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Advanced boundary element methods: treatment of boundary value, transmission and contact problems

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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  Read more...

Abstract: 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

Author(s): Gwinner, Joachim; Stephan, Ernst Peter
Series: Springer series in computational mathematics 52
Publisher: Springer
Year: 2018

Language: English
Pages: 661
Tags: Boundary element methods.;MATHEMATICS -- Calculus.;MATHEMATICS -- Mathematical Analysis.;Mathematics.;Integral Equations.;Numerical Analysis.;Partial Differential Equations.;Calculus of Variations and Optimal Control;Optimization.;Mathematical and Computational Engineering.;Theoretical, Mathematical and Computational Physics.;Numerical analysis.;Differential calculus & equations.;Calculus of variations.;Maths for engineers.;Mathematical physics.;Integral calculus & equations.

Content: Intro
Preface
Contents
1 Introduction
1.1 The Basic Approximation Problems
1.2 Convergence of Projection Methods
2 Some Elements of Potential Theory
2.1 Representation Formulas
2.2 Single- and Double-Layer Potential
2.2.1 Some Remarks on Distributions
2.2.2 Jump Relations
2.3 Mapping Properties of Boundary Integral Operators
2.4 Laplace's Equation in R3
2.4.1 Representation Formula
2.5 Calderon Projector
2.6 Use of Complex Function Theory
2.6.1 Representation Formula Again
2.6.2 Applicable Representation of the Hypersingular Integral Operator
3 A Fourier Series Approach. 4.4 Interface Problem in Linear Elasticity4.5 A Strongly Elliptic System for Exterior Maxwell's Equations
4.5.1 A Simple Layer Procedure
4.5.2 Modified Boundary Integral Equations
5 The Signorini Problem and More Nonsmooth BVPs and Their Boundary Integral Formulation
5.1 The Signorini Problem in Its Simplest Form
5.2 A Variational Inequality of the Second Kind Modelling Unilateral Frictional Contact
5.3 A Nonmonotone Contact Problem from Delamination
6 A Primer to Boundary Element Methods
6.1 Galerkin Scheme for Strongly Elliptic Operators. 6.2 Galerkin Methods for the Single-Layer Potential6.2.1 Approximation with Trigonometric Polynomials
6.2.2 Approximation with Splines
6.3 Collocation Method for the Single-Layer Potential
6.4 Collocation Methods-Revisited
6.4.1 Periodic Splines as Test and Trial Functions
6.4.2 Convergence Theorem for Projection Methods
6.5 BEM on Quasiuniform Meshes
6.5.1 Periodic Polynomial Splines
6.5.2 The Approximation Theorem
6.5.3 Stability and Inverse Estimates
6.5.4 Aubin-Nitsche Duality Estimateand Superapproximation
6.5.5 Numerical Quadrature
6.5.6 Local H( -1/2)-Error Estimates. 6.5.7 Local L2-Error Estimates6.5.8 The K-Operator-Method
6.5.9 L(∞)-Error Estimates for the Galerkin Approximation
6.6 A Discrete Collocation Method for Symm's Integral Equation on Curves with Corners
6.7 Improved Galerkin Method with Augmented BoundaryElements
6.8 Duality Estimates for Projection Methods
6.8.1 Application to Galerkin Methods
6.8.2 Application to Collocation Methods
6.9 A Collocation Method Interpreted as (GM)
6.10 Modified Collocation and Qualocation
6.11 Radial Basis Functions and Spherical Splines