In solid mechanics the finite element method (FEM) is widely used because the equations are dominated by elliptic equations. Fluid mechanics problems tend to be dominated by convection and diffusion and are thus hyperbolic or parabolic equations. FEM, Finite Volume Method (FVM) and Finite Differences Method (FDM) have all been used in fluid mechanics but FEM only shares a small portion of the solutions.
Both FEM and FVM are types of weighted residual methods and can be numerically identical but the simplified versions of FEM, typically used in common solid mechanics, impose a symmetry on the fluid problem that produces lots of problems. That is the resason why, in my humble opinion, we use FEM widely in solid mechanics while we almost never use it in fluid mechanics and gas dynamics.
It has been a pleasure to listen and learn from Professor Strang.