Both FE and FV methods are types of weighted residual methods.

I (and you) can write a FE method that is numerically identical to a FV method.

And vice versa. And people often do this this without realizing this is what they are doing. So the major difference between FE and FV is how the method is presented. (Math guys love FE, Engineers tend to love FV).

Still, at the very simplest level they are slightly different and that difference

answers your question. To discretize a PDE system you must approximate the solution AND the equations (both the unknowns AND equations must become finite). The simplest (Galerkin) FE methods approximate both

things using the SAME interpolation functions. This implies a great deal of symmetry. The PDE's of solid mechanics possess that symmetry (they are self-adjoint PDEs). So FE methods use an assumption that is consistent with those PDEs.

FV methods use a different interpolation for the equations than they do for the solution. Advective PDEs (Like Navier-Stokes) are NOT self-adjoint. So FV methods reflect that property. Simple FE methods impose a symmetry on the advective PDE which the PDE does not have (and therefore produce lots of problems).

However, as I said at the beginning, you can make a FE work for an advective PDE (like N.S.) if you work at it. All the FE solutions for advection boil down to making the approximation of the solution and equations different (just like FV is to begin with). Those are called Petrov-Galerkin FE methods. Petrov-Galerkin, and discontinuous Galerkin (DG) methods ARE FV methods. The FE guys are just ashamed to admit this fact - or they don't know it. For more info perhaps see my Annual Reviews of Fluid Mechanics paper that is downloadable from my website, and references therein.

I know that was a bit abstract. But it is best I can do in a few paragraphs.

In summary: (1) There is a real and fundamental reason for certain methods to work better for certain PDEs. and (2) the FE and FV methods are nearly identical despite all the hoopla suggesting otherwise (and can be easily morphed into each other).

COMSOL multiphysics package are based on FEM and they have CFD solver. There are some CFD work on free surface, simple flow in closed chanel and etc. based on FEM. For non-Newtonian fluids it seems that FE is quite frequent in that field. For incompressible flow codes written on FEM performs well, but when there are compressible flow - FV more common choice.

FVM is better for fluid simulation because of mathemathical background - values calculated on center of control volume helps to reduce computational costs.

According to some people - FVM and FDM are more easier to implement to code than FE and in my opinion FV interpretation is little bit more easier to understand than FE. (volume vs surface) Maybe that could be another reasons why FE is less popular in fluid flow simulation field.

Thanks Karolis for your answer. I agree with you concerning the physical interpretation of FVM by using the conservation laws when compared with FEM (mathematics). I didn't know that FEM have such frequent applications in fluid dynamics. Do you have any idea about the numerical method mostly used in multiphase flows (especially aerosols).

I think that there must be some advantages of FEM when it is applied to solid mechanics?

There could be FEM of FVM with additional equations for particle movement on it. If aerosols are treated like particles - try to look at discrete element method (DEM) or combined methods of DEM and FVM like there https://www.youtube.com/watch?v=UQVY_EE48ks

While FV methods are indeed a very popular choice for compressible flows due to their natural incorporation of discontinuous solutions and the stabilization through the flux function, their advantages can be combined with the spectral accuracy of high order ansatz functions in discontinuous FE methods. These are in fact a pretty active research topic currently.

I would not agree that FV is cheaper (it might be in terms of computing time per DOF), but a higher order FE /CG / DG formulation can give more accurate results for the same number of DOF.

You might also want to check out http://nek5000.mcs.anl.gov/ to take a look at an example of these ideas applied to fluid flow.

If you read about the arbitrary Lagransian Eulerian formulation and compare the Lagrance approach with the Eulerian approach you will be able to understand why, given that you comprehend the formulations of the FE, FV and FD methods. Nevertheless, scientists do use FE for solving CFD problems but as it was mentioned above FD and FV are the traditional methods used.If you are starting now with the field of CFD a very interesting numerical method is the particles method which can be used even for modeling concrete (with the use of proper boundary formulae).

FEM, FVM and FDM have all been used in fluid mechanics, even thought FEM only shares a small portion. However, some well-known commercial softwares (e.g. Fluent, Adina) still prefer FEM in their fluid simulations.

In my opinion, because of the existence of fluid surface, the deformations of mesh or grid in FEM are always large, which leads to inaccuracy results. In the recent decades, along with developments of the arbitrary Lagrangian-Eulerian (ALE) and the mixed Eulerian-Lagrangian (MEL), mesh or gird could be regenerated in each step. Therefore, the disadvantage of FEM mentioned above might be improved to a great extent.

By the way, an effective way to simulate multiphase flow is to use meshless numerical method (e.g SPH, Level-set). A calssics SPH simulation is attached.

One reason : Variable Mesh and/or Variables Properties versus Time.

In solid mechanics the finite element method is widely used because the equations are dominated by elliptic equations, so every part of the solution influences every other part. Fluid mechanics problems tend to be dominated by advection and are thus hyperbolic or parabolic in nature. While it is possible to develop finite element methods for theses problems types it is quite complex to do so, and may need to lower order solutions (as with discontinuous galerkin methods). Finite volume methods tend to be simpler to implement and can incorporate Godunov type techniques which respect the underlying physics.

Both FE and FV methods are types of weighted residual methods.

I (and you) can write a FE method that is numerically identical to a FV method.

And vice versa. And people often do this this without realizing this is what they are doing. So the major difference between FE and FV is how the method is presented. (Math guys love FE, Engineers tend to love FV).

Still, at the very simplest level they are slightly different and that difference

answers your question. To discretize a PDE system you must approximate the solution AND the equations (both the unknowns AND equations must become finite). The simplest (Galerkin) FE methods approximate both

things using the SAME interpolation functions. This implies a great deal of symmetry. The PDE's of solid mechanics possess that symmetry (they are self-adjoint PDEs). So FE methods use an assumption that is consistent with those PDEs.

FV methods use a different interpolation for the equations than they do for the solution. Advective PDEs (Like Navier-Stokes) are NOT self-adjoint. So FV methods reflect that property. Simple FE methods impose a symmetry on the advective PDE which the PDE does not have (and therefore produce lots of problems).

However, as I said at the beginning, you can make a FE work for an advective PDE (like N.S.) if you work at it. All the FE solutions for advection boil down to making the approximation of the solution and equations different (just like FV is to begin with). Those are called Petrov-Galerkin FE methods. Petrov-Galerkin, and discontinuous Galerkin (DG) methods ARE FV methods. The FE guys are just ashamed to admit this fact - or they don't know it. For more info perhaps see my Annual Reviews of Fluid Mechanics paper that is downloadable from my website, and references therein.

I know that was a bit abstract. But it is best I can do in a few paragraphs.

In summary: (1) There is a real and fundamental reason for certain methods to work better for certain PDEs. and (2) the FE and FV methods are nearly identical despite all the hoopla suggesting otherwise (and can be easily morphed into each other).

The generic FEM applied to fluid problems suffers from major mass conservation violations. Sure there has been advances in FEM for CFD in the last decade or two to deal with the mass conservation issue, but that comes at cost. And even if a new modification of FEM would make it better for CFD, it would take time for the CFD community to adapt and change the culture, i.e. the huge volume of literature behind the FDM/FVM for CFD.

To Ming: Fluent is a FV solver. Some other ANSYS solvers are FE but not Fluent.

FE conserves total mass and momentum by construction.

(The weight functions are a partition of unity).

Interestingly, it also conserve things locally (just like a FV method).

See Tom Hughes for FE versions of the proof.

See Matusiossi (sp ?) (JCP) for the FV looking proof.

Or my above referenced review on conservation.

In short: Take a bunch of different FV methods

that use different meshes/volumes but have the same unknown locations.

Each conserves locally. The FE method is a certain average of those

- so it also conserves locally.

I repeat. You all have been deceived by teachers who knew no better.

There is no 'us' versus 'them'. Or much much less than you all think.

There is a huge volume of literature for both methods.

It is just that each side has no ability to read the other's papers - so they don't

get cross referenced. Its just like politics ...

In general I agree with J.Perot and David Ingram. When we developed FV methods at Imperial with Spalding , Patankar ,Gosman etc , the idea was that FE was well-suited for linear problems or weakly non-linear. When convection is strong and particularly when turbulence is strong and you need to solve for Reynolds stresses as well I know of no FE method that will handle the strong non-linearities ( although of course FV is the marriage of FD and FE, and of course they are both weighted residual methods) at the same accuracy and at the same cost as FV. So FE for flow at the bottom of the sea where diffusion prevails and FV at the surface of the sea where convection prevails !!

Incidentally some insight may be gained by reading

N.C.Markatos,Dynamic Computer Modelling of Environmental Systems for decision making,risk assessment and design, Asia-Pacific Journal of Chemical Engineering,7(2),182-205,2012.

I have a comment on this issue talking about standard CFD software for simulations of solids and fluids. The Comsol Multiphysics software is based on finite elements. You can do simulations of the navier stokes equations with or without turbulence models but to my opinion it is not very fast. So for complex time resolved 3D LES and DNS solutions I would not advise to use Comsol but for laminar flows and for fundamental detailed interactions I would. For the former packages like Fluent and CFX are more appropriate. And if it becomes more fundamental I would advice to write a code yourself or use an academic one. If money is an issue then OpenFOAM might be an option if you have sufficient time. This because it is for free but not very well documented, from experience I know that you need a year to get a good level of knowledge about it.

I agree Mr.Bastiaans

Fluid mechanics is a very wide field and it is difficult to make some general statements. Speaking about Euler equations, where propagation of discontinuity is inherent to the solution, and could be a non linear effect (shock/discontinuity can arises also starting from a continuous initial condition), this can be more easily taken into account in the mathematical framework of the FDM or FVM (i.e. starting from Godunov method ) resulting in a easy computer Programming. To take into account such phenomena using FEM you should resort to "unsymmetrical" formulation like Petrov-Galerkin to be able to correctly model the shock resulting in more complicated programming and longer CPU time.

I don't. I use the lattice spring model for elasticity and the lattice Boltzmann method for fluids......

In the case of almost incompressible, low Mach number flows (for example, blood flow or thermal hydraulics in a pressurized water reactor), there do exist extremely efficient spectral element codes - for such problems, these codes tend to be far more efficient on massively parallel machines than any of the extant FV methods, even for complex geometries. One example of such a code is Nek5000; it has, for instance, won the Gordon Bell prize for this very reason. These types of codes also show excellent convergence characteristics, a feature shared with spectral codes in general. (Unlike pure spectral codes, however, spectral element codes have the ability to vary spatial resolution anywhere within the computational domain, something that can be done because both the spatial structure of the elements and the order of the spectral expansion can be adjusted across the computational domain.) The only (substantial) downside is that such codes cannot deal with sharp discontinuities within the computational domain, thus excluding any compressible effects beyond what one can handle via the Boussinesq approximation.

To Khaled Al-Salem, sorry for my mistake. Fluent is a FV solver.

If we don't want to look in depth, then weighted residuals methods in CFD can produce a kind of "nonlinear instabilty" in solution. While linear instability is easily recognized, the nonlinear case often can hardly be identified. Thus you can receive false solution which looks like something meaningfull, and it is necessary to control a lot of conservation properties while solving, resulting in increasing of computation time.

Also if you do not develop the commercial software but you study some scientific problem and do not want to deal with the "black box" then the coding time exceeds the finite difference case tremendously.

If you are trying to decide which method to use, then the best path is probably found by consulting the literature in the specific problem area.

FVM and FDM are generally considered easier to program than FEM, but opinions vary on this point.

FVM are generally expected to provide better conservation properties, but opinions vary on this point also.