This paper is devoted to the multigrid convergence analysis for the linear systems arising from the conforming linear finite element discretization of the second order elliptic equations with anisotropic diffusion. The multigrid convergence behavior is known to strongly depend on whether the discretization grid is aligned or non-aligned with the anisotropic direction and analyses in the paper will be mainly focused on two-level algorithms. For an aligned grid case, a lower bound is given for point-wise smoother which shows deterioration of convergence rate. In both aligned and non-aligned cases we show that for a specially designed block smoother the convergence is uniform with respect to both anisotropy ratio and mesh size in the energy norm. The analysis is complemented with numerical experiments which confirm the theoretical results
A cost-effective nonlinear extremum-preserving finite volume scheme for highly anisotropic diffusion on Cartesian grids, with application to radiation belt dynamics
An Efficient Parity-Blocked Method for Band-Structure Computation of 3D Anisotropic Phononic Crystals
Southeast University（东南大学）；Nanjing Normal University（南京师范大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；National Yang Ming Chiao Tung University（国立阳明交通大学）
An asymptotic preserving scheme for strongly anisotropic elliptic problems
Université de Toulouse, UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse（图卢兹大学，UPS，INSA，UT1，UTM，图卢兹数学研究所）；CNRS, Institut de Mathématiques de Toulouse UMR 5219（法国国家科学研究中心，图卢兹数学研究所 UMR 5219）；CMI/LATP, Université de Provence（数学中心/分析与应用数学实验室，普罗旺斯大学）
Block preconditioning methods for asymptotic preserving scheme arising in anisotropic elliptic problems
强各向异性椭圆型方程的高效、鲁棒迭代求解是一个极具挑战性的问题。本文提出了一种块预处理方法来求解一类微宏保渐近（MMAP）格式的线性代数系统。MMAP方法由Degond { \it et al.}其中离散矩阵具有$2\times2$块结构。利用近似Schur补构造了一系列块预条件子。我们首先分析一个自然的近似舒尔补，它是原始非AP离散化的系数矩阵。然而，它往往是奇异的非常小的各向异性参数。然后，我们改进它通过使用更合适的近似边界行的确切舒尔补。利用这些块预条件子，建立了求解离散方程组的预条件GMRES迭代法。一些数值试验表明，块预处理方法可以是一个强大的战略方面的网格细化和各向异性的强度。
A nonoverlapping spectral additive Schwarz method for interior penalty discontinuous Galerkin discretizations of Anisotropic Elliptic Problems
Massachusetts Institute of Technology（麻省理工学院）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；State University of New York Polytechnic Institute（纽约州立理工大学）；Worcester Polytechnic Institute（伍斯特理工学院）
