##
**Mixed and hybrid finite element methods.**
*(English)*
Zbl 0788.73002

Springer Series in Computational Mathematics. 15. New York etc.: Springer-Verlag. ix, 350 p. (1991).

This book is dedicated to the mathematical analysis of the mixed and hybrid finite element methods and written by two of the most well known contributors on the subject. It is written in the same spirit as Ciarlet’s book [P. G. Ciarlet, The finite element method for elliptic problems (1978; Zbl 0383.65058)] which was concerned with the classical theory of finite element methods and thus, it becomes its necessary complement for any reader who wants to have a complete mathematical presentation of finite element methods for elliptic problems. And, beyond this mathematical analysis, it is worth to underline that readers mainly interested in the applications, particularly in Dirichlet’s problem, Stokes problem, linear (compressible or incompressible) elasticity, thin or thick plates, will find very stimulating presentations of these problems including valuable comments on the way how to solve the associate discrete problems.

The contents of the book can be analyzed as follows:

Chapter I is dedicated to various formulations of basic continuous problems which are studied in details throughout the book: Laplace equation, linear elasticity, Stokes problem for viscous incompressible flow and deflection of a thin clamped plate. After the classical formulations of these problems, several dual formulations are detailed. Next the authors show how to formulate these problems in order to be solved by using domain decomposition methods, augmented variational methods and transposition methods.

Chapter II sets the general framework in which mixed and hybrid finite element methods can be studied. Since it is concerned with question of existence and uniqueness of solutions, it can be considered as the kernel of the book. Firstly, the simpler case associated to the minimization of a quadratic functional under linear constraints is examined, and then, it is extended to a more general case. Next the approximations of the solutions of these basic problems are considered, inf-sup sufficient conditions of convergence and associated error estimates are given. Various generalizations of these error estimates are considered: weaker coercivity conditions, use of nonconforming methods or/and use of numerical integration techniques, dual error estimates. This chapter is then completed by some considerations on the numerical properties of the discrete problems and by presentation of some methods to solve them (in particular, penalty methods, Uzawa’s algorithm).

Chapter III collects the properties of the spaces \(H^ m(\Omega)\); \(H(\text{div},\Omega)\) and of the associated spaces of traces of functions of these spaces, which are essential to construct finite element approximations. In particular, the construction of finite element approximations of \(H(\text{div};\Omega)\) is thoroughly described for simplicial and rectangular elements and the associated error estimates are derived.

Various examples of the theory developed in chapter II are now developed in chapter IV. Proofs of existence and uniqueness as well as error estimates for different approximation methods are obtained. More details and other applications are given in subsequent chapters. Firstly, the approximation of an elliptic, Laplacian-like equation in \({\mathbb{R}}^ n\) is successively developed by mixed finite element methods, by primal hybrid methods and by dual hybrid methods.

Chapter V starts with a discussion on the numerical techniques which can be used to solve the linear system associated to the mixed finite element method, essentially by introducing interelement Lagrange multipliers. Corresponding computational effort is analyzed. This technique of Lagrange multipliers generally allows to improve the approximation. Then, other error estimates using in particular \(L^ \infty\) and \(H^{-S}\) norms, are derived. Finally some examples of bad intuitive discretizations are done and applications of augmented formulations are presented.

Chapter VI is mainly concerned with the incompressible materials and, particularly, with Stokes problems. Firstly, the Stokes problem is examined in the general theoretical framework of chapter II. Then a survey of finite elements suitable for incompressible materials is given: their classification is based on the techniques required for their analysis. In particular, standard techniques of proof for the inf-sup condition are carefully detailed, including the macroelement technique, an alternative technique for the generalized Taylor-Hood elements, the reduced integration methods and their relation with penalty methods for nearly incompressible elasticity. Finally, the construction and the use of a divergence-free basis are analyzed.

The seventh and last chapter presents some other applications of mixed methods: linear thin plates (Love-Kirchhoff modelization), nearly incompressible linear elasticity and Reissner-Mindlin moderately thick plates.

This book is completely successful in giving a unified presentation of the mathematical theory of mixed and hybrid finite element methods. And, besides these mathematical properties, the reader will find the state-of- the-art of the most actual problems of computational mechanics including a clear and precise description of the way to follow to solve them efficiently. With no doubt this book will be a “bible” for applied mathematicians and, more generally, for all computational mechanicians.

The contents of the book can be analyzed as follows:

Chapter I is dedicated to various formulations of basic continuous problems which are studied in details throughout the book: Laplace equation, linear elasticity, Stokes problem for viscous incompressible flow and deflection of a thin clamped plate. After the classical formulations of these problems, several dual formulations are detailed. Next the authors show how to formulate these problems in order to be solved by using domain decomposition methods, augmented variational methods and transposition methods.

Chapter II sets the general framework in which mixed and hybrid finite element methods can be studied. Since it is concerned with question of existence and uniqueness of solutions, it can be considered as the kernel of the book. Firstly, the simpler case associated to the minimization of a quadratic functional under linear constraints is examined, and then, it is extended to a more general case. Next the approximations of the solutions of these basic problems are considered, inf-sup sufficient conditions of convergence and associated error estimates are given. Various generalizations of these error estimates are considered: weaker coercivity conditions, use of nonconforming methods or/and use of numerical integration techniques, dual error estimates. This chapter is then completed by some considerations on the numerical properties of the discrete problems and by presentation of some methods to solve them (in particular, penalty methods, Uzawa’s algorithm).

Chapter III collects the properties of the spaces \(H^ m(\Omega)\); \(H(\text{div},\Omega)\) and of the associated spaces of traces of functions of these spaces, which are essential to construct finite element approximations. In particular, the construction of finite element approximations of \(H(\text{div};\Omega)\) is thoroughly described for simplicial and rectangular elements and the associated error estimates are derived.

Various examples of the theory developed in chapter II are now developed in chapter IV. Proofs of existence and uniqueness as well as error estimates for different approximation methods are obtained. More details and other applications are given in subsequent chapters. Firstly, the approximation of an elliptic, Laplacian-like equation in \({\mathbb{R}}^ n\) is successively developed by mixed finite element methods, by primal hybrid methods and by dual hybrid methods.

Chapter V starts with a discussion on the numerical techniques which can be used to solve the linear system associated to the mixed finite element method, essentially by introducing interelement Lagrange multipliers. Corresponding computational effort is analyzed. This technique of Lagrange multipliers generally allows to improve the approximation. Then, other error estimates using in particular \(L^ \infty\) and \(H^{-S}\) norms, are derived. Finally some examples of bad intuitive discretizations are done and applications of augmented formulations are presented.

Chapter VI is mainly concerned with the incompressible materials and, particularly, with Stokes problems. Firstly, the Stokes problem is examined in the general theoretical framework of chapter II. Then a survey of finite elements suitable for incompressible materials is given: their classification is based on the techniques required for their analysis. In particular, standard techniques of proof for the inf-sup condition are carefully detailed, including the macroelement technique, an alternative technique for the generalized Taylor-Hood elements, the reduced integration methods and their relation with penalty methods for nearly incompressible elasticity. Finally, the construction and the use of a divergence-free basis are analyzed.

The seventh and last chapter presents some other applications of mixed methods: linear thin plates (Love-Kirchhoff modelization), nearly incompressible linear elasticity and Reissner-Mindlin moderately thick plates.

This book is completely successful in giving a unified presentation of the mathematical theory of mixed and hybrid finite element methods. And, besides these mathematical properties, the reader will find the state-of- the-art of the most actual problems of computational mechanics including a clear and precise description of the way to follow to solve them efficiently. With no doubt this book will be a “bible” for applied mathematicians and, more generally, for all computational mechanicians.

Reviewer: M.Bernadou (Le Chesnay)

### MSC:

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

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

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

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

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