The theory of finite elements and their applications is such a lively area that a third edition has become necessary to cover new material that is of interest for actual applications. At the same time we have taken the opportunity to correct some misprints.

The greatest changes are found in Chapter III. Saddle point problems and mixed methods are used now not only for variational problems with given constraints, but there is also an increasing interest in nonstandard saddle point methods. Their flexibility enables the construction of finite elements with special properties, e.g. they can soften specific terms of the energy functional in order to eliminate locking phenomena. The treatment of the Poisson equation in the setting of saddle point formulations can be regarded as a template for other examples and some of these are covered in the present edition.

Another nonstandard application is the construction of a new type of a posteriori error estimate for conforming elements. This has the advantage that there is no generic constant in the main term. Moreover, in this framework it is possible to shed light on other relations between conforming elements and mixed methods.

In Chapter VI the treatment of locking has been reworked. Such phenomena are well known to engineers, but the mathematical proof of locking is often more cumbersome than the remedy. In most cases we focus therefore on the use of appropriate finite elements and describe the negative results more briefly within the context of the general theory of locking.

Finite element methods are the most widely used tools for computing the deformations and stresses of elastic and inelastic bodies subject to loads. These types of problems involve systems of differential equations with the following special feature: the equations are invariant under translations and orthogonal transformations since the elastic energy of a body does not change under so-called rigid body motions.

Practical problems in structural mechanics often involve small parameters which can appear in both obvious and more subtle ways. For example, for beams, membranes, plates, and shells, the thickness is very small in comparison with the other dimensions. On the other hand, for a cantilever beam, the part of the boundary on which Dirichlet boundary conditions are prescribed is very small. Finally, many materials allow only very small changes in density. These various cases require different variational formulations of the finite element computations. Using an incorrect formulation leads to so-called locking. Often, mixed formulations provide a suitable framework for both the computation and a rigorous mathematical analysis.

Most of the characteristic properties appear already in the so-called linear theory, i.e., for small deformations where no genuine nonlinear phenomenon occurs. However, strictly speaking, there is no complete linear elasticity theory, since the above-mentioned invariance under rigid body motions cannot be completely modeled in a linear theory. For this reason, we don't restrict ourselves to the linear theory until later.

The mathematical treatment of the finite element method is based on the variational formulation of elliptic differential equations. Solutions of the most important differential equations can be characterized by minimal properties, and the corresponding variational problems have solutions in certain function spaces called Sobolev spaces. The numerical treatment involves minimization in appropriate finite-dimensional linear subspaces. A suitable choice for these subspaces, both from a practical and from a theoretical point of view, are the so-called finite element spaces.

For linear differential equations, it suffices to work with Hilbert space methods. In this framework, we immediately get the existence of so-called weak solutions. Regularity results, to the extent they are needed for the finite element theory, will be presented without proof.

This chapter contains a theory of the simple methods which suffice for the treatment of scalar elliptic differential equations of second order. The aim of this chapter are the error estimates in §7 for the finite element solutions. They refer to the L2-norm and to the Sobolev norm ∥ · ∥1. Some of the more general results presented here will also be used later in our discussion in Chapter III of other elliptic problems whose treatment requires additional techniques.

The paper of Courant [1943] is generally considered to be the first mathematical contribution to a finite element theory, although a paper of Schellbach [1851] had appeared already a century earlier. If we don't take too narrow a view, finite elements also appear in some work of Euler. The method first became popular at the end of the sixties, when engineers independently developed and named the method.

This book is based on lectures regularly presented to students in the third and fourth year at the Ruhr-University, Bochum. It was also used by the translater, Larry Schumaker, in a graduate course at Vanderbilt University in Nashville. I would like to thank him for agreeing to undertake the translation, and for the close cooperation in carrying it out. My thanks are also due to Larry and his students for raising a number of questions which led to improvements in the material itself.

Chapters I and II and selected sections of Chapters III and V provide material for a typical course. I have especially emphasized the differences with the numerical treatment of ordinary differential equations (for more details, see the preface to the German edition).

One may ask why I was not content with presenting only simple finite elements based on complete polynomials. My motivation for doing more was provided by problems in fluid mechanics and solid mechanics, which are treated to some extent in Chapter III and VI. I am not aware of other textbooks for mathematicians which give a mathematical treatment of finite elements in solid mechanics in this generality.

The English translation contains some additions as compared to the German edition from 1992. For example, I have added the theory for basic a posteriori error estimates since a posteriori estimates are often used in connection with local mesh refinements. This required a more general interpolation process which also applies to non-uniform grids. In addition, I have also included an analysis of locking phenomena in solid mechanics.

The method of finite elements is one of the main tools for the numerical treatment of elliptic and parabolic partial differential equations. Because it is based on the variational formulation of the differential equation, it is much more flexible than finite difference methods and finite volume methods, and can thus be applied to more complicated problems. For a long time, the development of finite elements was carried out in parallel by both mathematicians and engineers, without either group acknowledging the other. By the end of the 60's and the beginning of the 70's, the material became sufficiently standardized to allow its presentation to students. This book is the result of a series of such lectures.

In contrast to the situation for ordinary differential equations, for elliptic partial differential equations, frequently no classical solution exists, and we often have to work with a so-called weak solution. This has consequences for both the theory and the numerical treatment. While it is true that classical solutions do exist under approriate regularity hypotheses, for numerical calculations we usually cannot set up our analisis in a framework in which the existence of classical solutions is guaranteed.

One way to get a suitable framework for solving elliptic boundary-value problems using finite elements is to pose them as variational problems. It is our goal in Chapter II to present the simplest possible introduction to this approach. In Sections 1 – 3 we discuss the existence of weak solutions in Sobolev spaces, and explain how the boundary conditions are incorporated into the variational calculation.

In this chapter we discuss numerical methods based on structured grids for direct numerical simulations of multiphase flows involving solid particles. We will focus on numerical approaches that are designed to solve the governing equations in the fluid and the interaction between the phases at their interfaces.

In methods employing structured grids, there are two distinct possibilities for handling the geometric complexities imposed by the phase boundaries. The first approach is to precisely define the phase boundaries and use a body-fitted grid in one or both phases, as necessary. The curvilinear grid that conforms to the phase boundaries greatly simplifies the specification of interaction processes that occur across the interface. Furthermore, numerical methods on curvilinear grids are well developed and a desired level of accuracy can be maintained both within the different phases and along their interfaces. The main difficulty with this approach is grid generation; for example, for the case of flow in a pipe with several hundred suspended particles, obtaining an appropriate structured body-fitted grid around all the particles is a nontrival task. There are several structured grid generation packages that are readily available. Nevertheless, the computational cost of grid generation increases rapidly as geometric complexity increases. This can be severely restrictive, particularly in cases where the phase boundaries are in time-dependent motion. The cost of grid generation can then overwhelm the cost of computing the flow.

The alternative to a body-fitted grid that is increasingly becoming popular are Cartesian grid methods. Here, irrespective of the complex nature of the internal and external boundaries between the phases, only a regular Cartesian mesh is employed and the governing equations are solved on this simple mesh.

In Chapter 4 we described finite difference methods for the direct numerical simulations of fluid–solid systems in which the local flow field around moving particles is resolved numerically without modeling. Here we describe methods of the finite element type for the same purpose.

The first method is termed the ALE (arbitrary Lagrangian-Eulerian) particle mover. The ALE particle mover uses a technique based on a combined formulation of the fluid and particle momentum equations, together with an arbitrary Lagrangian–Eulerian moving unstructured finite element mesh technique to deal with the movement of the particles. It was first developed by Hu et al. (1992, 1996, 2001). The method has been used to solve particle motions in both Newtonian and viscoelastic fluids in two- and three-dimensional flow geometries. It also handles particles of different sizes, shapes, and materials.

The second method is based on a stabilized space-time finite element technique to solve problems involving moving boundaries and interfaces. In this method, the temporal coordinate is also discretized using finite elements. The deformation of the spatial domain with time is reflected simply in the deformation of the mesh in the temporal coordinate. Using this technique, Johnson and Tezduyar (1999) were able to simulate the sedimentation of 1000 spheres in a Newtonian fluid at a Reynolds number of 10.

The third numerical method is termed the DLM (distributed Lagrange multiplier) particle mover. The basic idea of the DLM particle mover is to extend the problem from a time-dependent geometrically complex domain to a stationary fictitious domain, which is larger but simpler.

In this chapter and the following three, we discuss numerical methods that have been used for direct numerical simulations of multiphase flow. Although direct numerical simulations, or DNS, mean slightly different things to different people, we shall use the term to refer to computations of complex unsteady flows where all continuum length and time scales are fully resolved. Thus, there are no modeling issues beyond the continuum hypothesis. The flow within each phase and the interactions between different phases at the interface between them are found by solving the governing conservation equations, using grids that are finer and time steps that are shorter than any physical length and time scale in the problem.

The detailed flow field produced by direct numerical simulations allows us to explore the mechanisms governing multiphase flows and to extract information not available in any other way. For a single bubble, drop, or particle, we can obtain integrated quantities such as lift and drag and explore how they are affected by free stream turbulence, the presence of walls, and the unsteadiness of the flow. In these situations it is possible to take advantage of the relatively simple geometry to obtain extremely accurate solutions over a wide range of operating conditions. The interactions of a few bubbles, drops, or particles is a more challenging computation, but can be carried out using relatively modest computational resources. Such simulations yield information about, for example, how bubbles collide or whether a pair of buoyant particles, rising freely through a quiescent liquid, orient themselves in a preferred way. Computations of one particle can be used to obtain information pertinent to modeling of dilute multiphase flows, and studies of a few particles allow us to assess the importance of rare collisions.

The previous chapter, with its direct simulation of the fluid flow and a modeling approach to the particle phase, may be seen as a transition between the methods for a fully resolved simulation described in the first part of this book and those for a coarse-grained description based on the averaging approach described in Chapter 8. We now turn to the latter, which in practice are the only methods able to deal with the complex flows encountered in most situations of practical interest such as fluidized beds, pipelines, energy generation, sediment transport, and others. This chapter and the next one are devoted to numerical methods for so-called two-fluid models in which the phases are treated as interpenetrating continua describing, e.g. a liquid and a gas, or a fluid and a suspended solid phase. These models can be extended to deal with more than two continua and, then, the denomination multifluid models might be more appropriate. For example, the commercial code OLGA (Bendiksen et al., 1991), widely used in the oil industry, recognizes three phases, all treated as interpenetrating continua: a continuous liquid, a gas, and a disperse liquid phase present as drops suspended in the gas phase. The more recent PeTra (Petroleum Transport, Larsen et al., 1997) also describes three phases: gas, oil, and water. Recent approaches to the description of complex boiling flows recognize four interpenetrating phases: a liquid phase present both as a continuum and as a dispersion of droplets, and a gas/vapor phase also present as a continuum and a dispersion of bubbles.

Brief history of the lattice Boltzmann method

In recent years, the lattice Boltzmann method (LBM) (Chen and Doolen, 1998; Succi, 2001) has become a popular numerical scheme for simulating fluid flows and modeling physics in fluids. The lattice Boltzmann method is based on a simplified mesoscopic equation, i.e. the discrete Boltzmann equation. By starting from mesoscopic modeling, instead of doing numerical discretizations of macroscopic continuum equations, the LBM can easily incorporate underlying physics into numerical solutions. By developing a simplified version of the kinetic equation, one avoids solving complicated kinetic equations such as the full Boltzmann equation. The kinetic feature of the LBM provides additional advantages of mesoscopic modeling, such as easy implementation of boundary conditions and fully parallel algorithms. This is arguably the major reason why the LBM has been quite successful in simulating multiphase flows. Furthermore, because of the availability of very fast and massively parallel computers, there is a current trend to use codes that can exploit the intrinsic features of parallelism. The LBM fulfills these requirements in a straightforward manner.

The lattice Boltzmann method was first proposed by McNamara and Zanetti in 1988 in an effort to reduce the statistical noise in lattice gas automaton (LGA) simulations (Rothman and Zaleski, 2004). The difference is that LBM uses real numbers to count particle population, while LGA only allows integers. The result has been phenomenal: the lattice Boltzmann model substantially reduced the statistic noise observed in LGA simulations.

This is a graduate-level textbook intended to serve as an introduction to computational approaches which have proven useful for problems arising in the broad area of multiphase flow. Each chapter contains references to the current literature and to recent developments on each specific topic, but the primary purpose of this work is to provide a solid basis on which to build both applications and research. For this reason, while the reader is expected to have had some exposure to graduate-level fluid mechanics and numerical methods, no extensive knowledge of these subjects is assumed. The treatment of each topic starts at a relatively elementary level and is developed so as to enable the reader to understand the current literature.

A large number of topics fall under the generic label of “computational multiphase flow,” ranging from fully resolved simulations based on first principles to approaches employing some sort of coarse-graining and averaged equations. The book is ideally divided into two parts reflecting this distinction. The first part (Chapters 2–5) deals with methods for the solution of the Navier-Stokes equations by finite difference and finite element methods, while the second part (Chapters 9–11) deals with various reduced descriptions, from point-particle models to two-fluid formulations and averaged equations. The two parts are separated by three more specialized chapters on the lattice Boltzmann method (Chapter 6), the boundary integral method for Stokes flow (Chapter 7), and on averaging and the formulation of averaged equation (Chapter 8).

This is a multi-author volume, but we have made an effort to unify the notation and to include cross-referencing among the different chapters.