose the functions into different levels. This operator is based on piecewise energy norm projection. Using the Xu–Zikatanov identity, we can show that the convergence rate is independent of the mesh size, the number of levels and the coefficients of equations. The main improvement of this paper is that we lose the restriction from previous works whereby the domain must be convex, and we prove the uniform convergence of the multigrid method for the problems defined on the domains which are constructed by finite rectangles, such as the L-shape domains.