An Introduction to Domain Decomposition Methods Algorithms, Theory, and Parallel Implementation

Domain decomposition methods are widely used for numerical simulations on parallel machines from tens to hundreds of thousands of cores. Contrary to direct methods, domain decomposition methods are naturally parallel. This book provides a detailed overview of the most popular domain decomposition methods for partial differential equations (PDEs), focusing on parallel linear solvers. The authors present all popular algorithms, both at the PDE level and the discrete level in terms of matrices, along with systematic scripts for sequential implementation in a free open-source finite element package as well as some parallel scripts. Also included is a new coarse space construction (two-level method) that adapts to highly heterogeneous problems. This book is beneficial to those working in domain decomposition methods, parallel computing and iterative methods, particularly those who need to implement parallel solvers for PDEs, as well as to mechanical, civil and aeronautical engineers, environmental scientists, and physicists.

Preface; 1. Schwarz methods; 2. Optimized Schwarz methods; 3. Krylov methods; 4. Coarse spaces; 5. Theory of two-level additive Schwarz methods; 6. Neumann–Neumann and FETI algorithms; 7. Robust coarse spaces via generalized eigenproblems: the GenEO method; 8. Parallel implementation of Schwarz methods; Bibliography; Index.

Victorita Dolean is a Reader in the Department of Mathematics and Statistics, University of Strathclyde. She has been a research assistant in CMAP (Center of Applied Mathematics) at the École Polytechnique in Paris, assistant professor at the University of Evry and the University of Nice, and visiting professor at the University of Geneva. Her research has been oriented toward practical and modern applications of scientific computing by developing interactions between academic and industrial partners and taking part in the life of the scientific community as a member of the Board of Directors of SMAI (Society of Industrial and Applied Mathematics in France).
Pierre Jolivet is a scientist at CNRS in the Toulouse Institute of Computer Science Research, France, working mainly in the field of parallel computing. Before that, he was an ETH Zürich Postdoctoral Fellow of the Scalable Parallel Computing Lab, Zürich, Switzerland. He received his PhD from the University of Grenoble, France, in 2014 for his work on domain decomposition methods and their applications on massively parallel architectures.
Frédéric Nataf is a senior scientist at CNRS in Laboratory J. L. Lions at Université de Paris VI (Pierre et Marie Curie), France. He is also part of an INRIA team. His field of expertise is in high performance scientific computing (domain decomposition methods/approximate factorizations), absorbing/PML boundary conditions, and inverse problems. He has coauthored nearly 100 papers and given several invited plenary talks on these subjects. He developed the theory of optimized Schwarz methods and, very recently, the GENEO coarse space. This last method enables the solving of very large highly heterogeneous problems on large scale computers.