##
**Boundary integral equations.**
*(English)*
Zbl 1157.65066

Applied Mathematical Sciences 164. Berlin: Springer (ISBN 978-3-540-15284-2/hbk). xx, 618 p. (2008).

The main goal of this book is to explain the mathematical foundation of the boundary element methods (BEMs) even though the boundary element discretizations are not discussed in this book. The BEM is well developed and widely used by engineers and scientists in applied mathematical computations for 40 years. Nevertheless, there was a gap between the theory of boundary integral equations and their computational applications.

This book attempts to bridge that gap. The book will be helpful not only for mathematicians who want to become familiar with the BIE but also for users of BEMs who want to understand mathematical background of the computational method. The employed mathematical language and explanation are acceptable also for engineers providing subsequent reading of the book. The systematic and comprehensive treatment can be documented by the fact that the authors confined themselves to elliptic boundary value problems even though the extent of the book is over 600 pages. The main principles and schemes can be generalized also to other kinds of initial and boundary value problems.

The central idea of eliminating the domain unknowns and reducing boundary value problems to equivalent equations coupling only the unknowns localized on the boundary is demonstrated via direct approaches. The direct procedure is based on Green’s representation formula for solutions of the boundary value problem, whereas the indirect approach needs an appropriate layer ansatz. Recall that the analysis and basic properties of the boundary integral operators are the same in both the approaches. Each boundary integral representation of the solution of a boundary value problem requires knowledge of the fundamental solution of the corresponding governing differential operator. If the fundamental solution is not available then the boundary value problem can still be reduced to a coupled system of domain and boundary integral equations using the fundamental solutions for simplified partial differential equation (PDE).

The basic mathematical properties that guarantee existence of solutions to the boundary integral equations are concerned and also the stability and convergence analysis of corresponding numerical procedures for boundary integral operators on appropriate function spaces. Applications of classical successive iteration procedure to a class of boundary integral equations of the second kind yields extension of these basic features intimately to the variational forms of the underlying elliptic boundary value problems, allowing to consider the boundary integral equations in the form of variational problems on the boundary manifold of the domain. It is natural to consider the boundary integral operators (Newton potentials) as pseudodifferential operators on the boundary manifold. The combination of the variational properties of the elliptic boundary value and transmission problems together with the strongly elliptic pseudodifferential operators is utilized in the analysis of a large class of elliptic value problems.

The book involves a selfconsistent treatment of classical pseudodifferential operators including the basic properties of Sobolev spaces; a construction of a parametrix for elliptic pseudodifferential operators; an iterative procedure to find Levi functions of arbitrary order for general elliptic systems of PDEs a representation of pseudodifferential operators as Hadamard’s finite part integral operators; transformation formulae and invariance properties under the change of coordinates; relationships between the classical pseudodifferential operators and boundary integral operators as well as jump relations of the potentials; concrete examples of boundary integral equations expressed as pseudodifferential operators on the boundary manifold; explicit calculations desired in typical boundary integral operators on closed surfaces; special features of Fourier series expansions of boundary integral operators on closed curves.

This book attempts to bridge that gap. The book will be helpful not only for mathematicians who want to become familiar with the BIE but also for users of BEMs who want to understand mathematical background of the computational method. The employed mathematical language and explanation are acceptable also for engineers providing subsequent reading of the book. The systematic and comprehensive treatment can be documented by the fact that the authors confined themselves to elliptic boundary value problems even though the extent of the book is over 600 pages. The main principles and schemes can be generalized also to other kinds of initial and boundary value problems.

The central idea of eliminating the domain unknowns and reducing boundary value problems to equivalent equations coupling only the unknowns localized on the boundary is demonstrated via direct approaches. The direct procedure is based on Green’s representation formula for solutions of the boundary value problem, whereas the indirect approach needs an appropriate layer ansatz. Recall that the analysis and basic properties of the boundary integral operators are the same in both the approaches. Each boundary integral representation of the solution of a boundary value problem requires knowledge of the fundamental solution of the corresponding governing differential operator. If the fundamental solution is not available then the boundary value problem can still be reduced to a coupled system of domain and boundary integral equations using the fundamental solutions for simplified partial differential equation (PDE).

The basic mathematical properties that guarantee existence of solutions to the boundary integral equations are concerned and also the stability and convergence analysis of corresponding numerical procedures for boundary integral operators on appropriate function spaces. Applications of classical successive iteration procedure to a class of boundary integral equations of the second kind yields extension of these basic features intimately to the variational forms of the underlying elliptic boundary value problems, allowing to consider the boundary integral equations in the form of variational problems on the boundary manifold of the domain. It is natural to consider the boundary integral operators (Newton potentials) as pseudodifferential operators on the boundary manifold. The combination of the variational properties of the elliptic boundary value and transmission problems together with the strongly elliptic pseudodifferential operators is utilized in the analysis of a large class of elliptic value problems.

The book involves a selfconsistent treatment of classical pseudodifferential operators including the basic properties of Sobolev spaces; a construction of a parametrix for elliptic pseudodifferential operators; an iterative procedure to find Levi functions of arbitrary order for general elliptic systems of PDEs a representation of pseudodifferential operators as Hadamard’s finite part integral operators; transformation formulae and invariance properties under the change of coordinates; relationships between the classical pseudodifferential operators and boundary integral operators as well as jump relations of the potentials; concrete examples of boundary integral equations expressed as pseudodifferential operators on the boundary manifold; explicit calculations desired in typical boundary integral operators on closed surfaces; special features of Fourier series expansions of boundary integral operators on closed curves.

Reviewer: Vladimir SlĂˇdek (Bratislava)

### MSC:

65N38 | Boundary element methods for boundary value problems involving PDEs |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

35J25 | Boundary value problems for second-order elliptic equations |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

31A10 | Integral representations, integral operators, integral equations methods in two dimensions |

31B10 | Integral representations, integral operators, integral equations methods in higher dimensions |

35C15 | Integral representations of solutions to PDEs |

35S15 | Boundary value problems for PDEs with pseudodifferential operators |