Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top

2016 | Book

The retarded layer potentials Read chapter Read first chapter

Retarded Potentials and Time Domain Boundary Integral Equations

A Road Map

Author: Francisco-Javier Sayas

Publisher: Springer International Publishing

Book Series : Springer Series in Computational Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Login to get access
insite
SEARCH

About this book

This book offers a thorough and self-contained exposition of the mathematics of time-domain boundary integral equations associated to the wave equation, including applications to scattering of acoustic and elastic waves. The book offers two different approaches for the analysis of these integral equations, including a systematic treatment of their numerical discretization using Galerkin (Boundary Element) methods in the space variables and Convolution Quadrature in the time variable. The first approach follows classical work started in the late eighties, based on Laplace transforms estimates. This approach has been refined and made more accessible by tailoring the necessary mathematical tools, avoiding an excess of generality. A second approach contains a novel point of view that the author and some of his collaborators have been developing in recent years, using the semigroup theory of evolution equations to obtain improved results. The extension to electromagnetic waves is explained in one of the appendices.

Table of Contents

Frontmatter
Chapter 1. The retarded layer potentials
Abstract
In this chapter we are going to introduce the basic concepts of time domain acoustic layer potentials and how they can be used to represent the solutions of scattering problems. All notions introduced in this chapter will be given at an intuitive level and with basically no formalization. The reader will find a precise sketch of the theory in the next chapters. For the sake of clarity, let me remark here:
  • Sections 1.1 through 1.3 deal with three-dimensional waves.
  • Just by looking at the mathematical expressions therein, it will be clear that Sections 1.4 through 1.6 are dimension-independent.
  • Section 1.7 revisits the particular three-dimensional case, using the specific formulas for the Huygens’ single layer potential.
  • Finally, Section 1.8 will present formulas for the two-dimensional case.
Francisco-Javier Sayas
Chapter 2. From time domain to Laplace domain
Abstract
A possible theoretical framework for the time domain layer potentials for the wave equation is that of vector-valued distributions. From this moment on, instead of thinking of functions of the space and time variables u(x, t) we will think of functions of the time variable with values on a space of functions of the space variables, which amounts to considering the functions u(t) = u( ⋅ , t) (we will not change the name). In principle, our distributions will be only allowed to take real values, but the test functions (and the Laplace transforms of the distributions) will take complex values.
Francisco-Javier Sayas
Chapter 3. From Laplace domain to time domain
Abstract
The task of identifying what functions F(s) are Laplace transforms of causal functions (distributions) is part of what is often called the Paley-Wiener Theorem, which is a collection of results related to holomorphic extensions of the Fourier transform that can be understood as two-sided Laplace transforms. Our presentation will first restrict the kind of symbols (Laplace transforms) that we want to invert. Part of the material that follows is adapted (and modified) from the PhD dissertation of Antonio Laliena [61].
Francisco-Javier Sayas
Chapter 4. Convolution Quadrature
Abstract
Convolution quadrature (CQ) is a discretization technique for causal convolutions and convolution equations. Coming from a mathematical argument that might seem bizarre at the beginning, this method ends up using data in the time domain but the Laplace transform of the operator. This mixture of Laplace and time domain looks somewhat unnatural but yields a general type of methods that can be easily used in black-box fashion. The method and much of its initial development are due to Christian Lubich. We will try not to be too heavy on notation for sequences by writing (a n ) to denote the sequence (a n ) n ≥ 0.
Francisco-Javier Sayas
Chapter 5. The discrete layer potentials
Abstract
In this chapter we explore fully discrete CQ-BEM methods (Convolution Quadrature in time, Galerkin Boundary Elements in space) for the exterior Dirichlet and Neumann problems associated with the wave equation. For the first one we will use a retarded single layer potential representation, and for the second one, a retarded double layer potential representation. We will detail the analysis of Galerkin semidiscretization error and some stability estimates that are needed to apply the results on Convolution Quadrature to the fully discrete scheme. It has already been mentioned, but let us repeat it here: the analysis of full discretizations for time domain integral equations in acoustics seems to be restricted to equations of the first kind, since the analysis is based on variational (energy) estimates. Not much is known about the behavior of the integral equations of the second kind, especially once they are discretized. The theory of full Galerkin discretizations is based on the weak weighted coercivity estimates given in Section 3.​7 In the way the method is used, space and time Galerkin can be understood as a Galerkin-in-time discretization of a semidiscrete Galerkin problem. From that point of view, all the results given here about semidiscretization in space could eventually be used for the fully discrete method.
Francisco-Javier Sayas
Chapter 6. A general class of second order differential equations
Abstract
In this short chapter we introduce some elementary tools of evolution equations which will be applied to the analysis of time domain potentials and integral operators. Most of these results in the abstract treatment of evolution equations can be obtained with elementary tools of the theory of strongly continuous semigroups of operators (of groups of isometries actually). An introduction to semigroup techniques applied to partial differential equations can be found in [58, Chapter 4] or [44], with a very general treatment given in [71]. The theory of semigroups of operators is rarely part of the mathematical toolbox for users of boundary integral equations. This is the reason why we include a self-contained approach in Appendix B using a simple point of view, namely the method of separation of variables.
Francisco-Javier Sayas
Chapter 7. Time domain analysis of the single layer potential
Abstract
In this chapter we develop a systematic approach to a pure time domain analysis of the single layer potential and operator, the Galerkin semidiscrete error operator and the Galerkin solver. Namely, we want to obtain estimates of
$$ \displaystyle{\mathcal{S}{\ast}\lambda,\qquad \mathcal{V}{\ast}\lambda,\qquad \mathcal{K}^{t}{\ast}\lambda,} $$
of its semidiscrete inverses
$$ \displaystyle{\mathcal{G}_{\lambda }^{h}{\ast}\beta,\qquad \mathcal{G}_{ u}^{h}{\ast}\beta = \mathcal{S}{\ast}\mathcal{G}_{\lambda }^{h}{\ast}\beta } $$
and of the error operators
$$ \displaystyle{\mathcal{E}_{\lambda }^{h}{\ast}\lambda = \mathcal{G}_{\lambda }^{h} {\ast}\mathcal{V}{\ast}\lambda -\lambda,\qquad \mathcal{E}_{ u}^{h}{\ast}\lambda = \mathcal{S}{\ast}\mathcal{E}_{\lambda }^{h} {\ast}\lambda.} $$
Francisco-Javier Sayas
Chapter 8. Time domain analysis of the double layer potential
Abstract
This chapter is the double layer counterpart of Chapter 7 We will start by studying
$$\displaystyle{\mathcal{D}{\ast}\varphi,\qquad \mathcal{K}{\ast}\varphi,\quad \mbox{ and}\quad \mathcal{W}{\ast}\varphi.}$$
We will next focus on the semidiscrete inversion of the operator \(\varphi \mapsto \mathcal{W}{\ast}\varphi\), namely, given a closed subspace \(Y _{h} \subset H^{1/2}(\Gamma )\) we will look for the causal solution \(\varphi ^{h}\) of
$$\displaystyle{\mathcal{W}{\ast}\varphi ^{h}-\alpha \in Y _{ h}^{\circ }}$$
and input it in a double potential representation
$$\displaystyle{u^{h} = \mathcal{D}{\ast}\varphi ^{h}.}$$
This process was studied in the Laplace domain in Sections 5.​5 and 5.​6
Francisco-Javier Sayas
Chapter 9. Full discretization revisited
Abstract
In this chapter we give some estimates for the fully discrete method (using Galerkin semidiscretization is space and CQ in time) for the single layer potential representation of the Dirichlet problem. Our goal is to prove all the bounds using time-domain techniques instead of estimates in the Laplace domain. The contents of this chapter follow (on a different example and with slightly different results) what appears in the last section of [13]. The techniques of this chapter mimic the traditional analysis of low order discretization methods for the wave equation. The context where these tools are used is, nevertheless, quite different from what is commonly seen in the literature.
Francisco-Javier Sayas
Chapter 10. Patterns, extensions, and conclusions
Abstract
In this short chapter we will go over the results of Chapters 7 and 8, trying to look for common arguments and patterns in the final estimates. It will be clear that problems are paired and that by looking at the type of transmission condition we can predict what kind of regularity the data needs to have. We will next give two simple extensions of this theory, to problems defined on screens and to problems on linear elasticity. The chapter will finish with an overview of part of the literature and mentioning some work in progress.
Francisco-Javier Sayas
Backmatter
Metadata
Title
Retarded Potentials and Time Domain Boundary Integral Equations
Author
Francisco-Javier Sayas
Copyright Year
2016
Electronic ISBN
978-3-319-26645-9
Print ISBN
978-3-319-26643-5
DOI
https://doi.org/10.1007/978-3-319-26645-9

Premium Partner

    Image Credits