Finite element analysis of acoustic scattering.

*(English)*Zbl 0908.65091
Applied Mathematical Sciences. 132. New York, NY: Springer. xiv, 224 p. (1998).

The rough contents of this beautiful book are as follows: Preface, 1. The governing equations of time-harmonic wave propagation, 2. Analytical and variational solutions of Helmholtz problems, 3. Discretization methods for exterior Helmholtz problems, 4. Finite element error analysis and control for Helmholtz problems, 5. Computational simulation of elastic scattering, References and a Subject index. Each chapter ends with a summary and some bibliographical remarks.

In the first chapter, under the unified concept of linear wave physics, the author considers acoustic waves, elastic waves, acoustic/elastic fluid-solid interaction and electromagnetic waves. The Sommerfeld condition (no waves are reflected from infinity) as well as boundary conditions on the scatterer, corresponding to various physical situations, are also displayed. The main interest of the author is in the numerical solution of exterior boundary value problems for the Helmholtz equation.

In this respect, in the second chapter, he takes into account some specific questions such as the variational formulation of Helmholtz problems on unbounded domains, the appropriate choice of test and trial spaces, the well-posedness of variational problems as well as the construction of variational methods (Galerkin and Ritz). The principal technique of tackling unbounded domain problems is the domain decomposition by introducing an artificial boundary around the obstacle (scatterer).

The author reviews some of the coupling strategies on these artificial boundaries in the third chapter. The emphasis is upon those methods based on series representation of the exterior solutions. He considers in turn Dirichlet-to-Neumann and other absorbing boundary conditions, a perfectly matched layer method and infinite elements.

The high sensitivity of numerical methods to large wave numbers, which characterizes the oscillatory behaviour of the exact solution, is the central issue of the fourth chapter. The author presents new estimates that characterize the error behaviour in the range of engineering computations and also investigates some a posteriori estimations mainly for the one-dimensional case.

A lot of computational results for three-dimensional elastic scattering from a sphere and a cylinder with spherical end-caps are reported in the final chapter.

The References contain 122 entries.

All in all, the book is recommended to mathematicians interested in numerical analysis of the exterior problems for elliptic equations as well as to physicists and engineers working on scattering problem. It is selfcontained and easily readable, and, at the same time, well written with clear and precise statements.

In the first chapter, under the unified concept of linear wave physics, the author considers acoustic waves, elastic waves, acoustic/elastic fluid-solid interaction and electromagnetic waves. The Sommerfeld condition (no waves are reflected from infinity) as well as boundary conditions on the scatterer, corresponding to various physical situations, are also displayed. The main interest of the author is in the numerical solution of exterior boundary value problems for the Helmholtz equation.

In this respect, in the second chapter, he takes into account some specific questions such as the variational formulation of Helmholtz problems on unbounded domains, the appropriate choice of test and trial spaces, the well-posedness of variational problems as well as the construction of variational methods (Galerkin and Ritz). The principal technique of tackling unbounded domain problems is the domain decomposition by introducing an artificial boundary around the obstacle (scatterer).

The author reviews some of the coupling strategies on these artificial boundaries in the third chapter. The emphasis is upon those methods based on series representation of the exterior solutions. He considers in turn Dirichlet-to-Neumann and other absorbing boundary conditions, a perfectly matched layer method and infinite elements.

The high sensitivity of numerical methods to large wave numbers, which characterizes the oscillatory behaviour of the exact solution, is the central issue of the fourth chapter. The author presents new estimates that characterize the error behaviour in the range of engineering computations and also investigates some a posteriori estimations mainly for the one-dimensional case.

A lot of computational results for three-dimensional elastic scattering from a sphere and a cylinder with spherical end-caps are reported in the final chapter.

The References contain 122 entries.

All in all, the book is recommended to mathematicians interested in numerical analysis of the exterior problems for elliptic equations as well as to physicists and engineers working on scattering problem. It is selfcontained and easily readable, and, at the same time, well written with clear and precise statements.

Reviewer: C.I.Gheorghiu (Cluj-Napoca)

##### MSC:

65Nxx | Numerical methods for partial differential equations, boundary value problems |

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

78A45 | Diffraction, scattering |

76Q05 | Hydro- and aero-acoustics |

74J20 | Wave scattering in solid mechanics |

74S05 | Finite element methods applied to problems in solid mechanics |

65Lxx | Numerical methods for ordinary differential equations |

76M10 | Finite element methods applied to problems in fluid mechanics |