# What can be done with Finite Element Method and not with the Finite Volume Method, and vice versa?

What are some applications where you would absolutely go for either FEM, but not FVM, or vice versa? What are some applications where both methods are equally suited?

I worked with the FEM so far and have written various codes, but I sometimes wonder whether I should also use FVM in future.

• I come more from a CFD background, but for CFD, both methods can work equally well, it's just that sometimes the implementation can be more complex in say FEM than FVM or vice versa, but both still work perfectly fine. You can do laminar visco-elastic flow in FVM and super sonic flows in FEM for instance... I fail to see an example in my field where you absolutely must go for one method or the other... – BlaB Jun 20 '17 at 12:49
• @BlaisB: That's also my opinion currently. I prefer FEM, because I'm familiar and implementation for my geometries are "reasonable easy". I didn't see any advantage of FVM over FEM currently, so I wonder whether there are some examples where one or the other method is prohibitive. – Michael Jun 20 '17 at 18:55
• The thing is, FEM and FVM can be so vast that there is bound to be a variation of one of the other that can compensate from some of their inherent flaws. For examples, having a locally and globally mass conservative scheme in a cell-centered FVM scheme is very, very easy, wheareas for FEM some elements (say stabilized P1-P1) are unable to achieve this. However, obtaining second order of derivation can be tricky in FVM, etc. I think one point where FVM truly dominates however is in its capacity to use generic polyhedral mesh with any number of vertices or face. I have never seen that in FEM. – BlaB Jun 20 '17 at 19:29
• Since most (if not all) FVM's are essentially a special case of some method in the FEM zoo, the question is kind of meaningless. There are authors which distinguish between those methods, but they often reduce the FEM to a particular subset of ansatz spaces and variational formulations. The FV derivation is often more accessible for teaching purposes, but this is the only real advantage that I see. You can do FEM on polyhedrals using discontinuous or hybrid ansatz spaces, you can introduce upwinding fluxes, limiters and so on, and the generalization to higher orders is often straightforward. – Christian Waluga Jun 20 '17 at 20:45
• BTW, @BlaisB, the stabilized P1-P1 FEM and also higher order nodal FEMs are locally conservative if you do the coupling to the transport part of your problem in the right way (which is where conservation really matters). I wrote two papers (in the context of incompressible flow) on exactly this issue. – Christian Waluga Jun 20 '17 at 20:55