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2011 | Buch

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Boundary Element Methods

verfasst von: Stefan A. Sauter, Christoph Schwab

Verlag: Springer Berlin Heidelberg

Buchreihe : Springer Series in Computational Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This work presents a thorough treatment of boundary element methods (BEM) for solving strongly elliptic boundary integral equations obtained from boundary reduction of elliptic boundary value problems in $\mathbb{R}^3$. The book is self-contained, the prerequisites on elliptic partial differential and integral equations being presented in Chapters 2 and 3. The main focus is on the development, analysis, and implementation of Galerkin boundary element methods, which is one of the most flexible and robust numerical discretization methods for integral equations. For the efficient realization of the Galerkin BEM, it is essential to replace time-consuming steps in the numerical solution process with fast algorithms. In Chapters 5-9 these methods are developed, analyzed, and formulated in an algorithmic way.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

Many physical processes can be described by systems of linear and non-linear differential and integral equations. Only in very few special cases can such equations be solved analytically, which is why numerical methods have to be developed for their solution. In light of the complexity of the problems that appear in practice, it is unrealistic to expect to find a numerical method that offers a black-box type of numerical method that is suitable for all these problems. A more reasonable approach is to develop special numerical methods for specific classes of problems in order to take advantage of the characteristic properties of these classes. These numerical methods should then be decomposed into isolated and elementary partial problems. It would then be possible to employ or develop efficient methods for these subproblems.

Stefan A. Sauter, Christoph Schwab

Chapter2. Elliptic Differential Equations

Abstract

Integral equations occur in many physical applications. We encounter some of the most important ones when we try to solve elliptic differential equations. These can be transformed into integral equations and can then be solved numerically by means of the boundary element method. The subject of this chapter is the formulation and analysis of scalar, elliptic boundary value problems.

Stefan A. Sauter, Christoph Schwab

Chapter 3. Elliptic Boundary Integral Equations

Abstract

Homogeneous, linear elliptic boundary value problems with constant coefficients can be transformed into boundary integral equations by using the integral equation method. In this chapter we will introduce the relevant boundary integral operators and we will derive the most important mapping properties and representations. We will also present the boundary integral equations for the boundary value problems from the previous chapter. Finally, we will prove the appropriate results on existence and uniqueness for these boundary integral equations.

Stefan A. Sauter, Christoph Schwab

Chapter 4. Boundary Element Methods

Abstract

In Chap. 3 we transformed strongly elliptic boundary value problems of second order in domains \( \Omega \subset \mathbb{R}^3\) into boundary integral equations. These integral equations were formulated as variational problems on a Hilbert space H:

Stefan A. Sauter, Christoph Schwab

Chapter 5. Generating the Matrix Coefficients

Abstract

In order to implement the Galerkin method for boundary integral equations, the approximation of the coefficients of the system matrix and the right-hand side becomes a primary task.

Stefan A. Sauter, Christoph Schwab

Chapter 6. Solution of Linear Systems of Equations

Abstract

The Galerkin boundary element method transforms the boundary integral equation to the linear system of equations

Stefan A. Sauter, Christoph Schwab

Chapter 7. Cluster Methods

Abstract

Partial differential equations can be directly discretized by means of difference methods or finite element methods (domain methods).

Stefan A. Sauter, Christoph Schwab

Chapter 8. p-Parametric Surface Approximation

Abstract

In practice, the description of the “true” physical surface might be very compli- 3 cated or even not available as an exact analytic function and has to be approximated 4 by using, e.g., pointwise measurements of the surface or some geometric mod- 5 elling software. In this chapter, we will address the question how to approximate 6 quite general surfaces in a flexible way by p-parametric boundary elements. Sur- 7 face approximations for integral equations and their influence on the discretization 8 error have first been studied systematically in [167]. Further papers on this topic are 9 [168], [80, Chap. XIII, Sect. 2], [84], [21], [63, Sect. 1.4].

Stefan A. Sauter, Christoph Schwab

Chapter 9. A Posteriori Error Estimation

Abstract

The error analysis for the Galerkin discretization exhibits the asymptotic conver- 3 gence rates for the boundary element method which depend on the regularity of the 4 underlying integral equation. These estimates are called a priori estimates because 5 they hold for large classes of problems which are characterized by their regular- 6 ity. They are important because they show the asymptotic quality of the Galerkin 7 boundary method. However, for a concrete problem these estimates could be by far 8 too pessimistic and do not allow answers to the following questions:

Stefan A. Sauter, Christoph Schwab

Backmatter

Titel: Boundary Element Methods
verfasst von: Stefan A. Sauter
Christoph Schwab
Copyright-Jahr: 2011
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-540-68093-2
Print ISBN: 978-3-540-68092-5
DOI: https://doi.org/10.1007/978-3-540-68093-2

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