A Finite Element Primer for Beginners: The Basics

A book that promises three times in the title that it is for beginners may be trying too hard. I think the present book is more a survey than a true introduction. It really does start at the beginning, but I think it introduces too many ideas too quickly and does not have enough worked examples. The true beginner would be better off studying the finite-element section of a full text on numerical analysis and then coming back to this book as a supplement if he needs more. The treatment in Greenbaum & Chartier’s Numerical Methods: Design, Analysis, and Computer Implementation of Algorithms is especially good.

The book originates in a mechanical engineering course, but it does give a good mathematical treatment of the subject and does not assume any engineering knowledge. It uses an elasticity problem (in one and three dimensions) as the model problem, but states rather than derives the differential equations of interest and works from there.

The book is primarily about Galerkin’s method. The first half of the book develops this for one-dimensional problems, and includes a good discussion of estimating the accuracy of the solution.  Galerkin’s method replaces the given differential equation with a system of linear equations, often with a large number of unknowns, and it may be computationally difficult to solve these exactly. Therefore the book includes a brief chapter on iterative methods for solving linear systems. The second half of the book is about three-dimensional problems, and also uses Galerkin’s method but does not repeat very much of the first half. A lot of the second half is about picking the elements, which is much more tricky in three dimensions than in one dimension. The last chapter is a brief survey of more advanced methods, with many references.

Very Good Features: two excellent appendices of problems, one a collection of exam problems from the author’s courses at Berkeley and one a set of computer projects.

Very Bad Features: (1) no index; (2) too expensive at $99 for an introductory text that’s only 135 pages (and that includes 25 pages of background appendices).

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.