Includes bibliographical references (p. 443-458) and index.

1. Introduction. 1.1. Dirichlet-Dirichlet domain decomposition methods in retrospect. 1.2. Two origins of domain decomposition methods -- 2. Fundamentals of the Schwarz methods. 2.1. Elliptic model problems and their discretizations. 2.2. Domain decomposition methods as preconditioning. 2.3. Main factors influencing convergence -- 3. Overlapping domain decomposition methods. 3.1. Construction principles. 3.2. Discretizations and generalized quasiuniformity conditions. 3.3. Algorithms with generous overlap. 3.4. Loss in convergence due to small overlap. 3.5. Multilevel versions -- 4. Nonoverlapping DD methods for h FE discretizations in 2d. 4.1. Schur complement algorithms for h discretizations. 4.2. Dirichlet-Dirichlet DD algorithms -- 5. BPS-type DD preconditioners for 3d elliptic problems. 5.1. DD algorithms and their main components. 5.2. Condition number and complexity estimates -- 6. DD Algorithms for discretizations with chaotically Piecewise variable orthotropism. 6.1. Single slim domain. 6.2. Schur complement preconditioning by DD. 6.3. Orthotropic discretizations with arbitrary aspect ratios on thin rectangles. 6.4. Discretizations with Piecewise variable orthotropism on domains composed of shape irregular rectangles -- 7. Nonoverlapping DD methods for hp discretizations of 2d elliptic equations. 7.1. Structure of DD preconditioners and its reflection in the relative condition number. 7.2. Prolongations and bounded extension splitting. 7.3. Square reference p-elements, their stiffness and mass matrices. 7.4. Preconditioning of stiffness and mass matrices by finite-difference matrices. 7.5. Schur complement preconditioners for reference elements -- 8. Fast Dirichlet solvers for 2d reference elements. 8.1. Fast Dirichlet solvers for hierarchical reference elements. 8.2. Numerical testing of DD solver for Dirichlet problem in a L-shaped domain. 8.3. Fast Dirichlet solvers for 2d spectral reference elements. 8.4. The numerical complexity of DD methods in two dimensions -- 9. Nonoverlapping Dirichlet-Dirichlet DD methods for hp discretizations of 3d elliptic equations. 9.1. General structure of DD and Schur complement preconditioners. 9.2. Reference elements and finite-difference preconditioners. 9.3. Fast preconditioner-solvers for internal and face problems.

Domain decomposition (DD) methods provide powerful tools for constructing parallel numerical solution algorithms for large scale systems of algebraic equations arising from the discretization of partial differential equations. These methods are well-established and belong to a fast developing area. In this volume, the reader will find a brief historical overview, the basic results of the general theory of domain and space decomposition methods as well as the description and analysis of practical DD algorithms for parallel computing. It is typical to find in this volume that most of the presented DD solvers belong to the family of fast algorithms, where each component is efficient with respect to the arithmetical work. Readers will discover new analysis results for both the well-known basic DD solvers and some DD methods recently devised by the authors, e.g., for elliptic problems with varying chaotically piecewise constant orthotropism without restrictions on the finite aspect ratios. The hp finite element discretizations, in particular, by spectral elements of elliptic equations are given significant attention in current research and applications. This volume is the first to feature all components of Dirichlet–Dirichlet-type DD solvers for hp discretizations devised as numerical procedures which result in DD solvers that are almost optimal with respect to the computational work. The most important DD solvers are presented in the matrix/vector form algorithms that are convenient for practical use.

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