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**Finite elements. Theory, fast solvers, and applications in solid mechanics. Transl. from the German by Larry L. Schumaker.**
*(English)*
Zbl 0894.65054

Cambridge: Cambridge Univ. Press. xvi, 323 p. (1997).

For reviews of the original ed. (Springer 1992) and the second ed. (Springer 1997) see Zbl 0754.65084 and Zbl 0870.65097.

(Translation of the original German review in Zbl 0886.73001): The present textbook is divided into three parts. Part 1 containing Chapters \(1,2,\) and 3 describes the basic ideas and essential aspects of the finite element method (FEM). Part 2, i.e. Chapters 4 and 5, is concerned with numerical algorithms for solving systems of equations occuring in FEM computations. Part 3, Chapter 6, provides a bridge to elasticity theory and computational mechanics.

This work covers a broad range of finite element methods, from the physical beginnings, to a detailed mathematical treatment including the latest numerical algorithms, up to applications in structural mechanics. This book has advantages when compared to the numerous competing books in FEM, both for students and researchers.

As to the contents of the book in detail: In the introductory Chapter 1, the method of finite differences is presented, and, for a subsequent comparison, the convergence is shown in the case of smooth solutions. The second chapter provides the standard theory of FEM for conforming elements in the framework of variational methods. Here the basic concepts, such as Sobolev spaces, the Lemma of Max-Milgram, trace theorems, and the Bramble-Hilbert Lemma are given. The convergence follows from Céa’s Lemma and the duality argument of Aubin-Nitsche. Chapter 3 is concerned with special aspects of FEM when variational crimes are admitted. In the center of this chapter are mixed methods and the Stokes equation. A detailed treatment of the Babuška-Brezzi condition and Fortin’s ideas is prepared by appropriate tools from functional analysis. The chapter closes with some a posteriori error estimates that are useful for applications to adaptive mesh refinements.

In the next two chapters, the author is in his own element: the fast solution of systems of linear equations that arise in finite element computations. In Chapter 4, he discusses the conjugate gradient method, including preconditioning and Uzawa’s algorithm for saddle point problems. In Chapter 5, he describes the multigrid method with many facettes, including convergence of the \(V\)-cycle and applications to nonlinear boundary value problems.

Chapter 6 is directed to applications in structural mechanics. It concludes with a comparison of the Kirchhoff- and the Mindlin-Reissner plate in the framework of mixed methods which is rare to find in the literature. The variational concepts such as the Hellinger-Reissner principle and the Hu-Washizu one are elucidated. The foundations of elasticity are provided to an extent such that a mathematician gets the background knowledge for treating these finite elements applications.

All in all, this is an excellent book that is appealing to both mathematicians and engineers.

(Translation of the original German review in Zbl 0886.73001): The present textbook is divided into three parts. Part 1 containing Chapters \(1,2,\) and 3 describes the basic ideas and essential aspects of the finite element method (FEM). Part 2, i.e. Chapters 4 and 5, is concerned with numerical algorithms for solving systems of equations occuring in FEM computations. Part 3, Chapter 6, provides a bridge to elasticity theory and computational mechanics.

This work covers a broad range of finite element methods, from the physical beginnings, to a detailed mathematical treatment including the latest numerical algorithms, up to applications in structural mechanics. This book has advantages when compared to the numerous competing books in FEM, both for students and researchers.

As to the contents of the book in detail: In the introductory Chapter 1, the method of finite differences is presented, and, for a subsequent comparison, the convergence is shown in the case of smooth solutions. The second chapter provides the standard theory of FEM for conforming elements in the framework of variational methods. Here the basic concepts, such as Sobolev spaces, the Lemma of Max-Milgram, trace theorems, and the Bramble-Hilbert Lemma are given. The convergence follows from Céa’s Lemma and the duality argument of Aubin-Nitsche. Chapter 3 is concerned with special aspects of FEM when variational crimes are admitted. In the center of this chapter are mixed methods and the Stokes equation. A detailed treatment of the Babuška-Brezzi condition and Fortin’s ideas is prepared by appropriate tools from functional analysis. The chapter closes with some a posteriori error estimates that are useful for applications to adaptive mesh refinements.

In the next two chapters, the author is in his own element: the fast solution of systems of linear equations that arise in finite element computations. In Chapter 4, he discusses the conjugate gradient method, including preconditioning and Uzawa’s algorithm for saddle point problems. In Chapter 5, he describes the multigrid method with many facettes, including convergence of the \(V\)-cycle and applications to nonlinear boundary value problems.

Chapter 6 is directed to applications in structural mechanics. It concludes with a comparison of the Kirchhoff- and the Mindlin-Reissner plate in the framework of mixed methods which is rare to find in the literature. The variational concepts such as the Hellinger-Reissner principle and the Hu-Washizu one are elucidated. The foundations of elasticity are provided to an extent such that a mathematician gets the background knowledge for treating these finite elements applications.

All in all, this is an excellent book that is appealing to both mathematicians and engineers.

Reviewer: N.Herrmann (Hannover)

### MSC:

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

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

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

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

65F10 | Iterative numerical methods for linear systems |

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |

65F35 | Numerical computation of matrix norms, conditioning, scaling |

74B05 | Classical linear elasticity |

74K20 | Plates |