區(qū)域分解算法

內(nèi)容概要

本書的目的是全面講述偏微分方程中的各種最成功、最通用的區(qū)域分解方法，有限元逼近和譜元素逼近預(yù)條件算子，內(nèi)容上做到白包含，而主要側(cè)重點(diǎn)在于算法和數(shù)學(xué)方面。本書詳細(xì)闡述的一些重要的方法如FETI、平衡Neumann-Neumann方法、譜元素方法等，都是第一次在數(shù)學(xué)專著中出現(xiàn)。

書籍目錄

1 Introduction  1.1 Basic Ideas of Domain Decomposition  1.2 Matrix and Vector Representations  1.3 Nonoverlapping Methods  1.4 The Schwarz Alternating Method  1.5 Block Jacobi Preconditioners  1.6 Some Results on Schwarz Alternating Methods2 Abstract Theory of Schwarz Methods  2.1 Introduction  2.2 Schwarz Methods  2.3 Convergence Theory  2.4 Historical Remarks  2.5 Additional Results  2.6 Remarks on the Implementation3 Two-Level Overlapping Methods  3.1 Introduction  3.2 Local Solvers  3.3 A Coarse Problem  3.4 Scaling and Quotient Space Arguments  3.5 Technical Tools  3.6 Convergence Results  3.7 Remarks on the Implementation  3.8 Numerical Results  3.9 Restricted Schwarz Algorithms  3.10 Alternative Coarse Problems4 Substructruing Methods: Introduction  4.1 Introduction  4.2 Problem Setting and Geometry  4.3 Schur Complement Systems  4.4 Discrete Harmonic Extensions  4.5 Condition Number of the Schur Complement  4.6 Technical Tools5 Primal Iterative Substructuring Methods  5.1 Introduction  5.2 Local Design and Analysia  5.3 Local Solvers  5.4 Coarse Spaces and Condition Number Estimates6 Neumann-Neumann and FETI Methods  6.1 Introduction  6.2 Balancing Neumann-Neumann Methods  6.3 One-Level FETI Methods  6.4 Dual-Primal FETI Methods7 Spectral Element Methods  7.1 Introduction  7.2 Deville-Mund Preconditioners  7.3 Two-Level Overlapping Schwarz Methods  7.4 Iterative Substructuring Methods  7.5 Remarks on p and hp Approximations8 Linear Elasticity  8.1 Introduction  8.2 A Two-Level Overlapping Method  8.3 Iterative Substructuring Methods  8.4 A Wire Basket Based Method  8.5 Neumann-Neumann and FETI Methods9 Preconditioners for Saddle Point Problems  9.1 Introduction  9.2 Bolck Preconditioners  9.3 Flows in Porous Media  9.4 The Stokes Problem and Almost Incompressible Elasticity10 Problems in H(div; )and H (curl;)  10.1 Overlapping Methods  10.2 Iterative Substructuring Methods11 Indefinite and Nonsymmetric Problems  11.1 Introduction  11.2 Algorithms on Overlapping Subregions  11.3 An Iterative Substructuring Method  11.4 Numerical Results  11.5 Additional TopicsA Elliptic Problems and Sobolev Spaces  A.1 Sobolev Spaces  A.2 Trace Spaces  A.3 Linear Operators  A.4 Poincare and Friedrichs Type Inequalities  A.5 Spaces of Vector-Valued Functions  A.6 Positive Definite Problems  A.7 Non-Symmetric and Indefinite Problems  A.8 Regularity ResultsB Galerkin Approximations  B.1 Finite Element Approximations  B.2 Spectral Element Approximations  B.3 Divergence and Curl Conforming Finite Elements  B.4 Saddle-Point Problems  B.5 Inverse Inequalities  B.6 Matrix Representation and Condition NumberC Solution of Algebraic Linear Systemse  C.1 Eigenvalues and Condition Number  C.2 Direct Methods  C.3 Richardson Method  C.4 Steepest Descent  C.5 Conjugate Gradient Method  C.6 Methods for Non-Symmetric and Indefinite SystemsReferencesIndex

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