Partial differential equations (PDEs) arise in various branches of science and engineering. In fact, mathematical modeling of various problems in fluid dynamics and control theory leads to nonlinear partial differential equations of a different kind. Obtaining exact closed-form solutions to these problems is almost impossible. Therefore, one has to seek some approximate solutions; in particular, numerical approximate solutions are the most realistic. The main contribution of this book is the numerical solution of the following nonlinear elliptic equations: the Lane-Emden type nonlinear eigenvalue problem, the Eikonal equation, and the Monge-Ampère equation. Overall, the book is excellent and very well written.
The description of the chapters of the books is as follows: Chapter 1 focuses on the basic concepts of variational problems in Hilbert spaces, like existence, uniqueness, and approximate properties of solutions. Various theorems and proofs have been provided. Discretizations of PDEs lead to systems of linear or nonlinear algebraic equations. In order to solve these equations, in chapter 2, Newton’s method and the conjugate gradient method are given, along with algorithms and a proof of convergence. Operator splitting and alternating direction methods are really efficient methods to solve time-dependent problems in higher dimensions; chapter 3 discusses these methods in detail and their application to nonlinear elliptic PDEs and to eigenvalue problems.
Augmented Lagrangian functional-based computational methods are used to solve linear and nonlinear variational problems that arise from mechanics; this method, along with the alternating direction method of multipliers, is studied thoroughly in chapter 4. In chapter 5, the conjugate gradient method is used to solve some nonlinear problems through the least-square type minimization problems. Further, the parameterized families of nonlinear problems in Hilbert spaces are solved by combining the least-square conjugate gradient method and the arc-length continuation techniques. The numerical solution of the obstacle problem arising from various applications is addressed in chapter 6, particularly for Bingham flows in cylinders and for “optimal control of distributed parameter systems modeled by parabolic variational inequalities of the obstacle type.”
The rest of the chapters are devoted to the study of numerical solutions for various nonlinear problems. In fact, chapter 7 deals with the numerical solution of Lane-Emden type nonlinear eigenvalue problems and the von Kármán equations. Chapter 8 discusses the Eikonal equations, and chapter 9 studies the numerical solution of Monge-Ampère equations in two dimensions. The proposed methods have been implemented in various numerical examples in each chapter, and the results are given in the form of tables and figures; this should really help readers understand the concepts clearly.
This is a good research monograph for people working on the numerical solution of elliptic partial differential equations and related areas. The book is well written with several problems arising in physics and fluid dynamics. This book provides the complete picture of variational methods, starting from the basic concepts and continuing to the advanced topics, along with a complete literature survey.
This well-written and lucid book will act as a useful state-of-the-art reference guide for researchers and students interested the numerical solution of nonlinear elliptic equations and the basics of variational methods for PDEs. Some portions of this book can be used as a graduate textbook for mathematics and other engineering disciplines. Also, this book may be one of the best reference books for people working on numerical methods, especially finite element methods for nonlinear elliptic problems.
