##
**Mathematical analysis and numerical methods for science and technology. (In six volumes). Volume 4: Integral equations and numerical methods. With the collaboration of Michel Artola, Philippe Bénilan, Michel Bernadou, Michel Cessenat, Jean-Claude Nédélec, Jacques Planchard, Bruno Scheurer. Transl. from the French by John C. Amson.**
*(English)*
Zbl 0784.73001

Berlin etc.: Springer-Verlag. x, 465 p. (1990).

This book, consisting of four chapters of unequal length, is focussed on three topics: Frankl and particularly Tricomi equations, integral equations, and numerical analysis (mainly based on finite element methods) of some stationary problems including approximation of integral equations.

The first concise chapter sketches the problem of the stationary flow of a compressible fluid around an obstacle. As an example, one may cite the air flow around an aircraft wing section. Under some simplifying assumptions, like the lack of viscosity and thermal conductivity, uniform flow at infinity, it is shown how to transform the set of basic equations into the second order Frankl equation in the hodograph plane. The Frankl equation is of a mixed type (elliptic-hyperbolic) depending upon the Mach number. The methods of solutions in both elliptic and hyperbolic regions are discussed, including the variational and energy approach.

The second chapter consists of two parts: A and B. In part A various methods of solving linear integral equations are presented, including some applications. The last comprise simple and double layer problems, the problem of a thin aerofoil profile, the determination of the charge density on the surface of a cylindrical body at potential \(V\) and the planar deformation problem in the classical elasticity. Part B can serve as an introduction to problems of integral representation of solutions to the interior and exterior Dirichlet and Neumann problems for the classical class of equations like the Laplace equation, the Helmholtz equation, the static linear and isotropic elasticity (a homogeneous material) equations and the Stokes system. The less studied Laplace- Beltrami operator on a regular surface is also introduced and its usefulness is exhibited. In all that part, when possible, the variational approach is employed what is important from the numerical point of view. The integral representations mentioned above are obtained in the form of a simple or a double layer potential. From the practical point of view, in order to apply the results presented, one has to be able to solve effectively the derived integral equations. The rather concisely written fourth chapter deals with the approximation by finite elements of integral equations on a surface in the three-dimensional space, associated with the Laplace equation in this space. Two types of surfaces are considered; the first one is a polyhedral surface. For such a surface, the classical mesh by triangles (or quadrilaterals) is easily performed. For a regular closed surface, defined by a finite number of charts, those charts are used to construct an approximate surface with the aid of finite elements. For both cases the error analysis is performed, though for details the reader has to refer to original papers. To better understand this chapter, the reader should be familiar with the preceding or main chapter of the book.

Chapter 3 is thought as a comprehensive introduction to rigorously treated finite element methods (FEMs) for stationary problems. To begin with, the principal features of the finite difference method and the FEM are assembled. It is worth noting that the FEM was initially invented to solve the problems of the mechanics of continuous media with complicated geometries. Consequently, the basic concepts of the FEM are developed in an example of essential importance in applications of solid mechanics, i.e., the problem of three-dimensional linear elasticity formulated on a polyhedral domain. The passage from the continuous to the discrete problem is carefully discussed, including numerical quadrature schemes. Examples of various finite elements, being now “classical”, illustrate well the introduced abstract notion of a finite element. Estimates of the error between the exact and the approximate solutions are carried out and the influence of the numerical integration is examined. To more practically oriented readers will appeal the description of the different steps of the implementation of a finite element method applied to the numerical solution of a three-dimensional elasticity problem. Being now equipped with all the basic notions and results related to polyhedral domains, the reader will have no difficulty in assimilating the more complicated case of curved boundaries. In the last case, the study is deliberately limited to the problem of the plane linear elasticity formulated on a domain with a curved boundary. Here the authors follow the methods introduced by M. Zlámal [Int. J. Numer. Methods Eng. 5, 367-373 (1973; Zbl 0254.65073); SIAM J. Numer. Anal. 10, 229-240 (1973; Zbl 0285.65067); SIAM J. numer. Analysis 11, 347-362 (1974; Zbl 0277.65064)] to investigate curved finite elements of class \(C^ 0\).

One has to do with nonconforming methods of finite elements when the finite dimensional space, to which a solution to the approximate problem belongs, is not included in the space to which an exact solution belongs. A nonconforming method is presented as the example of the two-dimensional problem of isotropic homogeneous elasticity by using the so-called Wilson finite element. Here the essential difficulty lies in proving that a solution to the discrete problem exists and in obtaining error estimates.

From the engineering point of view, one has to be able to solve boundary value problems arising in structural mechanics. Structures, like plates and shells, are usually reduced by imposing suitable constraints, to two- dimensional models. The principal results concerning the approximation of solutions for thin linear elastic plates and shells made of isotropic and homogeneous materials are stated without proofs. We observe, however, that plates are described by the operator of order 4; the shell operator is a system of order 2 with respect to the first two unknowns and of order 4 with respect to the third one. Consequently, new difficulties appear in the approximation of solutions to relevant boundary value problems.

The basic ideas of approximation of eigenvalues and eigenvectors are expounded by the example of the eigenvalue problem for a three- dimensional linear elastic body. The respective operator is self-adjoint and compact. Consequently, one can readily extend the formulated (and proved) results concerning the estimation of errors to a larger class of linear, self-adjoint and compact operators. From among numerous physical and mechanical problems involving non self-adjoint operators, the case of neutron economy of nuclear reactors is studied in detail. In particular, having recalled the neutron diffusion equations, the critical calculation theory is presented first with two energy groups (stationary problem). In this specific case approximation using finite elements of order 1 is examined. The critical calculation theory is then extended to a number of neutron energy groups greater than two. The chapter ends up with the study of the eigenvalue problem connected with the evolution problem in the case of two energy groups.

The theory of singular integral equations is presented in a quite comprehensive appendix adduced to the main body of the text. The presentation is rigorous and somewhat abstract and includes the Calderon- Zygmund theorem as well as the basic results related to Marcinkiewicz spaces. The latter are needed for the study of the singular integral operator on the space \(L^ 1\) of integrable functions.

Though being a collaborative work, the book is rather well written, in a manner typical for the famous French school of applied mathematics. It contains a lot of information and treats in mathematically elegant fashion several classes of important physical and mechanical problems. The list of references is by no means exhaustive; to add a few: N. F. Morozov, Mathematical questions in the theory of cracks, Nauka, Moskva (1984; Zbl 0566.73079); B. A. Szabo and I. Babuška, Finite element analysis, J. Wiley \(\&\) Sons, New York (1991); A. Ženišek, Nonlinear elliptic and evolution problems and their finite element approximations, Academic Press, London (1990; Zbl 0731.65090) and the reference cited therein.

Some faults and misprints will easily be perceived by the carefull reader. The reviewer recommends strongly the book to applied mathematicians including those involved in the numerical analysis as well as to researchers working in mathematical physics. The volume should be of particular interest to researchers and graduate students elaborating finite element methods in solid and structural mechanics.

The first concise chapter sketches the problem of the stationary flow of a compressible fluid around an obstacle. As an example, one may cite the air flow around an aircraft wing section. Under some simplifying assumptions, like the lack of viscosity and thermal conductivity, uniform flow at infinity, it is shown how to transform the set of basic equations into the second order Frankl equation in the hodograph plane. The Frankl equation is of a mixed type (elliptic-hyperbolic) depending upon the Mach number. The methods of solutions in both elliptic and hyperbolic regions are discussed, including the variational and energy approach.

The second chapter consists of two parts: A and B. In part A various methods of solving linear integral equations are presented, including some applications. The last comprise simple and double layer problems, the problem of a thin aerofoil profile, the determination of the charge density on the surface of a cylindrical body at potential \(V\) and the planar deformation problem in the classical elasticity. Part B can serve as an introduction to problems of integral representation of solutions to the interior and exterior Dirichlet and Neumann problems for the classical class of equations like the Laplace equation, the Helmholtz equation, the static linear and isotropic elasticity (a homogeneous material) equations and the Stokes system. The less studied Laplace- Beltrami operator on a regular surface is also introduced and its usefulness is exhibited. In all that part, when possible, the variational approach is employed what is important from the numerical point of view. The integral representations mentioned above are obtained in the form of a simple or a double layer potential. From the practical point of view, in order to apply the results presented, one has to be able to solve effectively the derived integral equations. The rather concisely written fourth chapter deals with the approximation by finite elements of integral equations on a surface in the three-dimensional space, associated with the Laplace equation in this space. Two types of surfaces are considered; the first one is a polyhedral surface. For such a surface, the classical mesh by triangles (or quadrilaterals) is easily performed. For a regular closed surface, defined by a finite number of charts, those charts are used to construct an approximate surface with the aid of finite elements. For both cases the error analysis is performed, though for details the reader has to refer to original papers. To better understand this chapter, the reader should be familiar with the preceding or main chapter of the book.

Chapter 3 is thought as a comprehensive introduction to rigorously treated finite element methods (FEMs) for stationary problems. To begin with, the principal features of the finite difference method and the FEM are assembled. It is worth noting that the FEM was initially invented to solve the problems of the mechanics of continuous media with complicated geometries. Consequently, the basic concepts of the FEM are developed in an example of essential importance in applications of solid mechanics, i.e., the problem of three-dimensional linear elasticity formulated on a polyhedral domain. The passage from the continuous to the discrete problem is carefully discussed, including numerical quadrature schemes. Examples of various finite elements, being now “classical”, illustrate well the introduced abstract notion of a finite element. Estimates of the error between the exact and the approximate solutions are carried out and the influence of the numerical integration is examined. To more practically oriented readers will appeal the description of the different steps of the implementation of a finite element method applied to the numerical solution of a three-dimensional elasticity problem. Being now equipped with all the basic notions and results related to polyhedral domains, the reader will have no difficulty in assimilating the more complicated case of curved boundaries. In the last case, the study is deliberately limited to the problem of the plane linear elasticity formulated on a domain with a curved boundary. Here the authors follow the methods introduced by M. Zlámal [Int. J. Numer. Methods Eng. 5, 367-373 (1973; Zbl 0254.65073); SIAM J. Numer. Anal. 10, 229-240 (1973; Zbl 0285.65067); SIAM J. numer. Analysis 11, 347-362 (1974; Zbl 0277.65064)] to investigate curved finite elements of class \(C^ 0\).

One has to do with nonconforming methods of finite elements when the finite dimensional space, to which a solution to the approximate problem belongs, is not included in the space to which an exact solution belongs. A nonconforming method is presented as the example of the two-dimensional problem of isotropic homogeneous elasticity by using the so-called Wilson finite element. Here the essential difficulty lies in proving that a solution to the discrete problem exists and in obtaining error estimates.

From the engineering point of view, one has to be able to solve boundary value problems arising in structural mechanics. Structures, like plates and shells, are usually reduced by imposing suitable constraints, to two- dimensional models. The principal results concerning the approximation of solutions for thin linear elastic plates and shells made of isotropic and homogeneous materials are stated without proofs. We observe, however, that plates are described by the operator of order 4; the shell operator is a system of order 2 with respect to the first two unknowns and of order 4 with respect to the third one. Consequently, new difficulties appear in the approximation of solutions to relevant boundary value problems.

The basic ideas of approximation of eigenvalues and eigenvectors are expounded by the example of the eigenvalue problem for a three- dimensional linear elastic body. The respective operator is self-adjoint and compact. Consequently, one can readily extend the formulated (and proved) results concerning the estimation of errors to a larger class of linear, self-adjoint and compact operators. From among numerous physical and mechanical problems involving non self-adjoint operators, the case of neutron economy of nuclear reactors is studied in detail. In particular, having recalled the neutron diffusion equations, the critical calculation theory is presented first with two energy groups (stationary problem). In this specific case approximation using finite elements of order 1 is examined. The critical calculation theory is then extended to a number of neutron energy groups greater than two. The chapter ends up with the study of the eigenvalue problem connected with the evolution problem in the case of two energy groups.

The theory of singular integral equations is presented in a quite comprehensive appendix adduced to the main body of the text. The presentation is rigorous and somewhat abstract and includes the Calderon- Zygmund theorem as well as the basic results related to Marcinkiewicz spaces. The latter are needed for the study of the singular integral operator on the space \(L^ 1\) of integrable functions.

Though being a collaborative work, the book is rather well written, in a manner typical for the famous French school of applied mathematics. It contains a lot of information and treats in mathematically elegant fashion several classes of important physical and mechanical problems. The list of references is by no means exhaustive; to add a few: N. F. Morozov, Mathematical questions in the theory of cracks, Nauka, Moskva (1984; Zbl 0566.73079); B. A. Szabo and I. Babuška, Finite element analysis, J. Wiley \(\&\) Sons, New York (1991); A. Ženišek, Nonlinear elliptic and evolution problems and their finite element approximations, Academic Press, London (1990; Zbl 0731.65090) and the reference cited therein.

Some faults and misprints will easily be perceived by the carefull reader. The reviewer recommends strongly the book to applied mathematicians including those involved in the numerical analysis as well as to researchers working in mathematical physics. The volume should be of particular interest to researchers and graduate students elaborating finite element methods in solid and structural mechanics.

Reviewer: J.J.Telega (Warszawa)

### MSC:

74-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of deformable solids |

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

74Kxx | Thin bodies, structures |

76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |

74B05 | Classical linear elasticity |

45Exx | Singular integral equations |

45C05 | Eigenvalue problems for integral equations |

### Keywords:

Tricomi equations; compressible fluid; aircraft wing section; Frankl equation; linear integral equations; thin aerofoil; interior and exterior Dirichlet and Neumann problems; Laplace equation; Helmholtz equation; Stokes system; variational approach; integral representations; double layer potential; three-dimensional linear elasticity; quadrature schemes; nonconforming methods; structural mechanics; plates; shells; eigenvalues; eigenvectors; self-adjoint operators; nuclear reactors; neutron diffusion equations; singular integral equations; Calderon-Zygmund theorem; Marcinkiewicz spaces
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\textit{R. Dautray} and \textit{J.-L. Lions}, Mathematical analysis and numerical methods for science and technology. (In six volumes). Volume 4: Integral equations and numerical methods. With the collaboration of Michel Artola, Philippe Bénilan, Michel Bernadou, Michel Cessenat, Jean-Claude Nédélec, Jacques Planchard, Bruno Scheurer. Transl. from the French by John C. Amson. Berlin etc.: Springer-Verlag (1990; Zbl 0784.73001)