We may know that the Finite Volume Method(FVM) is usually superior to the Finite Element Method(FEM) to treat shock-involved flows. Due to its properties, most fluid flows are analyzed by the FVM in the literature. Is it true?

If so, can anyone clear why the FVM is superior to the FEM for treating shock-involved flows? If there is a good reference mentioning this superiority, could you let me know?

Thanks in advance!

Hi,

I have very limited experience in this area too, but I think that if you look the FEM uses the conservation equations in differential form, and the FVM uses the equations in integral form. The two form are equivalent if the variables are smooth continuous variables. But when you have a shock, you have a jump in the density, pressure, etc etc at the shock interface, so the differential form is no longer valid. The integral form is however always valid.

I suggest reading "Riemann Solvers and Numerical Methods for Fluid Dynamics" by E.F. Toro and/or "Finite Volume Methods for Hyperbolic Problems" by R.J. Leveque

Thanks for commenting. Although the FEM is based on the differential equation, it is rewritten in the integral form by the weighted residual method. How do you think about this conversion?

Although the FEM is based on the differential equation, it is rewritten in the integral form by the weighted residual method. How do you think about this conversion?

An interesting trick... Develop along the lines of FEM, then take the weighting to be the cell area ie. locally fixed... compare the end result to the FVM.

For reference - read Patankar's classic book.

mw...

"Develop along the lines of FEM, then take the weighting to be the cell area ie. locally fixed... compare the end result to the FVM"

If, instead of that, you take the weighting functions as Heavisides, you'll have something like FVM being a particular case of FEM, when weighting functions are Heaviside instead of the usual polynomials in the usual case of Galerkin Formulations. Is this the reason of the local conservation properties of the FVM? And also, according to what Luke posted, FEM would be locally conservative for, say, low Re flows?