Any great *Introductory* books for Finite (Element/Difference) Methods

Solution 1:

I would start by learning the FEM for elliptic problems as this is the easiest. The book Numerical Solution of Partial Differential Equations by the Finite Element Method by Claes Johnson is a fairly good introductory book if you are mainly interested in implementing and using the finite element method. It skips most of the Hilbert space theory needed to make the arguments rigorous. It also skips most of the technical difficulties. On the other hand, if you want a rigorous treatment I recommend just reading a book on the Hilbert space approach to PDE as a complementary text. I thought Introduction to PDE by Renardy and Rogers (especially chapters 6-8) to be a gentle introduction to this topic. There is also a great deal of work done on constructing approximation spaces and finding error approximations in p-norms rather than the Sobolev estimates the FEM gives you via Cea's Lemma. I found this to be the hardest part and would leave it for last. As for you second question, no, I don't think learning the FEM before the FDM will cause major problems.