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.