Numerical Approximation of Partial Differential Equations

This book aims to provide a thorough introduction to the construction, analysis, and implementation of finite element methods for model problems arising in continuum mechanics. Finite element methods for approximating partial differential equations have reached a high degree of maturity and are an indispensable tool in science and technology. The first part of the book discusses elementary properties of linear partial differential equations along with their basic numerical approximation, the functional-analytical framework for rigorously establishing existence of solutions, and the construction and analysis of basic finite element methods. The second part is devoted to the optimal adaptive approximation of singularities and the fast iterative solution of linear systems of equations arising from finite element discretization. In the third part, the mathematical framework for analyzing and discretizing saddle-point problems is formulated, corresponding finite element methods are analyzed, and particular applications including incompressible elasticity, thin elastic objects, electromagnetism, and fluid mechanics are addressed. The book includes theoretical problems and practical projects for all chapters and an introduction to the implementation of finite element methods.

Covers Matlab implementations to illustrate the devised methods Provides a thorough introduction to the construction, analysis, and implementation of finite element methods Includes theoretical and physical backgrounds of mathematical models

Autorentext

Sören Bartels is Professor of Applied Mathematics at the University of Freiburg, Germany. His primary research interest is in the development and analysis of approximation schemes for nonlinear partial differential equations with applications in the simulation of modern materials. Professor Bartels has published the Springer textbook Numerik 3x9 and the monograph Numerical methods for nonlinear partial differential equations in the Springer Series in Computational Mathematics.

Inhalt

Preface.- Part I Finite differences and finite elements.- Elliptic partial differential equations.- Finite Element Method.- Part II Local resolution and iterative solution.- Local Resolution Techniques.- Iterative Solution Methods.- Part III Constrained and singularly perturbed problems.- Saddled-point Problems.- Mixed and Nonstandard methods.- Applications.- Problems and Projects.- Implementation aspects.- Notations, inequalities, guidelines.- Index.