Theory and Practice of Finite Elements

by Ern, Alexandre; Guermond, Jean-Luc

• ISBN13: 9780387205748
• ISBN10: 0387205748
• Format: Hardcover
• Copyright: 2004-04-29
• Publisher: Springer Nature

Summary

"This book presents the mathematical theory of finite elements, starting from basic results on approximation theory and finite element interpolation and building up to more recent research topics, such as Discontinuous Galerkin, subgrid viscosity stabilization, and a posteriori error estimation." "The book will be useful to researchers and graduate students in mathematics, computer science, and engineering."--BOOK JACKET.

Table of Contents

Part I Theoretical Foundations
1 Finite Element Interpolation	3	(78)
1.1 One-Dimensional Interpolation	3	(16)
1.2 Finite Elements: Definitions and Examples	19	(12)
1.3 Meshes: Basic Concepts	31	(7)
1.4 Approximation Spaces and Interpolation Operators	38	(20)
1.5 Interpolation of Smooth Functions	58	(10)
1.6 Interpolation of Non-Smooth Functions	68	(7)
1.7 Inverse Inequalities	75	(2)
1.8 Exercises	77	(4)
2 Approximation in Banach Spaces by Galerkin Methods	81	(30)
2.1 The Banach-Necas-Babuka (BNB) Theorem	81	(7)
2.2 Galerkin Methods	88	(5)
2.3 Error Analysis	93	(6)
2.4 Saddle-Point Problems	99	(7)
2.5 Exercises	106	(5)
Part II Approximation of PDEs
3 Coercive Problems	111	(64)
3.1 Scalar Elliptic PDEs: Theory	111	(9)
3.2 Scalar Elliptic PDEs: Approximation	120	(21)
3.3 Spectral Problems	141	(9)
3.4 Continuum Mechanics	150	(14)
3.5 Coercivity Loss	164	(6)
3.6 Exercises	170	(5)
4 Mixed Problems	175	(44)
4.1 Mathematical Study of the Stokes Problem	175	(8)
4.2 Mixed Finite Element Approximation	183	(18)
4.3 Galerkin/Least-Squares (GaLS) Approximation	201	(7)
4.4 Linear Algebra	208	(6)
4.5 Exercises	214	(5)
5 First-Order PDEs	219	(60)
5.1 Standard Galerkin Approximation in One Dimension	219	(6)
5.2 First-Order PDEs in L²	225	(9)
5.3 Least-Squares Formulation and Variants	234	(8)
5.4 Galerkin/Least-Squares (GaLS) Approximation	242	(9)
5.5 Subgrid Viscosity Approximation	251	(14)
5.6 Discontinuous Galerkin (DG) Approximation	265	(6)
5.7 Non-Standard Galerkin Approximation	271	(4)
5.8 Exercises	275	(4)
6 Time-Dependent Problems	279	(58)
6.1 Parabolic Problems	279	(21)
6.2 Time-Dependent Mixed Problems	300	(12)
6.3 Evolution Equations Without Coercivity	312	(20)
6.4 Exercises	332	(5)
Part III Implementation
7 Data Structuring and Mesh Generation	337	(20)
7.1 Data Structuring	337	(3)
7.2 Mesh Generators	340	(6)
7.3 Example: Delaunay Triangulations	346	(8)
7.4 Exercises	354	(3)
8 Quadratures, Assembling, and Storage	357	(26)
8.1 Quadratures: Theory and Implementation	357	(11)
8.2 Assembling	368	(4)
8.3 Storage of Sparse Matrices	372	(4)
8.4 Non-Homogeneous Dirichlet Boundary Conditions	376	(4)
8.5 Exercises	380	(3)
9 Linear Algebra	383	(38)
9.1 Conditioning	383	(10)
9.2 Reordering	393	(8)
9.3 Iterative Solution Methods	401	(12)
9.4 Introduction to Parallel Implementation	413	(4)
9.5 Exercises	417	(4)
10 A Posteriori Error Estimates and Adaptive Meshes	421	(42)
10.1 Residual-Based Error Estimates	421	(14)
10.2 Hierarchical Error Estimates	435	(10)
10.3 Duality-Based Error Estimates	445	(8)
10.4 Adaptive Mesh Generation	453	(4)
10.5 Exercises	457	(6)
Part IV Appendices
A Banach and Hilbert Spaces	463	(14)
A.1 Basic Definitions and Results	463	(5)
A.2 Bijective Banach Operators	468	(9)
B Functional Analysis	477	(16)
B.1 Lebesgue and Lipschitz Spaces	477	(3)
B.2 Distributions	480	(2)
B.3 Sobolev Spaces	482	(11)
Nomenclature	493	(4)
References	497	(16)
Author Index	513	(4)
Subject Index	517