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**Natural boundary integral method and its applications. Translated from the 1993 Chinese original.**
*(English)*
Zbl 1028.65129

Mathematics and its Applications (Dordrecht). 539. Dordrecht: Kluwer Academic Publishers. Beijing: Science Press. xviii, 539 p. (2002).

This very interesting book presents a systematical introduction to the natural boundary integral method which was suggested and developed by Kang Feng and the author of the book. The book is a translated edition of the Chinese original (published in 1993 by Science Press, Beijing) but also it contains many new research results.

The book contains of seven chapters as follows:

In Chapter 1 the author starts with summarizing the direct and indirect boundary reductions and boundary element methods and presenting the basic ideas of the natural boundary reduction, the numerical computation of hypersingular integrals and convergence and error estimates for the natural boundary element solutions.

Chapter 2 deals with boundary value problem for the harmonic equation. In this chapter the author studies the representation of a solution of the harmonic equation by complex variable functions, the principal of its natural boundary reduction and the corresponding natural integral equations, the natural boundary reduction for general simply connected domains, the natural integral operators and their inverse operators and numerical solutions of the natural integral equations.

Chapter 3 is devoted to the boundary value problem for the biharmonic equation. Here the author gives representation of the solutions by complex variable functions, the principle of the natural boundary reduction for the biharmonic equation, the natural integral operators and their inverse, the numerical solution of the natural integral equations, etc.

Chapter 4 and Chapter 5 deal with the plane elasticity problem and the Stokes problem, respectively. In these chapters the author considers the representation of the respective solutions by complex variable functions, presents the natural boundary reduction for the problems under consideration and study in details some concrete domains.

In Chapter 6 the author considers the coupling of the natural boundary elements and the finite elements for solving the harmonic equation, biharmonic equation, plane elasticity problem and the Stokes problem. The last Chapter 7 deals with the domain decomposition methods (DDM) based on the natural boundary reduction and here the author presents overlapping and non-overlapping DDM based on natural boundary reduction as well as considers the Steklov-Poincaré operators and their inverse operators.

The book is very well written, the exposition is clear, readable and very systematical. The book is of a great value for the researchers in numerical analysis and its related fields.

The book contains of seven chapters as follows:

In Chapter 1 the author starts with summarizing the direct and indirect boundary reductions and boundary element methods and presenting the basic ideas of the natural boundary reduction, the numerical computation of hypersingular integrals and convergence and error estimates for the natural boundary element solutions.

Chapter 2 deals with boundary value problem for the harmonic equation. In this chapter the author studies the representation of a solution of the harmonic equation by complex variable functions, the principal of its natural boundary reduction and the corresponding natural integral equations, the natural boundary reduction for general simply connected domains, the natural integral operators and their inverse operators and numerical solutions of the natural integral equations.

Chapter 3 is devoted to the boundary value problem for the biharmonic equation. Here the author gives representation of the solutions by complex variable functions, the principle of the natural boundary reduction for the biharmonic equation, the natural integral operators and their inverse, the numerical solution of the natural integral equations, etc.

Chapter 4 and Chapter 5 deal with the plane elasticity problem and the Stokes problem, respectively. In these chapters the author considers the representation of the respective solutions by complex variable functions, presents the natural boundary reduction for the problems under consideration and study in details some concrete domains.

In Chapter 6 the author considers the coupling of the natural boundary elements and the finite elements for solving the harmonic equation, biharmonic equation, plane elasticity problem and the Stokes problem. The last Chapter 7 deals with the domain decomposition methods (DDM) based on the natural boundary reduction and here the author presents overlapping and non-overlapping DDM based on natural boundary reduction as well as considers the Steklov-Poincaré operators and their inverse operators.

The book is very well written, the exposition is clear, readable and very systematical. The book is of a great value for the researchers in numerical analysis and its related fields.

Reviewer: Emil Minchev (Chiba)

### MSC:

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

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

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

35J40 | Boundary value problems for higher-order elliptic equations |

35Q30 | Navier-Stokes equations |

31B30 | Biharmonic and polyharmonic equations and functions in higher dimensions |

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

65N15 | Error bounds for boundary value problems involving PDEs |

74B05 | Classical linear elasticity |

74S15 | Boundary element methods applied to problems in solid mechanics |