I sort of have two questions on the same subject so I just thought to ask both in one thread.
I am looking to do some research on Finite Element Methods (FEM) and I am recently started looking into the related different methods such as Finite Difference Methods (FDM) and Finite Volume Methods (FVM). Assuming I will do the research on (FEM), I wanted to ask what were some good books that will be suitable for a beginner in this subject field topic with more so a electrical engineering and math background? I am not sure if there are any applications for these methods to use in the field of electrical engineering or similar topics, but this is what type of books I am most concerned to look for. I am also interested in the traditional subject field topics these methods are used for such as (viscosity, collisions, structure analysis,$\;\ldots$ etc.), from more of a Applied Math stand point/background.
My second question is, I am still unsure which method would be most suitable for a beginning learner on this topic and most beneficial for the electrical applications (preferably). I hear that the (FEM) is the most difficult method of the three for one who is just starting to pursue interest in this field of study. I hear that the (FDM) and (FVM) are the two that are more accessible and easy to pick up on for the first time then (FEM). Any our your suggestions would be greatly appreciated whether they are from just reading or from actual personal experience or someone you may know experience when it came to dealing with this in real life practice. So again, to summarize, my question is what method do you feel would be best to start off with learning with the preferred topics of interest stating above which are electrical phenomena or the latter (viscosity,$~\ldots$). In the long run I would like to have experience with each method (as a side note).
Thank You.
Best Answer
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.