The Mathematical Theory of Finite Element Methods

0.0

A rigorous and thorough mathematical introduction to the foundations of the subject A clear and concise treatment of modern fast solution techniques Third edition substantially updated with new material on Additive Schwarz Preconditioners and Adaptive Meshes Adaptable text that can be used in the context of differing course paths This book develops the basic mathematical theory of the finite element method, the most widely used technique for engineering design and analysis. The third edition contains four new sections: the BDDC domain decomposition preconditioner, convergence analysis of an adaptive algorithm, interior penalty methods and Poincara\'e-Friedrichs inequalities for piecewise W^1_p functions. New exercises have also been added throughout. The initial chapter provides an introducton to the entire subject, developed in the one-dimensional case. Four subsequent chapters develop the basic theory in the multidimensional case, and a fifth chapter presents basic applications of this theory. Subsequent chapters provide an introduction to: multigrid methods and domain decomposition methods mixed methods with applications to elasticity and fluid mechanics iterated penalty and augmented Lagrangian methods variational "crimes" including nonconforming and isoparametric methods, numerical integration and interior penalty methods error estimates in the maximum norm with applications to nonlinear problems error estimators, adaptive meshes and convergence analysis of an adaptive algorithm Banach-space operator-interpolation techniques The book has proved useful to mathematicians as well as engineers and physical scientists. It can be used for a course that provides an introduction to basic functional analysis, approximation theory and numerical analysis, while building upon and applying basic techniques of real variable theory. It can also be used for courses that emphasize physical applications or algorithmic efficiency. Table of Contents 1. Preface(3rdEd) 2. Preface(2ndEd) 3. Preface(1stED) 4. Basic Concepts 5. Sobolev Spaces 6. Variational Formulation of Elliptic Boundary Value Problems 7. The Construction of a Finite Element of Space 8. Polynomial Approximation Theory in Sobolev Spaces 9. n-Dimensional Variational Problems 10. Finite Element Multigrid Methods 11. Additive Schwarz Preconditioners 12. Max-norm Estimates 13. Adaptive Meshes 14. Variational Crimes 15. Applications to Planar Elasticity 16. Mixed Methods 17. Iterative Techniques for Mixed Methods 18. Applications of Operator-Interpolation Theory