##
**Finite element solution of boundary value problems. Theory and computation.**
*(English)*
Zbl 0537.65072

Computer Science and Applied Mathematics. Orlando etc.: Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers). XVIII, 432 p. $ 59.00 (1984).

This book provides an introduction to both theoretical and computational aspects of the finite element method for linear elliptic boundary value problems (BVP). It is written at a graduate level in the areas of numerical analysis, mathematics, and computer science. The book contains 7 chapters and an appendix on Chebyshev polynomials.

Chapter I deals with minimization methods for quadratic functionals in \(R^ N\) (steepest descent, conjugate gradient, and preconditioning techniques). Chapter II is concerned with the classical variational formulation of BVP. The Euler-Lagrange equation for one-dimensional problems is presented first and the distinction between essential and natural boundary conditions is discussed. Problems in 2 and 3 space dimensions are analysed and examples from physics are given (the field equation is especially involved in this book). An introduction to the modern treatment of BVP by use of functional analysis in Hilbert spaces can be found in Chapter III (Sobolev spaces, the Lax-Milgram theorem with examples, distributions). Chapter IV deals with the Ritz method (error estimates) and with the more general Galerkin method (error estimates and applications to noncoercive problems). The subject of Chapter V is the finite element method: fundamental properties of finite element meshes and basis functions, details on the construction of the Ritz-Galerkin matrix, isoparametric basis functions, error analysis using the theory of interpolation in Sobolev spaces, analysis of the spectral condition number of the Ritz-Galerkin matrix. Chapter VI is devoted to direct (noniterative) methods for solving an algebraic system of equations (Gaussian elimination and the Cholesky method). Much of this chapter is concerned with various strategies for ordering the nodes and their corresponding computational features. The influence of data errors and of rounding errors on the accuracy of the computed solution is also investigated. Chapter VII deals with iterative methods for the finite element system of equations: the symmetric successive overrelaxation method and incomplete factorization methods. A comparison between direct and iterative methods and an introduction to multgrid methods are given.

Every chapter is followed by references and by many interesting exercises going from mathematical results to written computer codes. The material in this book is detailed and some of it is new in book form (part of chapters I, VI, VII). Concluding, a nice book from both points of view: theoretical and computational.

Chapter I deals with minimization methods for quadratic functionals in \(R^ N\) (steepest descent, conjugate gradient, and preconditioning techniques). Chapter II is concerned with the classical variational formulation of BVP. The Euler-Lagrange equation for one-dimensional problems is presented first and the distinction between essential and natural boundary conditions is discussed. Problems in 2 and 3 space dimensions are analysed and examples from physics are given (the field equation is especially involved in this book). An introduction to the modern treatment of BVP by use of functional analysis in Hilbert spaces can be found in Chapter III (Sobolev spaces, the Lax-Milgram theorem with examples, distributions). Chapter IV deals with the Ritz method (error estimates) and with the more general Galerkin method (error estimates and applications to noncoercive problems). The subject of Chapter V is the finite element method: fundamental properties of finite element meshes and basis functions, details on the construction of the Ritz-Galerkin matrix, isoparametric basis functions, error analysis using the theory of interpolation in Sobolev spaces, analysis of the spectral condition number of the Ritz-Galerkin matrix. Chapter VI is devoted to direct (noniterative) methods for solving an algebraic system of equations (Gaussian elimination and the Cholesky method). Much of this chapter is concerned with various strategies for ordering the nodes and their corresponding computational features. The influence of data errors and of rounding errors on the accuracy of the computed solution is also investigated. Chapter VII deals with iterative methods for the finite element system of equations: the symmetric successive overrelaxation method and incomplete factorization methods. A comparison between direct and iterative methods and an introduction to multgrid methods are given.

Every chapter is followed by references and by many interesting exercises going from mathematical results to written computer codes. The material in this book is detailed and some of it is new in book form (part of chapters I, VI, VII). Concluding, a nice book from both points of view: theoretical and computational.

Reviewer: V.Arnautu

### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

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

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

65F05 | Direct numerical methods for linear systems and matrix inversion |

65F10 | Iterative numerical methods for linear systems |

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

35-04 | Software, source code, etc. for problems pertaining to partial differential equations |