Mixed and hybrid finite element methods are extensively used in computational engineering and scientific computing. Loosely speaking, a finite element method is called “mixed” when independent approximations are used for both the dependent variable and its derivatives. For example, in elasticity problems, displacements as well as stresses are usually computed and, of course, the stresses require the derivatives of the displacement field. On the other hand, hybrid methods arise when independent approximations are used for the dependent variable on the interior of an element and for its traces on the boundary. In a domain decomposition problem this may allow, for instance, for a relaxation of the continuity requirements of the approximations across the domain interfaces.
A finite element method always requires some variational principle on a suitable function space. For example, the classical setting involves the minimization of a functional (such as an energy functional) on some Hilbert space of admissible functions. Generally, for a given physical problem, many different variational formulations may be constructed, which lead to different settings for the finite element approximation. The methods analyzed in this book have in common that they are developed for variational principles that express an equilibrium or saddle-point condition rather than a minimization principle.
In recent years, the mathematical properties of mixed and hybrid finite element methods have been thoroughly investigated, and a general theory is beginning to emerge. This book is intended to give a unified presentation of the general framework of the theoretical developments and of the central results, coupled with an introduction to some of the major applications of the methods.
Chapter 1 introduces a number of model problems for use throughout the book. Included are several examples of duality methods, domain decomposition methods, augmented variational formulations, and transposition methods. Chapter 2 is, in essence, the heart of the book, where the general, abstract framework for the study of the various methods is developed. From a simple case involving the minimization of a linearly constrained quadratic functional, the presentation advances to more general saddle point problems and then to the approximation of their solution and the numerical properties of the discretized problems.
Chapter 3 discusses the properties of the function spaces needed for the application of the abstract theory. In particular, properties of the Sobolev spaces Hm(&OHgr;) and of H(div ;&OHgr;) are addressed together with their finite element approximations. Then chapter 4 presents various examples, including nonstandard methods for Dirichlet’s problem, the Stokes problem, elasticity problems, a mixed fourth-order problem, and dual hybrid methods for plate bending problems. This discussion continues in chapter 5 with some further results on the application of the mixed finite element method to linear elliptic problems, including certain numerical aspects, related error analysis and error estimations, and an application to an equation from semiconductor theory.
Chapter 6 applies the general theory of mixed methods to the approximation of problems involving incompressible materials and flows. After introducing the problems, the authors give examples of elements and proof techniques and then discuss the solution by penalty methods. Finally, chapter 7 considers a few further applications, including mixed methods for linear thin plates and linear elasticity problems and an introduction to the theory of moderately thick plates.
The authors have been in the forefront of the development of the theory of nonstandard finite element methods for years, and the book reflects their wide knowledge of these methods and their applications. The book is not concerned with practical implementation details. Instead, as the authors formulate it, the goal “was to provide an analysis of the methods in order to understand their properties as thoroughly as possible.” They have certainly succeeded in this and have written a work that provides a definitive state-of-the-art introduction to this important area of finite element analysis.
