The Finite Element Method for Elliptic Problems

P.G. Ciarlet

• Publisher: North Holland
• Year: 1978
• ISBN:
  9780080875255
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

The objective of this book is to analyze within reasonable limits (it is not a treatise) the basic mathematical aspects of the finite element method. The book should also serve as an introduction to current research on this subject.

On the one hand, it is also intended to be a working textbook for advanced courses in Numerical Analysis, as typically taught in graduate courses in American and French universities. For example, it is the author's experience that a one-semester course (on a three-hour per week basis) can be taught from Chapters 1, 2 and 3 (with the exception of Section 3.3), while another one-semester course can be taught from Chapters 4 and 6.

On the other hand, it is hoped that this book will prove to be useful for researchers interested in advanced aspects of the numerical analysis of the finite element method. In this respect, Section 3.3, Chapters 5, 7 and 8, and the sections on "Additional Bibliography and Comments" should provide many suggestions for conducting seminars.

Chapter 1

Elliptic Boundary Value Problems

Introduction

Many problems in elasticity are mathematically represented by the following minimization problem: The unknown u, which is the displacement of a mechanical system, satisfies

where the set U of admissible displacements is a closed convex subset of a Hilbert space V, and the energy J of the system takes the form

where a(·, ·) is a symmetric bilinear form and f is a linear form, both defined and continuous over the space V. In Section 1.1, we first prove a general existence result (Theorem 1.1.1), the main assumptions being the completeness of the space V and the V-ellipticity of the bilinear form. We also describe other formulations of the same problem (Theorem 1.1.2), known as its variational formulations, which, in the absence of the assumption of symmetry for the bilinear form, make up variational problems on their own. For such problems, we give an existence theorem when U = V (Theorem 1.1.3), which is the well-known Lax–Milgram lemma.

All these problems are called abstract problems inasmuch as they represent an “abstract” formulation which is common to many examples, such as those which are examined in Section 1.2.

From the analysis made in Section 1.1, a candidate for the space V must have the following properties: It must be complete on the one hand, and it must be such that the expression J(v) is well-defined for all functions v ∈ V on the other hand (V is a “space of finite energy”). The Sobolev spaces fulfill these requirements. After briefly mentioning some of their properties (other properties will be introduced in later sections, as needed), we examine in Section 1.2 specific examples of the abstract problems of Section 1.1, such as the membrane problem, the clamped plate problem, and the system of equations of linear elasticity, which is by far the most significant example. Indeed, even though throughout this book we will often find it convenient to work with the simpler looking problems described at the beginning of Section 1.2, it must not be forgotten that these are essentially convenient model problems for the system of linear elasticity.

Using various Green’s formulas in Sobolev spaces, w...

Table of contents

• Cover image
• Title page
• Table of Contents
• Studies in Mathematics and its Applications
• Copyright page
• Dedication
• Preface
• General Plan and Interdependence Table
• Chapter 1: Elliptic Boundary Value Problems
• Chapter 2: Introduction to the Finite Element Method
• Chapter 3: Conforming Finite Element Methods for Second-Order Problems
• Chapter 4: Other Finite Element Methods For Second-Order Problems
• Chapter 5: Application of the finite Element Method to Some Nonlinear Problems
• Chapter 6: Finite Element Methods for The Plate Problem
• Chapter 7: A Mixed Finite Element Method
• Chapter 8: Finite Element Methods for Shells
• Epilogue: Some “real-life” finite element model examples
• Bibliography
• Glossary of Symbols
• Index