Schwarz Methods and Multilevel Preconditioners for Boundary Element Methods

Preface
Acknowledgements
Contents
Part I General Theory
Chapter 1 Introduction
	1.1 Model Problems
		1.1.1 Screen problems
		1.1.2 Crack problems in elasticity
	1.2 Galerkin Boundary Element Methods
		1.2.1 The h-version
		1.2.2 The p-version
		1.2.3 The hp-version
		1.2.4 Graded meshes
		1.2.5 Adaptive schemes
	1.3 Condition Numbers
Chapter 2 General Framework of Preconditioners
	2.1 Finite Dimensional Problems
		2.1.1 Galerkin equations and matrix systems
		2.1.2 Solution by CG
		2.1.3 Solution by GMRES
	2.2 Preconditioners
		2.2.1 Preconditioned conjugate gradient method
		2.2.2 Preconditioned GMRES
		2.2.3 Preconditioning and linear iteration
	2.3 Schwarz Operators
		2.3.1 Required properties of a preconditioner
		2.3.2 Ingredients of the Schwarz operators
		2.3.3 Schwarz operators
	2.4 Additive Schwarz Preconditioners
		2.4.1 Matrix forms of P_j and P_
ad
		2.4.2 Additive Schwarz preconditioner: the matrix form
		2.4.3 Additive Schwarz preconditioner: the variational form
		2.4.4 Additive Schwarz preconditioner: the operator form
	2.5 Multiplicative Schwarz Preconditioners
		2.5.1 Multiplicative Schwarz preconditioner: the matrix form
		2.5.2 Multiplicative Schwarz preconditioner: the operator form
		2.5.3 Symmetric multiplicative Schwarz
	2.6 Convergence Theory for Preconditioners with PCG
		2.6.1 Extremal eigenvalues of preconditioned matrices
		2.6.2 Condition numbers of Schwarz operators
		2.6.3 Stability of decomposition – A lower bound for λ_min(P_ad)
		2.6.4 Coercivity of decomposition – An upper bound for λ_max(P_ad)
		2.6.5 Strengthened Cauchy–Schwarz inequalities and local stability
		2.6.6 An upper bound for λ_max(P_smu)
		2.6.7 A lower bound for λ_min(P_smu)
		2.6.8 Summary
	2.7 Convergence Theory for Preconditioners with GMRES
		2.7.1 Preconditioned GMRES with the B-inner product
		2.7.2 Preconditioned GMRES with the C-inner product
		2.7.3 Preconditioned GMRES with block matrices
	2.8 Other Krylov Subspace Methods
		2.8.1 General Krylov subspace methods
		2.8.2 The generalised three-term CG methods
		2.8.3 The hybrid modified conjugate residual (HMCR) method
	2.9 Convergence Theory for Linear Iterative Methods with Preconditioners
		2.9.1 Symmetric preconditioner
		2.9.2 Non-symmetric preconditioner
Part II Two-Dimensional Problems
Chapter 3 Two-Level Methods: the h-Version
	3.1 Additive Schwarz Methods
		3.1.1 Non-overlapping methods
		3.1.2 Overlapping methods
	3.2 Multiplicative Schwarz Methods
		3.2.1 Non-overlapping methods
		3.2.2 Overlapping methods
		3.2.3 A special case
	3.3 Numerical Results
Chapter 4 Two-Level Methods: the p-Version
	4.1 Additive Schwarz Methods
		4.1.1 Non-overlapping methods
		4.1.2 Overlapping methods
	4.2 Multiplicative Schwarz Methods
		4.2.1 Non-overlapping methods
		4.2.2 Overlapping methods
	4.3 Numerical Results
Chapter 5 Multilevel Methods: the h-Version
	5.1 Additive Schwarz Methods
		5.1.1 The hypersingular integral equation
		5.1.2 The weakly-singular integral equation
	5.2 Multiplicative Schwarz methods
		5.2.1 The hypersingular integral equation
		5.2.2 The weakly-singular integral equation
	5.3 Numerical Results
Chapter 6 Additive Schwarz Methods for the hp-Version
	6.1 Preconditioners with Quasi-uniform Meshes
		6.1.1 A two-level non-overlapping method
		6.1.2 A two-level overlapping method
		6.1.3 Multilevel methods
	6.2 Preconditioners with Geometric Meshes
		6.2.1 A two-level preconditioner
		6.2.2 A multilevel preconditioner
	6.3 Results for the Weakly-Singular Integral Equation
	6.4 Numerical Results
Chapter 7 A Fully Discrete Method
	7.1 The Boundary Integral Equation and a Fully Discrete Method
	7.2 The Fully-Discrete and Symmetric Method
	7.3 Two-level Methods
		7.3.1 A non-overlapping method
		7.3.2 An overlapping method
	7.4 A Multilevel Method
	7.5 Numerical Experiments
		7.5.1 Implementation issues
		7.5.2 Numerical results
Chapter 8 Indefinite Problems
	8.1 General Theory for Indefinite Problems
		8.1.1 Assumptions
		8.1.2 Additive Schwarz operators
	8.2 Hypersingular Integral Equation
		8.2.1 The h-version
		8.2.2 The p-version
	8.3 Weakly-Singular Integral Equation
		8.3.1 The h-version
		8.3.2 The p-version
	8.4 Numerical Results
	8.5 Indefinite Problems in Three-Dimensions
Chapter 9 Implementation Issues and Numerical Experiments
	9.1 Implementation Issues
	9.2 Numerical Results for the h-Version
	9.3 Numerical Results for the p-Version
	9.4 Numerical Results for the hp-Version
Part III Three-Dimensional Problems
Chapter 10 Two-Level Methods: the hp-Version on Rectangular Elements
	10.1 Preliminaries
		10.1.1 Two-level meshes
		10.1.2 Shape functions
		10.1.3 Boundary element spaces
	10.2 The Hypersingular Integral Equation
		10.2.1 A non-overlapping method
		10.2.2 An overlapping method
	10.3 The Weakly-Singular Integral Equation
	10.4 Numerical Results
		10.4.1 The hypersingular integral equation
		10.4.2 The Lamé equation
Chapter 11 Two-Level Methods: the hp-Version on Triangular Elements
	11.1 Preliminaries
		11.1.1 Sobolev spaces of functions vanishing on a part of the boundary of a domain
		11.1.2 Extension operators
		11.1.3 Construction of basis functions
		11.1.4 Change of basis functions and change of the wire basket space
		11.1.5 The wire basket in three dimensions
		11.1.6 Properties of the interpolation operators IbW and IbW (bW)
	11.2 Preconditioners for the Hypersingular Integral Equation
		11.2.1 Subspace decomposition
		11.2.2 Preconditioner I
		11.2.3 Preconditioner II
	11.3 Numerical Results
Chapter 12 Diagonal Scaling Preconditioner and Locally-Refined Meshes
	12.1 Problem Setting
	12.2 Preconditioning by Diagonal Scaling
	12.3 Shape-Regular Mesh Refinements
		12.3.1 Coercivity of the decomposition
		12.3.2 Stability of the decomposition
		12.3.3 Bounds for the condition numbers
	12.4 Anisotropic Mesh Refinements
		12.4.1 Technical results
		12.4.2 Coercivity of the decomposition
		12.4.3 Stability of the decomposition
		12.4.4 Bounds for the condition numbers
	12.5 Numerical Results
Chapter 13 Multilevel Preconditioners with Adaptive Mesh Refinements
	13.1 Preliminaries
		13.1.1 Mesh refinements and hierarchical structures
		13.1.2 Level functions and uniform mesh refinements
	13.2 Multilevel Preconditioners for the Hypersingular Integral Equation
		13.2.1 Local multilevel diagonal preconditioner (LMD preconditioner)
		13.2.2 Global multilevel diagonal preconditioner (GMD preconditioner)
	13.3 Numerical Experiments
	13.4 A Remark on the Weakly-Singular Integral Equation
Part IV FEM–BEM Coupling
Chapter 14 FEM-BEM Coupling
	14.1 The Interface Problem
		14.1.1 The symmetric FEM-BEM coupling
		14.1.2 Non-symmetric coupling methods
	14.2 Preconditioning for the Symmetric Coupling
		14.2.1 Preconditioning with HMCR
		14.2.2 Preconditioning with GMRES
	14.3 Preconditioning for the Non-Symmetric Coupling Methods
	14.4 Numerical Results
Part V Problems on the Sphere
Chapter 15 Pseudo-differential Equations with Spherical Splines
	15.1 Pseudo-differential Operators and Sobolev Spaces
		15.1.1 Sobolev spaces on the sphere
		15.1.2 Pseudo-differential operators on the sphere
	15.2 Solving Pseudo-differential Equations by Spherical Splines
		15.2.1 Spherical splines
		15.2.2 Approximate solutions and error estimates
	15.3 Additive Schwarz Methods for Equations on the Sphere
		15.3.1 Decomposition of the finite element space
		15.3.2 The hypersingular integral equation
	15.4 Numerical Results
Chapter 16 Pseudo-differential Equations with Radial Basis Functions
	16.1 Radial Basis Functions
		16.1.1 Positive-definite kernels
		16.1.2 Spherical radial basis functions
		16.1.3 Native space and reproducing kernel property
	16.2 Solving Pseudo-differential Equations by Radial Basis Functions
	16.3 Additive Schwarz Methods
		16.3.1 Subspace decomposition
		16.3.2 Coercivity of the decomposition
		16.3.3 Stability of the decomposition and bounds for the minimum eigenvalue of P
		16.3.4 Bounds for the condition number
	16.4 Numerical Results
		16.4.1 An algorithm
		16.4.2 Numerical results
Part VI Appendices
Appendix A Interpolation Spaces and Sobolev Spaces
	A.1 Real Interpolation Spaces
		A.1.1 Compatible couples and intermediate spaces
		A.1.2 The K-functional
	A.2 Sobolev Spaces
		A.2.1 Notations
		A.2.2 Sobolev spaces on R^d
		A.2.3 Sobolev spaces on a Lipschitz domain
		A.2.4 Extension operators
		A.2.5 Equivalence of norms
		A.2.6 The weighted norms
		A.2.7 Scaling properties
		A.2.8 Important results
		A.2.9 Comparison of global and local norms
		A.2.10 The special case s = 1/2
		A.2.11 Sobolev spaces on curves and surfaces
		A.2.12 A generalised antiderivative operator in Sobolev spaces
Appendix B Boundary Integral Operators
	B.1 Boundary Integral Operators and Pseudo-differential Operators on the Sphere
		B.1.1 Boundary potentials and boundary integral operators
		B.1.2 Representation of harmonic functions by potentials
		B.1.3 Dirichlet-to-Neumann and Neumann-to-Dirichlet operators
		B.1.4 The weakly-singular and hypersingular bilinear forms
		B.1.5 Representations of solutions to the Laplace equation by spherical harmonics
		B.1.6 Representations of boundary integral operators by spherical harmonics
	B.2 Discretised Operators
		B.2.1 Natural embedding operators and biorthonormal bases
		B.2.2 Discretised operators and matrix representations
		B.2.3 Compositions of operators
Appendix C Some Additional Results
	C.1 Conditioning of Matrices
		C.1.1 The hierarchical basis functions for the p-version
		C.1.2 The condition numbers of the weakly-singular and hypersingular stiffness matrices
		C.1.3 Extremal eigenvalues of equivalent matrices
		C.1.4 Eigenvalues of block matrices
	C.2 Norms of Nodal Basis Functions
	C.3 Further Properties of the Hierarchical Basis Functions for the p-Version
	C.4 Some Properties of Polynomials
		C.4.1 Inverse properties
		C.4.2 Some useful bounds for the h and p finite element functions
		C.4.3 Polynomials of low energies
		C.4.4 Discrete harmonic functions and discrete harmonic extension
	C.5 Some Useful Projections or Projection-like Operators
		C.5.1 The L^2-projection
		C.5.2 The standard interpolation operator
		C.5.3 Some useful operators
		C.5.4 Clément’s interpolation
		C.5.5 The Scott–Zhang interpolation
	C.6 Gauss–Lobatto Quadrature
	C.7 Additional Technical Lemmas
References
Index
	Index of Notation: Function Spaces
	Index of Notation: Norms
	Index of Notation: Operators and Other Symbols