Inspired by the philosophy to Leibnitz, the authors have written an ambitious text for undergraduate/graduate engineers and scientists which combines mathematical modelling, analysis and computation.

In order to plot a course through the vast landscape of material dealing with computation, the authors have chosen a particular method and used it to present a unified treatment of problems ranging from the fundamental theorem of calculus to partial differential equations.

After 20 chapters of rather concrete, computationally oriented discussion, the authors state and prove the Lax-Milgram lemma and illustrate its application with several examples. This mode of exposition is contrary to that of most mathematics texts and is quite effective.

The authors' writing style is clear and crisp. There is an abundance of quotations, many of them from Leibnitz, but also from sources ranging from Aristotle to Hank Williams.

Because of its very definite point of view, this book does not fit the mold of the standard numerical analysis text or of the standard numerical analysis course. However, it is provocative and should be taken seriously by all faculty, not just those in applied mathematics. It takes a fresh look at some of the standard topics in undergraduate analysis, and the ideas of the text could be used in many courses in analysis and computation. While not without its limitations, this book provides a vision of computation and analysis that may become a model for the future.

(excerpted from Mathematics of Computation. Full text of review can be read here )

The goal of this book, as stated at the preface, is to provide a student with the essential theoretical and computational tools that make it possible to use differential equations in science and engineering effectively. ...the backbone of this book is a unified presentation of numerical solution techniques for differential equations based on Galerkin methods.

...the authors introduce the subject from a historical point of view, and motivate the reading of their book by arguing that to interface calculus and numerical computation is both useful and necessary.

The authors' enthusiasm in writing this text is apparent, and is clearly reflected in the manner by which they introduce new concepts; intellectual curiosity takes priority over the need to build an apparatus. An example is the ingenious introduction of the Cauchy sequence in Chapter 3 entitled a Review of Calculus. ...Each subsequent chapter concisely but clearly introduces and explains a key idea or class of methods, often by example and always on a need-to-know basis.

This book is not about practical numerical computation but is intended to explain the framework upon which the computation relies. It is addressed to students of, or researchers with a sound background on, applied mathematics. Frequent error analyses and proofs of convergence are reminders of this focus. The careful reader will learn the fundamentals of Galerkin and finite-element methods, and should then be in a position to use them rigorously and skilfully to solve particular problems, carrying out error and convergence analyses if so desired.

One cannot find the heart to criticize any aspect of this delightful and illuminating book which I recommend for general reading and one's continuing education. The book could be used as a text in an advanced undergraduate course or introductory graduate course in curricula of applied mathematics. It could also be used as a text in corresponding courses in a number of disciplines of applied science and engineering, but the students must have a strong mathematics background as well as carry a certain amount of intellectual curiosity, or they may complain about the authors' rightful demand for attention to the fundamentals.

(excerpted from Journal on Fluid Mechanics. Full text of review can be read here )

‘ ... the aim of the book is successfully attained, viz. to present a unified approach to computational mathematical modelling based on differential equations combining aspects of mathematics, computation and application.’

G. Kirlinger, International Mathematical News

‘This is a well-written and highly readable book that should be useful for advanced undergraduate in Mathematics or for other scientists new to the field.’

Mark Baldwin and Pijush Bhattacharyya, Mathematics Today

The book ... provides a readable, up-to-date, easily accessible, and mathematically sound introduction to finite element methods.

With some exceptions, the level of mathematics technicalities is kept low to make the book accessible to a wide audience in mathematics, science, and engineering students.

Priori knowledge of differential equations could be minimal: as each new problem class is introduced, there is a brief description of applications, qualitative properties, and analytic solution techniques.

Indeed, the authors advocate a unified approach wherein the analytic and numerical study of differential equations proceed step-in-step and the present book is written with this viewpoint in mind. However, the main focus of the book remains on numerical computations...

Two particular features distinguish this book. From the beginning, equal emphasis is placed on a priori and a posteriori error estimates, the latter being motivated by their use in error control and adaptive mesh refinement... Also, equal emphasis is placed on all types of differential equations... These emphases reflect advances in finite element methods made over the last two decades and in particular much of the authors' own research.

This book provides a clear elucidation of the fundamentals of the finite element method at an elementary level. There are many examples and numerical results throughout, and there is a good selection of exercises. The emphasis on error control and adaptivity is both important in practice and provides students with a good example of the practical application of theoretical analysis.

(Extracted from SIAM Book Reviews, 1998. Full text of review can be read here )

[This] book is an update of Johnson's book that has been very successful during the past years. It appears that the same verdict can be given to this extended new book. It is nicely crafted and full of interesting details.

(Extracted from ITW Nieuws, 1997. Full text of review can be read here )

The book ... provides a thoroughgoing treatise on computational mathematical modelling in which relevant aspects of mathematical analysis are judiciously combined with numerical methods based on Galerkin's method... The book is written in an accessible style and is well suited to serve as a basis for courses in mathematics, science, and engineering.

(Extracted from Aslib Book Guide, 1997. Full text of review can be read here )

This text on computational differential equations, which ultimately centers on the Galerkin method for linear partial differential equations, starts at a quite elementary level... Subsequently, initial and boundary value problems are investigated in one and several dimensions within the Galerkin framework for the most important types of linear equations. ...Due to the book's extensive discussion, it will also be useful for the scientist and engineer who needs to employ the methods.

(Extracted from Monatshafte for Math, 1998. Full text of review can be read here )

The goal of [this book] is to present a unified approach to computational mathematical modelling with differential equations, based on a principle of a fusion of mathematics and computation.

The material is often centered around specific examples, with generalizations coming as additional material and worked out in exercises.

The authors ... show that many numerical algorithms have their roots in theorems and proofs and demonstrate how these algorithms work when applied to actual mathematical models. The book includes exercises of various degrees of complexity and many examples and is well illustrated. The text is accompanied by very nice historical surveys concerning the subject treated.

(Extracted from A.M.S. Mathematical Reviews, 1999. Full text of review can be read here.)

Part I: Introduction

1. The Vision of Leibniz

2. A Brief History
A timeline for Calculus - A timeline for the computer - An important mathematical controversy - Some important personalities

3. A Review of Calculus
Real numbers - Functions - Differential equations - The Fundamental Theorem of Calculus - A look ahead

4. A Short Review of Linear Algebra
Linear combinations, linear independency, and basis - Norms, inner products, and orthogonality - Linear transformations and matrices - Eigenvalues and eigenvectors - Norms of matrices - Vector spaces of functions

5. Polynomial Approximation
Vector spaces of polynomial - Polynomial interpolation - Vector spaces of piecewise polynomials - Interpolation by piecewise polynomials - Quadrature - The L₂ projection into a space of polynomials - Approximation by trigonometric polynomials

6. Galerkin's Method
Galerkin's method with global polynomials - Galerkin's method with piecewise polynomials - Galerkin's method with trigonometric polynomials - Comments on Galerkin's method - Important issues

7. Solving Linear Algebraic Systems
Stationary systems of masses and springs - Direct methods - Direct methods for special systems - Iterative methods - Estimating the error of the solution

Part II: The archetypes

8. Two-Point Boundary Value Problems
The finite element method - Error estimates and adaptive error control - Data and modeling errors - Higher order finite element methods - The elastic beam - A comment on a priori and a posteriori analysis

9. Scalar Initial Value Problems
A model in population dynamics - Galerkin finite element methods - A posteriori error analysis and adaptive error control - A priori error analysis - Quadrature errors - The existence of solutions

10. Initial Value Problems for Systems
Model problems - The existence of solutions and Duhamel's formula - Solutions of autonomous problems - Solutions of non-autonomous problems - Stability - Galerkin finite element methods - Error analysis and adaptive error control - The motion of a satellite

11. Calculus of Variations
Minimization problems - Hamilton's principle

12. Computational Mathematical Modeling

Part III: Problems in Several Dimensions

13. Calculus of Several Variables
Functions - Partial derivatives - Integration - Representation by potentials - A model of heat conduction - Calculus of variations in several dimensions - Mechanical models

14. Piecewise Polynomials in Several Dimensions
Meshes in several dimensions - Vector spaces of piecewise polynomials - Error estimates for piecewise linear interpolation - Quadrature in several dimensions

15. The Poisson Equation
The finite element method for the Poisson equation - Energy norm error estimates - Adaptive error control - Dealing with different boundary conditions - Error estimates in the L₂ norm

16. The Heat Equation
Maxwell's equations - The basic structure of solutions of the heat equation - Stability - A finite element method for the heat equation - Error estimates and adaptive error control - Proofs of the error estimates

17. The Wave Equation
Transport in one dimension - The wave equation in one dimension - The wave equation in higher dimensions - A finite element method - Error estimates and adaptive error control

18. Stationary Convection-Diffusion Problems
A basic model - The stationary convection-diffusion problem - The streamline diffusion method - A frame-work for an error analysis - A posteriori error analysis in one dimension - Error analysis in two dimensions - 79 A.D.

19. Time Dependent Convection-Diffusion Problems
Euler and Lagrange coordinates - The characteristic Galerkin method - The streamline diffusion method on an Euler mesh - Error analysis

20. The Eigenvalue Problem for an Elliptic Operator
Computation of the smallest eigenvalue - On computing larger eigenvalues - The Schroedinger equation for the hydrogen atom - The special functions of mathematical physics

21. The Power of Abstraction
The abstract formulation - The Lax-Milgram theorem - The abstract Galerkin method - Applications - A strong stability estimate for Poisson's equation