This book represents a modern introduction to the numerical analysis of partial differential equations and to optimization techniques. The goal is to introduce the reader to the world of mathematical modelling and numerical simulation. It contains finite difference as well as finite element methods for numerical solution of stationary and non-stationary problems. Moreover, the book also treats optimization and operational research techniques. The first chapter introduces the reader to the area of mathematical modelling and numerical simulation. It contains examples of problems leading to partial differential equations of various types and explains some basic questions and obstacles met in their numerical solution. Chapter 2 is concerned with the finite difference method. In chapter 3 the variational formulation of stationary boundary value problems is introduced. Chapter 4 is devoted to the explanation of the concept and basic properties of Sobolev spaces. Chapter 5 presents the qualitative theory of elliptic problems. Chapters 3 to 5 form the basis for the treatment of the finite element method in chapter 6.
Chapter 7 is concerned with eigenvalue problems. In chapter 8, basic qualitative properties and the numerical solution of evolution problems are treated. Chapters 9 to 11 are devoted to optimization techniques. In these chapters, the motivation and examples are given and optimality conditions and basic optimization algorithms are discussed. Finally, methods of operational research are presented. The book contains two appendices: ‘Review of Hilbert spaces’ and ‘Matrix numerical analysis’. The book represents an interesting modern treatment of classical numerical techniques. It contains a number of examples accompanied by results of computations and figures showing solutions of various problems. A positive feature of the book is the fact that it needs only facts from the first few years of university study. Therefore, it can be recommended to students of mathematics and engineering sciences, applied and numerical mathematicians, engineers, physicists and specialists dealing with modelling and numerical simulation of various structures and processes.