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Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scienti?c disciplines and a resurgence of interest in the modern as well as the cl- sical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and to encourage the teaching of new courses. TAMwillpublishtextbookssuitableforuseinadvancedundergraduate and beginning graduate courses, and will complement the Applied Mat- matical Sciences (AMS) series, which will focus on advanced textbooks and research-level monographs. Pasadena, California J.E. Marsden Providence, Rhode Island L. Sirovich College Park, Maryland S.S. Antman Preface to the Third Edition This edition contains four new sections on the following topics: the BDDC domain decomposition preconditioner (Section 7.8), a convergent ad- tive algorithm (Section 9.5), interior penalty methods (Section 10.5) and 1 Poincar´ e-Friedrichs inequalities for piecewise W functions (Section 10.6).



0. Basic Concepts

The finite element method provides a formalism for generating discrete (finite) algorithms for approximating the solutions of differential equations. It should be thought of as a black box into which one puts the differential equation (boundary value problem) and out of which pops an algorithm for approximating the corresponding solutions. Such a task could conceivably be done automatically by a computer, but it necessitates an amount of mathematical skill that today still requires human involvement. The purpose of this book is to help people become adept at working the magic of this black box. The book does not focus on how to turn the resulting algorithms into computer codes, but this topic is being pursued by several groups. In particular, the FEniCS project (on the web at utilizes the mathematical structure of the finite element method to automate the generation of finite element codes.

1. Sobolev Spaces

This chapter is devoted to developing function spaces that are used in the variational formulation of differential equations. We begin with a review of Lebesgue integration theory, upon which our notion of “variational” or “weak” derivative rests. Functions with such “generalized” derivatives make up the spaces commonly referred to as Sobolev spaces. We develop only a small fraction of the known theory for these spaces—just enough to establish a foundation for the finite element method.

2. Variational Formulation of Elliptic Boundary Value Problems

This chapter is devoted to the functional analysis tools required for developing the variational formulation of differential equations. It begins with an introduction to Hilbert spaces, including only material that is essential to later developments. The goal of the chapter is to provide a framework in which existence and uniqueness of solutions to variational problems may be established.

3. The Construction of a Finite Element Space

4. Polynomial Approximation Theory in Sobolev Spaces

We will now develop the approximation theory appropriate for the finite elements developed in Chapter 3. We take a constructive approach, defining an averaged version of the Taylor polynomial familiar from calculus. The key estimates are provided by some simple lemmas from the theory of Riesz potentials, which we derive. As a corollary, we provide a proof of Sobolev's inequality, much in the spirit given originally by Sobolev.

5. n-Dimensional Variational Problems

We now give several examples of higher-dimensional variational problems that use the theory developed in previous chapters. The basic notation is provided by the Sobolev spaces developed in Chapter 1. We combine the existence theory of Chapter 2 together with the approximation theory of Chapters 3 and 4 to provide a complete theory for the discretization process. Several examples will be fully developed in the text, and several others are found in the exercises. Throughout this chapter, we assume that the domain Ω is bounded.

6. Finite Element Multigrid Methods

The multigrid method provides an optimal order algorithm for solving elliptic boundary value problems. The error bounds of the approximate solution obtained from the full multigrid algorithm are comparable to the theoretical bounds of the error in the finite element method, while the amount of computational work involved is proportional only to the number of unknowns in the discretized equations.

7. Additive Schwarz Preconditioners

The symmetric positive definite system arising from a finite element discretization of an elliptic boundary value problem can be solved efficiently using the preconditioned conjugate gradient method (cf. (Saad 1996)). In this chapter we discuss the class of additive Schwarz preconditioners, which has built-in parallelism and is particularly suitable for implementation on parallel computers. Many well-known preconditioners are included in this class, for example the hierarchical basis and BPX multilevel preconditioners, the two-level additive Schwarz overlapping domain decomposition preconditioner, the BPS, Neumann-Neumann, and BDDC nonoverlapping domain decomposition preconditioners.

8. Max—norm Estimates

The finite element approximation is essentially defined by a mean-square projection of the gradient. Thus, it is natural that error estimates for the gradient of the error directly follow in the L2 norm. It is interesting to ask whether such a gradient-projection would also be of optimal order in some other norm, for example L∞. We prove here that this is the case. Although of interest in their own right, such estimates are also crucial in establishing the viability of approximations of nonlinear problems (Douglas & Dupont 1975) as we indicate in Sect. 8.7. Throughout this chapter, we assume that the domain Ω ⊂ IR d is bounded and polyhedral.

9. Adaptive Meshes

In Section 0.8, we demonstrated the possibility of dramatic improvements in approximation power resulting from adaptive meshes. In current computer simulations, meshes are often adapted to the solution either using a priori information regarding the problem being solved or a posteriori after an initial attempt at solution (Babuška et al. 1983 & 1986). The resulting meshes tend to be strongly graded in many important cases, no longer being simply modeled as quasi-uniform. Here we present some basic estimates that show that such meshes can be effective in approximating difficult problems. For further references, see (Eriksson, Estep, Hansbo and Johnson 1995), (Verfürth 1996), (Ainsworth and Oden 2000), (Becker and Rannacher 2001), (Babuška and Strouboulis 2001), (Dörfler and Nochetto 2002), (Bangerth and Rannacher 2003), (Neittaanmäki and Repin 2004), (Han 2005), (Carstensen 2005) and (Carstensen, Hu and Orlando 2007).

10. Variational Crimes

11. Applications to Planar Elasticity

In most physical applications, quantities of interest are governed by a system of partial differential equations, not just a single equation. So far, we have only considered single equations (for a scalar quantity), although much of the theory relates directly to systems. We consider one such system coming from solid mechanics in this chapter.

12. Mixed Methods

The name “mixed method” is applied to a variety of finite element methods which have more than one approximation space. Typically one or more of the spaces play the role of Lagrange multipliers which enforce constraints. The name and many of the original concepts for such methods originated in solid mechanics where it was desirable to have a more accurate approximation of certain derivatives of the displacement. However, for the Stokes equations which govern viscous fluid flow, the natural Galerkin approximation is a mixed method.

13. Iterative Techniques for Mixed Methods

Equations of the form (12.3.1) or (12.5.14) are indefinite and require special care to solve. We will now consider one class of algorithms which involve a penalty method to enforce the second equation in (12.3.1) or (12.5.14). These algorithms transform the linear algebra to positive-definite problems in many cases. Moreover, the number of unknowns in the algebraic system can also be significantly reduced.

14. Applications of Operator-Interpolation Theory

Interpolation spaces are useful technical tools. They allow one to bridge between known results, yielding new results that could not be obtained directly. They also provide a concept of fractional-order derivatives, extending the definition of the Sobolev spaces used so far. Such extensions allow one to measure more precisely, for example, the regularity of solutions to elliptic boundary value problems.


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