I have been looking at finite element methods for numerically solving PDEs. The approach and software are very impressive, however I was trying to better understand the underlying intuition of the method.
So in finite volume methods, we usually have this notion of "Flux" where there are processes of energy transfer across cell boundaries, such as through diffusion or advection. Then there is a bookkeeping game to keep track of all of the fluxes from different sources and then appropriately applying them at each time step.
In finite elements, I don't have as clear a notion of flux or how energy is passed between cells in a mesh. I think the nearest idea is stress or strain. But the underlying machinery of finite elements seems to be function approximation and interpolation. Hence I was trying to understand--from a numerical standpoint--what is the corresponding intuition for how stress or strain is passed between cells. Is that stress and strain transfer encoded in the weak form and subsequently the coefficients in the element matrix--I guess they call it the bilinear term. I just wanted to make sure I have a clear sense of the logical continuity between finite volume and finite element methods.
In my view, while finite volume methods are developed around the notion of flux between cells, there is no equivalent approach in finite elements.
The core notion in finite elements is to start from the weak form of a PDE. Typically, a certain function $u \in W$ is a weak solution to the problem if $a(u,v)=l(v), \forall v \in W$. The space $W$, the bilinear form $a(\cdot, \cdot)$ and the linear form $l(\cdot)$ will be specific to the differential equation and type of boundary/initial conditions.
The mesh and the chosen finite elements are just a way of producing a finite dimensional version of the weak form of the equation, ultimately leading to the solution of (eventually very large) linears systems.