##
**Finite element methods for Maxwell’s equations.**
*(English)*
Zbl 1024.78009

Numerical Mathematics and Scientific Computation. Oxford: Oxford University Press. xiv, 450 p. (2003).

The purpose of the book is to present fundamentals of the mathematical theory of the celebrated Maxwell equations relevant to the finite element method. The author starts with a discussion of mathematical models of electromagnetism introducing the reader to the subject. Chapters 2 and 3 are concerned with background material and present basic facts from functional analysis along with abstract finite element convergence theory and fundamental information on Sobolev spaces, vector function spaces, and regularity.

Equipped with these facts, the reader proceeds in Chapter 4 with a discussion of a simple model for Maxwell’s equations, a cavity problem on a bounded domain with boundary conditions motivated by scattering applications. A standard variational formulation for the cavity problem is derived and analyzed with the help of function spaces introduced in the previous chapter. Motivated by interesting advances in finite element theory and function space theory for Maxwell’s equations which had taken place in the context of Lipschitz polyhedral domains, the author focuses his attention to Lipschitz polyhedra which allow the use of standard tetrahedral measures.

Since the edge elements of Nédélec are particularly well suited for discretization of the Maxwell system, Chapters 5 and 6 are devoted to a detailed presentation of these spaces, along with an associated scalar space for the electrostatic potential and related spaces. Special attention is paid to the discrete de Rham diagram which summarizes the relationship between the relevant function spaces, their finite element discretizations, and interpolation operators.

Finite element discretization is discussed in Chapter 7, where two proofs of convergence for this method are presented in detail. Another family of elements, also due to Nédélec, is considered in Chapter 8, where basic facts on approximations of curved boundaries and a brief introduction to \(hp\) finite element methods for Maxwell’s equations can be also found.

Classical scattering by a sphere is the subject of Chapter 9, where the famous integral representation of the solution of Maxwell’s equation (Stratton-Chu formula for the scattered field in a homogeneous isotropic exterior domain) is derived together with classical series representations of the solution. These formulae are used in the next chapter for constructing a semi-discrete method for the scattering problem based on the electromagnetic equivalent of the Dirichlet to Neumann map called the electric-to-magnetic Calderón operator. A fully discrete domain-decomposed version of this algorithm is suggested and examined in Chapter 11.

The drawback of the methods presented in Chapters 10 and 11 lies in the fact that they need a truncated domain with a spherical truncation boundary which results in high computational cost for high aspect ratio scatterers. Thus a method for approximating the electromagnetic field scattered from objects embedded in a non-uniform background medium is proposed in Chapter 12. The Stratton-Chu formula is used to represent the solution outside the scatterer and simultaneously the finite element method is used on a truncated domain extending outside the scatterer. This leads to the existence of an overlapping region where both the finite element method and integral representation give an approximation to the electromagnetic field.

Chapter 13 deals with issues related to practical aspects of solving Maxwell’s equations. In particular, solution of the large and sparse linear system resulting from the discretization of the Maxwell system is addressed, along with sensitivity of the error in the calculation of the frequency of the radiation and extraction of the far field pattern of the scattered wave from the knowledge of the near field.

The final chapter provides a brief introduction to inverse problems which are the author’s main reason for studying scattering theory. The bibliography collects many basic papers and monographs in the field and consists of 302 items.

The book is well-written, and its core part consisting of Chapters 4, 5, and 7 along with material from Chapter 13 can be used for teaching a graduate course for students with solid mathematical background in functional analysis and Sobolev space theory. It is a valuable reference for specialists interested in the mathematical theory of Maxwell’s equations relevant to numerical analysis.

Equipped with these facts, the reader proceeds in Chapter 4 with a discussion of a simple model for Maxwell’s equations, a cavity problem on a bounded domain with boundary conditions motivated by scattering applications. A standard variational formulation for the cavity problem is derived and analyzed with the help of function spaces introduced in the previous chapter. Motivated by interesting advances in finite element theory and function space theory for Maxwell’s equations which had taken place in the context of Lipschitz polyhedral domains, the author focuses his attention to Lipschitz polyhedra which allow the use of standard tetrahedral measures.

Since the edge elements of Nédélec are particularly well suited for discretization of the Maxwell system, Chapters 5 and 6 are devoted to a detailed presentation of these spaces, along with an associated scalar space for the electrostatic potential and related spaces. Special attention is paid to the discrete de Rham diagram which summarizes the relationship between the relevant function spaces, their finite element discretizations, and interpolation operators.

Finite element discretization is discussed in Chapter 7, where two proofs of convergence for this method are presented in detail. Another family of elements, also due to Nédélec, is considered in Chapter 8, where basic facts on approximations of curved boundaries and a brief introduction to \(hp\) finite element methods for Maxwell’s equations can be also found.

Classical scattering by a sphere is the subject of Chapter 9, where the famous integral representation of the solution of Maxwell’s equation (Stratton-Chu formula for the scattered field in a homogeneous isotropic exterior domain) is derived together with classical series representations of the solution. These formulae are used in the next chapter for constructing a semi-discrete method for the scattering problem based on the electromagnetic equivalent of the Dirichlet to Neumann map called the electric-to-magnetic Calderón operator. A fully discrete domain-decomposed version of this algorithm is suggested and examined in Chapter 11.

The drawback of the methods presented in Chapters 10 and 11 lies in the fact that they need a truncated domain with a spherical truncation boundary which results in high computational cost for high aspect ratio scatterers. Thus a method for approximating the electromagnetic field scattered from objects embedded in a non-uniform background medium is proposed in Chapter 12. The Stratton-Chu formula is used to represent the solution outside the scatterer and simultaneously the finite element method is used on a truncated domain extending outside the scatterer. This leads to the existence of an overlapping region where both the finite element method and integral representation give an approximation to the electromagnetic field.

Chapter 13 deals with issues related to practical aspects of solving Maxwell’s equations. In particular, solution of the large and sparse linear system resulting from the discretization of the Maxwell system is addressed, along with sensitivity of the error in the calculation of the frequency of the radiation and extraction of the far field pattern of the scattered wave from the knowledge of the near field.

The final chapter provides a brief introduction to inverse problems which are the author’s main reason for studying scattering theory. The bibliography collects many basic papers and monographs in the field and consists of 302 items.

The book is well-written, and its core part consisting of Chapters 4, 5, and 7 along with material from Chapter 13 can be used for teaching a graduate course for students with solid mathematical background in functional analysis and Sobolev space theory. It is a valuable reference for specialists interested in the mathematical theory of Maxwell’s equations relevant to numerical analysis.

Reviewer: Yuri V.Rogovchenko (Famagusta)

### MSC:

78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |

78-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to optics and electromagnetic theory |

78A25 | Electromagnetic theory (general) |

78A46 | Inverse problems (including inverse scattering) in optics and electromagnetic theory |

78A45 | Diffraction, scattering |

65Z05 | Applications to the sciences |

35Q60 | PDEs in connection with optics and electromagnetic theory |