We introduce and analyze a two-level overlapping additive Schwarz preconditioner for discontinuous Galerkin discretizations of virtual element spaces on general polygonal meshes in two dimensions, for a second-order elliptic problem whose coefficient is constant on each subdomain and may have large jumps across the interface. The discretization is a weighted symmetric interior penalty formulation: the averages carry the coefficient and the penalty is scaled by its harmonic mean across each edge, which keeps the method consistent and makes the constants of the analysis independent of the coefficient. The coarse space is obtained from a virtual, continuous partition of unity with a single degree of freedom per subdomain vertex, and is therefore applicable to the irregular subdomains produced by a graph partitioner; the discrete harmonic extension used to build it may be replaced by a cheaper projection onto polynomials of low degree. We prove that the condition number of the preconditioned system is bounded by $C(1+\log(H/h))(1+H/δ)$, where $H$, $h$ and $δ$ denote the subdomain diameter, the mesh size and the overlap, and where $C$ is independent of the number of subdomains and of the contrast of the coefficient, provided the latter is quasi-monotone. This is the same bound known for the conforming case and the analysis allows for hanging nodes. We also show, by an explicit example, that the quasi-monotonicity assumption cannot be dispensed with. Numerical experiments with structured triangular, hexagonal and Voronoi meshes, and with subdomains obtained both by uniform partitioning and by METIS, confirm the theory.
An Overlapping Domain Decomposition Preconditioner for the Helmholtz equation
State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences（中国科学院计算数学与科学工程计算重点实验室）；Department of Mathematics, University of South Carolina（南卡罗来纳大学数学系）
A FETI-DP preconditioner of discontinuous Galerkin method for multiscale problems in high constrast media
University of Bergen（卑尔根大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Bergen University College（卑尔根大学学院）
The analysis of FETI-DP preconditioner for full DG discretization of elliptic problems
Warsaw University（华沙大学）；Universidad Nacional de Colombia（哥伦比亚国立大学）；Worcester Polytechnic Institute（伍斯特理工学院）；Instituto Nacional de Matemática Pura e Aplicada (IMPA)（巴西纯粹与应用数学国家研究所）
Subdomain Deflation Combined with Local AMG: a Case Study Using AMGCL Library
Kazan Branch of Joint Supercomputer Center, Russian Academy of Sciences（俄罗斯科学院联合超级计算中心喀山分院）；Kazan Federal University（喀山联邦大学）；Universitat Politècnica de Catalunya (UPC)（加泰罗尼亚理工大学）；CIMNE — Centre Internacional de Mètodes Numèrics en Enginyeria（国际数值工程研究中心）
Additive Schwarz preconditioner for the general finite volume element discretization of symmetric elliptic problems
University of Warsaw（华沙大学）；Bergen University College（卑尔根大学学院）；University of Bergen（卑尔根大学）；University of South Bohemia（南波希米亚大学）；Institute of Information Theory and Automation of the ASCR（捷克科学院信息论与自动化研究所）
Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption
Sorbonne Université, Université Paris-Diderot SPC, CNRS, Inria, Laboratoire Jacques-Louis Lions（索邦大学，巴黎第七大学SPC，法国国家科学研究中心，法国国家信息与自动化研究所，雅克-路易·利翁斯实验室）；Department of Mathematics and Statistics, University of Strathclyde（斯特拉斯克莱德大学数学与统计系）；Université Côte d’Azur, CNRS, Laboratoire J-A Dieudonné（蔚蓝海岸大学，法国国家科学研究中心，J-A 迪厄多内实验室）；Department of Mathematical Sciences, University of Bath（巴斯大学数学科学系）
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）
Additive average Schwarz with adaptive coarse spaces: scalable algorithms for multiscale problems
Faculty of Mathematics, Univeristy of Warsaw（华沙大学数学系）；Department of Computing, Mathematics and Physics, Western Norway University of Applied Sciences（西挪威应用科学大学计算、数学与物理系）
An algebraic multiscale framework for two-level overlapping Schwarz preconditioners
Delft Institute of Applied Mathematics, Delft University of Technology（代尔夫特应用数学研究所，代尔夫特理工大学）；Department of Geoscience and Engineering, Delft University of Technology（地球科学与工程学院，代尔夫特理工大学）
