Boundary Element Methods with Applications to Nonlinear Problems

✍ Scribed by Goong Chen; Jianxin Zhou

• Publisher: Atlantis Studies in Mathematic
• Year: 2010
• Tongue: English
• Leaves: 733
• Edition: Illustrated
• Category: Library

✦ Synopsis

Boundary Element Methods have become a major numerical tool in scientific and engineering problem-solving, with particular applications to numerical computations and simulations of partial differential equations in engineering. Boundary Element Methods provides a rigorous and systematic account of the modern mathematical theory of Boundary Element Methods, including the requisite background on general partial, differential equation methods, Sobolev spaces, pseudo-differential and Fredholm operators and finite elements. It aims at the computation of many types of elliptic boundary value problems in potential theory, elasticity, wave propagation, and structural mechanics. Also presented are various methods and algorithms for nonlinear partial differential equations. This second edition has been fully revised and combines the mathematical rigour necessary for a full understanding of the subject, with extensive examples of applications illustrated with computer graphics. This book is intended as a textbook and reference for applied mathematicians, physical scientists and engineers at graduate and research level. It will be an invaluable sourcebook for all concerned with numerical modeling and the solution of partial differential equations.

✦ Table of Contents

Cover
Atlantis Studies in Mathematics for Engineering and Science
Preface to the 1st edition
Dedication
Acknowledgements for the 1st edition
Preface to the 2nd edition
Notation and Abbreviations
Contents
Chapter 1: Introduction)
1.1 How boundary element methods work
1.2 An example of implementation
1.3 Comparison between BEM and FEM
Chapter 2: Some Basic Properties of Sobolev Spaces)
2.1 Definition and imbedding theorems
2.2 The trace theorems
Chapter 3: Theory of Distributions)
3.1 Test functions and generalized functions. Regularization of divergent integrals
3.2 The pseudofunctions
3.3 The distributions
3.4 Regularizing divergent integrals in
3.5 The Fourier transform of tempered distributions
3.6 Examples of Fourier transforms of tempered distributions
Chapter 4: Pseudodifferential Operators and Their Fredholm Properties)
4.1 Symbol class
4.2 Products and adjoints of pseudo-differential operators. Asymptotic expansions of a symbol
4.3 Elliptic operators
4.4 Calculation of the principal symbols of boundary integral operators representing multiple-layer potentials
4.5 The Calder`on projector
4.6 Fredholm operators
4.7 Applications to BIE of elliptic BVP with Neumann boundary conditions
Chapter 5: Finite-Element Methods: Spaces and Properties)
5.1 Minimization of a quadratic functional. The Ritz variational formulation
5.2 Error bounds for internal approximations
5.3 Finite-element computation of BVP: an example
5.4 (t,m)-systems of approximating subspaces)
5.5 Polynomial splines in one dimension
5.6 Barycentric coordinates
5.7 Finite elements in two dimensions
5.8 Finite elements in three dimensions
5.9 Computation of element matrices
5.10 Curved boundary and isoparametric transformations
5.11 Accuracy of finite-element approximations)
5.12 The Aubin–Nitsche lemma
5.13 Inverse inequalities
Chapter 6: The Potential Equation)
6.1 The occurrence of the potential equation
6.2 The fundamental solution of the Laplace equation
6.3 The volume and boundary potentials
6.4 Geometry of hypersurfaces
6.5 Regularity of the layer potentials and the jump property
6.6 The two-dimensional case
6.7 Regularity of solutions of the potential BVP
6.8 Simple-layer representations for interior BVP in
6.9 Simple-layer representations for exterior BVP in
6.10 Double-layer representations for interior BVP in
6.11 Double-layer representations for exterior BVP in
6.12 Simple-layer representations for BVP in
6.13 Double-layer representations for two-dimensional BVP
6.14 Multiconnected domains
6.15 Direct formulation of BIE based upon Green’s formula
6.16 Numerical example (I): an interior Dirichlet problem in
6.17 Numerical example (II): an interior Neumann problem in
6.18 Numerical example (III): an exterior Neumann problem in
Chapter 7: The Helmholtz Equation)
7.1 Background
(a) Time harmonic acoustic scattering
(b) Eigenfunctions and eigenvalues of a vibrating membrane
(c) Aeroacoustics
(d) TE and TM waves in electromagnetic wave guides
7.2 The fundamental solution of the Helmholtz equation
7.3 Regularity of the layer potentials and jump properties
7.4 Solution of BVP in scattering theory by layer potentials
7.5 Asymptotics and uniqueness of solutions to the Helmholtz equation Theorem 7.1.
7.6 BIE solutions to the exterior Dirichlet and Neumann BVP
7.7 Existence and uniqueness of solutions to the exterior impedance BVP (Imk
7.8 Solutions to the interior BVP (Imk
7.9 Modified integral equations approach to exterior scattering problems (Imk
7.10 Numerical example (I): computation of eigenfunctions of the Laplacian
7.11 Numerical example (II): scattering of a plane wave by a 2D elliptic obstacle)
7.12 Numerical example (III): minimizing the reflection of waves by boundary impedance
Chapter 8: The Thin Plate Equation)
8.1 The Kirchhoff thin static plate model subject to pure bending
8.2 Existence, uniqueness and regularity of solutions to the plate equation
8.3 Multilayer potentials for the plate BVP
8.4 BIE for interior plate BVP
8.5 Other multilayer representations of biharmonic functions
8.6 BIE for exterior plate BVP
8.7 Numerical computations and examples (I): exterior BVP
8.8 Numerical computations and examples (II): interior BVP
Chapter 9: Linear Elastostatics)
9.1 Derivations of equations in linear elasticity
9.2 Kelvin’s fundamental solution to the linear elastostatic equation
9.3 BVP in linear elastostatics
9.4 The Betti–Somigliana formula. Simpleand double-layers
9.5 Solutions of the interior BVP by simpleand double-layer potentials
9.6 BIE for exterior problems in linear elastostatics
9.7 Simple-layer representation for the exterior displacement elastostatic problem
9.8 Simple-layer solution for the exterior traction BVP
9.9 Solutions of exterior BVP by double-layer potentials
9.10 Direct formulations of BIE based upon the Betti–Somigliana formula
(I) Interior displacement BVP (I.1)
(II) Interior traction BVP (II.1)
(III) Exterior displacement BVP (III.1)
(IV) Exterior traction BVP
9.11 Numerical example (I): comparison of direct formulation and simplelayer approaches
9.12 Numerical example (II): a ball subject to vertical compression
9.13 Numerical example (III): a ball subject to twisting traction
Chapter 10: Some Error Estimates for Numerical Solutions of Boundary Integral Equations)
10.1 Convergence and error estimates of the Galerkin method
10.2 Convergence and error estimates for collocation by odd degree splines with uniform meshes
10.3 Collocation estimates by Fourier analysis techniques for both evenand odd-degree splines
10.4 Convergence and error estimates for collocation of augmented systems of BIE
Chapter 11: Boundary Element Methods for Semilinear Elliptic Partial Differential Equations (I): The Monotone Iteration Scheme and Error Est)
11.1 Introduction
11.2 A straightforward iteration scheme and the monotone iteration scheme
11.3 Formulation of boundary integral equations based on the simple-layer representation
11.4 Error estimates for the Galerkin boundary element scheme
11.5 Higher than regular-order error estimates for nonlinearities that are separable
11.6 Neumann and Robin boundary conditions
11.7 Numerical examples
11.8 Quasimonotone iteration for coupled
Chapter 12: Boundary Element Methods for Semilinear Elliptic Partial Differential Equations (II): Algorithms and Computations for Unstable S)
12.1 Introduction
12.2 Iterative algorithms and numerical methods
12.2.1 The mountain–pass algorithm (MPA))
12.2.2 The scaling iterative algorithm (SIA))
12.2.3 The direct iteration algorithm (DIA) and the monotone iteration algorithm (MIA))
12.2.4 A boundary element numerical elliptic solver based on the simple-layer and volume potentials)
12.3 Graphics for visualization of the Dirichlet problem of
12.3.1 The unit disk)
12.3.2 Nonconcentric annular domains)
12.3.3 A “pathological” annulus, with boundary formed by two tangent circles)
12.3.4 The radially symmetric annulus)
12.3.5 A dumbbell-shaped domain)
12.3.6 A starshaped domain degenerated from a dumbbell)
12.3.7 Dumbbell-shaped domains with cavities lacking symmetry)
12.3.8 Sign-changing solutions)
12.4 The singularly perturbed Dirichlet problem
12.4.1 The unit disk)


12.5 Other variant semilinear elliptic Dirichlet problems
12.5.1 Henon’s equation)
12.5.2 Chandrasekhar’s equation)

12.6 The sublinear Dirichlet problem
12.6.1 Solutions of (12.103) by direct iteration)
12.6.2 A consequence of visualization: monotonicity of solutions of (12.103) with respect to p)
Appendix A
A.1 Integration by parts and the Gauss–Green formulas
A.2 Banach spaces. Linear operators and linear functionals. Reflexivity
A.3 The basic principles of linear analysis
A.4 Hilbert spaces. The Riesz representation theorem
A.5 Compactness. Completely continuous operators
A.6 Quotient spaces
A.7 Direct sums. Projection operators
A.8 The Cauchy–Schwarz inequality and the H¨older–Young inequality
Bibliography
Subject Index