This book is like any other FEA book in regards to some things have been done very well, some things have been treated poorly, and some things are noticeably missing. Overall, this one is in the middle of the road.

If you are more mathematically inclined, I would say start with this book before you delve into the gems by Ciarlet The Finite Element Method for Elliptic Problems (Classics in Applied Mathematics) or Brenner and Scott The Mathematical Theory of Finite Element Methods (Texts in Applied Mathematics). As one of the other reviewers mentioned, the theory part of the book is crammed together and kind of lax; well, to me, this would be the perfect place to start into the theory. He covers most of the relevant theory without bogging you down with the technical details to be 100% correct. In a first pass through learning the theory, you probably don't need to be inundated with all of the finer points of Hilbert spaces.

Download the MATLAB code for the book and read through it as you work your way through the book. Gockenbach's discussion of "implementing" the finite element method is very mathematical and not very practical (I don't know why I was expecting anything else from a SIAM book). Examining the code is where you will learn how to implement FEA on your own; the book alone is not sufficient. Better yet, pick up the Cook book Concepts and Applications of Finite Element Analysis, 4th Edition if you want some real instruction on applications and implementation.

The title should have included the words "for elliptic PDEs", as that is all that is covered. The last half of the book isn't very well organized. Direct and iterative solvers are briefly covered, as well as multigrid methods and error estimations...sometimes out of order, sometimes belaboring moot points, sometimes just deferring the reader to a reference. You would be better served to get a book or paper on the individual topics. And to put some trust in direct solvers (Gockenbach certainly doesn't). The example problems have nothing to do with a real world problem you would come across. I've yet to come across a material or domain that has a nice, continuous function of only spatial dimensions that describes thermal conductivity...or even a perfectly square domain. Circles have come up, but I digress... Symmetry is missing. The patch test is missing. Too much faith is placed in triangular elements, without really discussing any of their problems and drawbacks (ESPECIALLY with the linear triangle).

Overall, the book does have its merits, but also its drawbacks. I'm ambivalent about recommending it, so I settle on the middle- three stars. I don't find anything in the book saying "oooh, you MUST have this on your bookshelf!", but that is just my humble opinion.