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E-raamat: Advanced Boundary Element Methods: Treatment of Boundary Value, Transmission and Contact Problems

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Advanced Boundary Element Methods: Treatment of Boundary Value, Transmission and Contact ProblemsSuurem pilt
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This book is devoted to the mathematical analysis of the numerical solution of boundary integral equations treating boundary value, transmission and contact problems arising in elasticity, acoustic and electromagnetic scattering. It serves as the mathematical foundation of the boundary element methods (BEM) both for static and dynamic problems. The book  presents a systematic approach to the variational methods for boundary integral equations including the treatment with variational inequalities for contact problems. It also features adaptive BEM, hp-version BEM, coupling of finite and boundary element methods efficient computational tools that have become extremely popular in applications.

Familiarizing readers with tools like Mellin transformation and pseudodifferential operators as well as convex and nonsmooth analysis for variational inequalities, it concisely presents efficient, state-of-the-art boundary element approximations and points to up-to-date research.



The authors are well known for their fundamental work on boundary elements and related topics, and this book is a major contribution to the modern theory of the BEM (especially for error controlled adaptive methods and for unilateral contact and dynamic problems) and is a valuable resource for applied mathematicians, engineers, scientists and graduate students.

Arvustused

The book can be recommended to researches in the field of mathematics and engineering and to graduate students to familiarize themselves with the state-of-the-art of boundary element methods. (Dana erná, zbMATH 1429.65001, 2020) The book can be recommended as a comprehensive set of results for the state of the art in BEM, and in the applications considered by the authors. (Michael J. Carley, Mathematical Reviews, August, 2019)

1 Introduction 1(8)
1.1 The Basic Approximation Problems
1(3)
1.2 Convergence of Projection Methods
4(5)
2 Some Elements of Potential Theory 9(34)
2.1 Representation Formulas
9(7)
2.2 Single- and Double-Layer Potential
16(9)
2.2.1 Some Remarks on Distributions
17(4)
2.2.2 Jump Relations
21(4)
2.3 Mapping Properties of Boundary Integral Operators
25(5)
2.4 Laplace's Equation in R3
30(4)
2.4.1 Representation Formula
32(2)
2.5 Calderon Projector
34(2)
2.6 Use of Complex Function Theory
36(7)
2.6.1 Representation Formula Again
36(3)
2.6.2 Applicable Representation of the Hypersingular Integral Operator
39(4)
3 A Fourier Series Approach 43(20)
3.1 Fourier Expansion-The Sobolev Space Hs[ 0, 2π]
43(5)
3.2 The Sobolev Space Hs (Γ)
48(1)
3.3 Interior Dirichlet Problem
49(3)
3.4 The Boundary Integral Operators in a Scale of Sobolev Spaces
52(5)
3.4.1 The Operators V and W
52(3)
3.4.2 The Operators K and K'
55(2)
3.5 Solution of Exterior Dirichlet Problem by BIE
57(3)
3.6 A First Garding Inequality
60(3)
4 Mixed BVPs, Transmission Problems and Pseudodifferential Operators 63(32)
4.1 Mixed Boundary Value Problems
63(7)
4.2 The Helmholtz Interface Problems
70(11)
4.3 Screen Problems
81(3)
4.4 Interface Problem in Linear Elasticity
84(5)
4.5 A Strongly Elliptic System for Exterior Maxwell's Equations
89(6)
4.5.1 A Simple Layer Procedure
89(2)
4.5.2 Modified Boundary Integral Equations
91(4)
5 The Signorini Problem and More Nonsmooth BVPs and Their Boundary Integral Formulation 95(20)
5.1 The Signorini Problem in Its Simplest Form
95(7)
5.2 A Variational Inequality of the Second Kind Modelling Unilateral Frictional Contact
102(4)
5.3 A Nonmonotone Contact Problem from Delamination
106(9)
6 A Primer to Boundary Element Methods 115(108)
6.1 Galerkin Scheme for Strongly Elliptic Operators
116(3)
6.2 Galerkin Methods for the Single-Layer Potential
119(7)
6.2.1 Approximation with Trigonometric Polynomials
119(2)
6.2.2 Approximation with Splines
121(3)
6.3 Collocation Method for the Single-Layer Potential
124(2)
6.4 Collocation Methods-Revisited
126(14)
6.4.1 Periodic Splines as Test and Trial Functions
128(3)
6.4.2 Convergence Theorem for Projection Methods
131(9)
6.5 BEM on Quasiuniform Meshes
140(32)
6.5.1 Periodic Polynomial Splines
140(1)
6.5.2 The Approximation Theorem
141(6)
6.5.3 Stability and Inverse Estimates
147(4)
6.5.4 Aubin-Nitsche Duality Estimate and Superapproximation
151(4)
6.5.5 Numerical Quadrature
155(4)
6.5.6 Local H-1/2-Error Estimates
159(4)
6.5.7 Local L2-Error Estimates
163(2)
6.5.8 The K-Operator-Method
165(3)
6.5.9 Linfinity-Error Estimates for the Galerkin Approximation
168(4)
6.6 A Discrete Collocation Method for Symm's Integral Equation on Curves with Corners
172(10)
6.7 Improved Galerkin Method with Augmented Boundary Elements
182(3)
6.8 Duality Estimates for Projection Methods
185(7)
6.8.1 Application to Galerkin Methods
186(3)
6.8.2 Application to Collocation Methods
189(3)
6.9 A Collocation Method Interpreted as (GM)
192(6)
6.10 Modified Collocation and Qualocation
198(7)
6.11 Radial Basis Functions and Spherical Splines
205(18)
7 Advanced BEM for BVPs in Polygonal/Polyhedral Domains: h- and p-Versions 223(46)
7.1 The Dirichlet Problem
224(12)
7.1.1 Regularity on a Polygon
225(1)
7.1.2 BEM on a Polygon
226(4)
7.1.3 Regularity on a Polyhedron
230(6)
7.2 The Neumann Problem
236(6)
7.2.1 Regularity on a Polyhedron
240(2)
7.3 1D-Approximation Results
242(7)
7.3.1 hp-Method with Quasiuniform Mesh on Polygons
242(5)
7.3.2 Approximation of the Normal Derivative on a One Dimensional Boundary-The h-Version on a Graded Mesh
247(2)
7.4 2D-Approximation Results
249(15)
7.4.1 Approximation of the Normal Derivative on a Two-dimensional Boundary-The h-Version on a Graded Mesh
250(7)
7.4.2 Approximation of the Trace on a Two-Dimensional Boundary-The h-Version on a Graded Mesh
257(7)
7.5 Augmented BEM for Screen/Crack Problems
264(5)
8 Exponential Convergence of hp-BEM 269(26)
8.1 The hp-Version of BEM on Polygons
270(12)
8.1.1 Application to Acoustic Scattering
279(3)
8.2 The hp-Version of BEM on Surfaces
282(6)
8.3 The hp-Version of BEM on a Geometrical Mesh for Mixed BVP on a Polygonal Domain
288(7)
9 Mapping Properties of Integral Operators on Polygons 295(38)
9.1 Mellin Symbols
295(11)
9.1.1 Mapping Properties in Weighted Sobolev Spaces
299(7)
9.2 Properties of the Mellin Transformation
306(7)
9.2.1 Local Regularity at Vertices
310(3)
9.3 A Direct Boundary Element Method for Interface Crack Problems
313(4)
9.4 Mixed BVP of Potential Theory on Polygons
317(6)
9.5 Boundary Integral Operators in Countably Normed Spaces
323(10)
10 A-BEM 333(56)
10.1 General Frame for A Posteriori Error Estimates for Boundary Element Methods
334(3)
10.1.1 Symm's Integral Equation
336(1)
10.2 Adaptive Boundary Element Methods
337(12)
10.2.1 Reliability of A Posteriori BEM Error Estimates
340(3)
10.2.2 Efficiency of A Posteriori BEM Error Estimates (2D)
343(6)
10.3 The Weakly Singular Integral Equation in 3D
349(8)
10.3.1 Adaptive Algorithms
353(2)
10.3.2 Numerical Example
355(2)
10.4 The Hypersingular Integral Equation in 3D
357(5)
10.5 Two-Level Adaptive BEM for Laplace, Lame, Helmholtz
362(15)
10.5.1 A Stable Two-Level Subspace Decomposition for the Hypersingular Operator
370(7)
10.6 Two-Level Subspace Decomposition for the p-Version BEM
377(4)
10.7 Convergence of Adaptive BEM for Estimators Without h-Weighting Factor
381(8)
11 BEM for Contact Problems 389(62)
11.1 h-BEM for the Signorini Problem
390(5)
11.1.1 Discretization of the Boundary Variational Inequality
390(2)
11.1.2 The Convergence Result
392(3)
11.2 hp-BEM with Hierarchical Error Estimators for Scalar Signorini Problems
395(8)
11.3 hp-BEM for a Variational Inequality of the Second Kind Modelling Unilateral Contact and Friction
403(17)
11.3.1 The hp-Version Galerkin Boundary Element Scheme
405(7)
11.3.2 A Cea-Falk Lemma for Variational Inequalities of the Second Kind
412(2)
11.3.3 A Priori Error Estimate for hp-Approximation
414(6)
11.4 Mixed hp-BEM for Frictional Contact Problems
420(16)
11.4.1 Boundary Integral Formulation for Contact Problem
420(2)
11.4.2 hp-Boundary Element Procedure with Lagrange Multiplier and Fast Solver
422(3)
11.4.3 Error Controlled hp-Adaptive Schemes
425(5)
11.4.4 Stabilized hp-Mixed Method-A Priori Error Estimate
430(1)
11.4.5 A Priori Error Estimates for hp-Penalty-BEM for Contact Problems in Elasticity
431(5)
11.5 h-Version BEM for a Nonmonotone Contact Problem from Delamination
436(7)
11.6 hp-BEM for Delamination Problems
443(8)
12 FEM-BEM Coupling 451(86)
12.1 Abstract Framework of Some Saddle Point Problems
452(3)
12.2 Galerkin Approximation of Saddle Point Problems
455(9)
12.2.1 Symmetric FE/BE Coupling for a Nonlinear Interface Problem
459(5)
12.3 Symmetric FE/BE Coupling-Revisited
464(27)
12.3.1 Convergence Analysis
468(5)
12.3.2 Adaptive FE/BE Coupling: Residual Based Error Indicators
473(5)
12.3.3 Adaptive FE/BE Coupling with a Schur Complement Error Indicator
478(10)
12.3.4 Convergence of Adaptive FEM-BEM Couplings
488(1)
12.3.5 Other Coupling Methods
489(2)
12.4 Least Squares FEM/BEM Coupling for Transmission Problems
491(8)
12.4.1 The Discretized Least Squares Formulation
497(2)
12.5 FE/BE Coupling for Interface Problems with Signorini Contact
499(7)
12.5.1 Primal Method
499(4)
12.5.2 Dual Mixed Method
503(3)
12.6 Coupling of Primal-Mixed FEM and BEM for Plane Elasticity
506(9)
12.7 Adaptive FE/BE Coupling for Strongly Nonlinear Interface Problems with Tresca Friction
515(4)
12.8 Adaptive FE-BE Coupling for the Eddy-Current Problem in R3
519(14)
12.8.1 p-Hierarchical Estimator
530(3)
12.9 Parabolic-Elliptic Interface Problems
533(4)
13 Time-Domain BEM 537(26)
13.1 Integral Equations and Anisotropic Space-Time Sobolev Spaces
538(5)
13.2 A Priori and A Posteriori Error Estimates
543(6)
13.2.1 Adaptive Mesh Refinements
547(2)
13.3 Time Domain BEM for Contact Problems
549(3)
13.4 Algorithmic Aspects of Time Domain BEM
552(5)
13.4.1 MOT Algorithm
552(2)
13.4.2 An hp-Composite Quadrature of Matrix Elements
554(3)
13.5 Screen Problems and Graded Meshes
557(6)
A Linear Operator Theory 563(6)
B Pseudodifferential Operators 569(12)
C Convex and Nonsmooth Analysis, Variational Inequalities 581(34)
C.1 Convex Optimization, Lagrange Multipliers
581(12)
C.1.1 Convex Quadratic Optimization in Finite Dimensions
582(4)
C.1.2 Convex Quadratic Optimization in Hilbert Spaces
586(3)
C.1.3 Lagrange Multipliers for Some Inequality Constrained Variational Inequalities
589(4)
C.2 Nonsmooth Analysis
593(8)
C.2.1 Nonsmooth Analysis of Locally Lipschitz Functions
593(2)
C.2.2 Regularization of Nonsmooth Functions
595(6)
C.3 Existence and Approximation Results for Variational Inequalities
601(7)
C.3.1 Existence Results for Linear VIs
601(4)
C.3.2 Approximation of Linear VIs
605(3)
C3.3 Pseudomonotone VIs-Existence Result
608(7)
C.3.4 Mosco Convergence, Approximation of Pseudomonotone VIs
610(1)
C.3.5 A Hemivariational Inequality as a Pseudomonotone VI
611(4)
D Some Implementations for BEM 615(16)
D.1 Symm's Equation on an Interval
615(1)
D.2 The Dirichlet Problem in 2D
616(2)
D.3 Symm's Equation on a Surface Piece
618(13)
D.3.1 Implementation of hp-BEM on Surfaces
622(9)
References 631(20)
Index 651
Joachim Gwinner is retired professor of mathematics at Bundeswehr  University Munich. His research interests span from optimization to numerical and  variational analysis with  applications in continuum mechanics.





Ernst Peter Stephan is retired professor of mathematics at Leibniz University Hannover. His research covers numerical methods for partial differential equations and boundary integral equations together with their analysis.
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