# Finite Difference and Finite Volume as special cases of Finite Element

I have heard this many times that "FD is a special case of FE" and separately, that "FV is a special case of FE".

However I have never come across a brief document that substantiates that claim, through derivation, examples, proofs, whatever.

I understand the basics of FE and the 3 manifestations: Collocation, Least-squares, Galerkin.

I also understand that these three can be considered as various kinds of "weighted residual" schemes, and the unifying principle is MWR (method of weighted residuals) which is a variational method.

Are there two easy-to-read resources, one for FD and one for FV, that can make the connection to either FE or MWR or something like that?

• Actually, that's backwards: MWR is arguably the most general; collocation and Galerkin are special cases; FD is a special case of collocation and FE and FV are special cases of Galerkin. The difference is how you chose the finite-dimensional space for the solution and the finite-dimensional space for the test (or weight) function. (Not everyone would agree on this ordering, depending on your precise definition of Galerkin and MWR. I don't see much value in nailing this down to the level of a formal theorem, either, but it is helpful to realize the common features of the three approaches.) – Christian Clason Nov 17 '17 at 23:27
• I'm not primarily looking for a theorem and proofs. In fact I would prefer if there are as little, or no, theorems and proofs. "Easy to read" means for a scientist/engineer (since I'm not a mathematician, and especially not very good with functional analysis). However, a combination of elaboration, derivations, examples, and putting things in context, is highly desirable. Otherwise those are empty claims/assertions. – Fi Zixer Nov 17 '17 at 23:32
• So what specifically are you looking for (assuming it's not just idle curiosity)? The problem is that your premise (first sentence) does not hold up if you look at it in any detail; at most you can say that FE methods are Galerkin methods, some FV methods can be interpreted as generalized (discontinuous) Galerkin methods, and (with some violence), so can some FD methods. – Christian Clason Nov 17 '17 at 23:40
• I saw an explanation of this once upon a time in a big, thick CFD book, but for the life of me, i can’t remember the title... i’ll seeif i can track it down. – Paul Nov 18 '17 at 2:00