# Finite Volume Method vs. Finite Element Method for Eulerian and Lagrangian Reference Frames

As far as I can tell, finite volume methods are primarily for eulerian reference frames (such as in fluid dynamics simulations) while finite element methods are easily ameanable to lagrangian reference frames (such as in solid mechanics simulations). I've always taken it for granted, but I never really thought about why this is the case.

What is it about finite volume method that makes it favorable for eulerian but not lagrangian frames of references? Conversely, what is it about the lagrangian frame of reference that make finite element methods better suited than finite volume methods?

In general, methods that conform to interfaces allow use of fewer degrees of freedom to accurately represent the nonsmoothness at the interface than interface tracking methods. Thus, when it is practical to conform to an interface, that is often preferable. FE methods for solid mechanics are almost always written for unstructured meshes, and the interfaces usually do not become too deformed. We may simulate until some structure fails, but many analyses do not continue simulating the dynamic crumpling after failure. In the fluids world, interfaces more frequently change topology very irregularly, such as when simulating vigorously rising bubbles. Meshing all those interfaces and transfering solutions to new meshes would be a headache, so is usually avoided. Other interfaces, such as along the edge of a wing on an aircraft, are extremely anisotropic (aspect ratio up to $\sim 10^6$) and either static or with static topology (though certainly moving in applications such as aeroacoustics). CFD simulations usually conform to boundaries like the wing, but use interface capturing or cut-cell technology for interfaces like the rising bubbles.