Highly heterogeneous, anisotropic coefficients, e.g. in the simulation of carbon-fibre composite components, can lead to extremely challenging finite element systems. Direct solvers for the resulting large and sparse linear systems suffer from severe memory requirements and limited parallel scalability, while iterative solvers in general lack robustness. Two-level spectral domain decomposition methods can provide such robustness for symmetric positive definite linear systems, by using coarse spaces based on independent generalized eigenproblems in the subdomains. Rigorous condition number bounds are independent of mesh size, number of subdomains, as well as coefficient contrast. However, their parallel scalability is still limited by the fact that (in order to guarantee robustness) the coarse problem is solved via a direct method. In this paper, we introduce a multilevel variant in the context of subspace correction methods and provide a general convergence theory for its robust convergence for abstract, elliptic variational problems. Assumptions of the theory are verified for conforming, as well as for discontinuous Galerkin methods applied to a scalar diffusion problem. Numerical results illustrate the performance of the method for two- and three-dimensional problems and for various discretization schemes, in the context of scalar diffusion and linear elasticity.
Adaptive-Multilevel BDDC and its parallel implementation
University of Southern California（南加州大学）；Institute of Thermomechanics, Academy of Sciences of the Czech Republic（捷克科学院热力学研究所）；Institute of Mathematics, Academy of Sciences of the Czech Republic（捷克科学院数学研究所）；University of Colorado Denver（科罗拉多大学丹佛分校）
Hierarchical Gaussian Random Fields for Multilevel Markov Chain Monte Carlo: Coupling Stochastic Partial Differential Equation and The Karhunen-Loève Decomposition
A-posteriori-steered $p$-robust multigrid and domain decomposition methods with optimal step-sizes for mixed finite element discretizations of elliptic problems
TU Wien（维也纳工业大学）；Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）；Inria（法国国家数字科学与多领域研究所）；Ecole nationale des ponts et chaussées（国立路桥学校）；University of Pittsburgh（匹兹堡大学）
Institute of Mathematics, Academy of Sciences of the Czech Republic（捷克科学院数学研究所）；Department of Mathematical and Statistical Sciences, University of Colorado Denver（科罗拉多大学丹佛分校数学与统计科学系）；Department of Aerospace and Mechanical Engineering, University of Southern California（南加州大学航空航天与机械工程系）；Department of Mathematics, Faculty of Mechanical Engineering, Czech Technical University（捷克理工大学机械工程学院数学系）
Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods, Part I: Geometrically Conforming Meshes
RWTH Aachen（亚琛工业大学）；CNRS, UMR 7598, Laboratoire Jacques-Louis Lions（法国国家科学研究中心，联合研究单位7598，雅克-路易·利翁斯实验室）；UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions（巴黎第六大学，联合研究单位7598，雅克-路易·利翁斯实验室）；Politecnico di Torino（都灵理工大学）
A highly parallel multilevel Newton-Krylov-Schwarz method with subspace-based coarsening and partition-based balancing for the multigroup neutron transport equations on 3D unstructured meshes
Neural network-driven domain decomposition for efficient solutions to the Helmholtz equation
Department of Mathematics and Computer Science, Eindhoven University of Technology（数学与计算机科学系，埃因霍温理工大学）；Atlantis project-team, Inria Research Center at Université Côte d'Azur（Atlantis项目组，法国国家信息与自动化技术研究院（Inria）在阿尔卑斯大学（Université Côte d'Azur）的研究中心）；MACARON project-team, Université de Strasbourg, CNRS, Inria, IRMA（MACARON项目组，斯特拉斯堡大学，法国国家科学研究中心（CNRS），法国国家信息与自动化技术研究院（Inria），IRMA）
