It is well documented in the link below:

https://en.wikipedia.org/wiki/Computational_fluid_dynamics#Finite_element_method

Dear Rayudu,

Discretization of Stokes' equations can be done with the finite element method. The use of the Uzawa's method, combined with the conjugate gradient method, can lead to solve the equations systems provided by finite element method.

The finite element method is still a popular method for solving Newtonian and non-Newtonian fluids flow.

Several flows can be modelled using the equations Navier-Stokes or, in simpler cases, Stokes equations. In most cases, as we can not get accurate solutions to these partial differential equations, numerical methods are used as the finite element method. This method leads to solve systems of linear equations very large in size. With computers becoming more powerful than is available today, it is now possible to solve problems of this magnitude, especially on very complex geometries. In recent years it has worked on problems in 2D (two dimensions). There is still work to do regarding the problems in 3D.

To get the most from the computer memory, one must take advantage from the structure of the matrices obtained in the discretization. Often we can do not use direct methods which are very expensive in terms of memory and must rather use iterative methods. The article by Carey et al. (1989) presents iterative methods, with penalties, applied to Newtonian fluid and Newtonian fluid generalized. For a more general reference to the iterative methods, one may consult Greenbaurn (1997).

With my best regards

Prof. Bachir ACHOUR

Howdy,

A major reason is similar to FDM, mainly that both FDM and FEM use local information to approximate the solution in a region of the computational domain. You can see this mainly in the case of FDM method due to the sparseness  of the systems you have to solve. Spectral methods are preferred because calculating the solution at a point locally requires knowledge of the entire computational domain (including boundary conditions). With this global interpolate, one finds spectral accuracy, and more relaxed stability conditions (typically) . The spectral accuracy implies that you have to have a less refined grid on your computational domain to obtain the same accuracy.  Lastly, many fluid problems have periodic boundary conditions, using spectral methods on these problems results in faster convergence to the true solution if you are using numerical linear algebra techniques to solve your problem(assuming you have a linear problem).

In more general terms, for irregular geometries, we would like to use some concept of FEM, but also have the good behavior of spectral methods. Spectral element method is the best for these problems. 

Hi Nithin,

In CFD, we have three main methods to solve the governing partial differential equations numerically: FDM, FEM and FVM. In all the three methods we convert a ‘continuous’ problem (represented by a p.d.e.) into a ‘discrete’ algebraic problem of the form [A]{x} = {b}, and then solve this algebraic problem essentially as {x} = [A]-1{b}, Where {x} is the solution vector. This is the CRUX OF CFD in simple words. FEM notwithstanding its limitations, has become popular in fluid dynamics due to its strong mathematical foundations.

In the past, finite element method was synonymous with Galerkin finite element method (GFEM). GFEM which is based on method of weighted residuals was highly successful in solid/ structural mechanics, the field of its origin. GFEM worked equally well in other situations like heat conduction that is governed by diffusion type equations. The reason for this success was, when applied to problems governed by self-adjoint elliptic or parabolic partial differential equations, GFEM leads to symmetric stiffness matrices. In such cases the difference between finite element solution and the exact solution is minimized with respect to the energy norm. Practically Galerkin formulation is optimal in problems governed by self-adjoint equations. In such cases, there exists a quadratic functional the minimum of which corresponds to satisfying the partial differential equation governing the problem. For example, in structural mechanics the equilibrium position of a structure corresponds to the minimum of the quadratic functional expressing the total potential energy of the structure. Similarly in heat conduction, the thermal equilibrium achieved by solving the Laplace or Poisson equation actually corresponds to the minimum of the quadratic functional expressed in terms of thermal flux which is representation of the total energy of the system at that moment. This success of GFEM in above mentioned fields provided strong case for its use in fluid dynamics. Similar success was expected in fluid dynamics too.

Finite element methods were introduced to fluid mechanics with great expectations in 1970s. The great expectation that the significant success of GFEM in structural dynamics and heat conduction problems would be replicated in fluid dynamics did not realize. Finite element methods make use of a spatial discretization and a weighted residual formulation to reach a system of equations. GFEM is the most widely used among these weighted residual formulations, where weight and interpolation functions are identical. But in fluid dynamics, convection operators present in the non-Lagrangian formulation (i.e. Eulerian) of the governing equations render the system of equation non-symmetric and the best approximation property in energy norm which made GFEM a success in structural mechanics is lost. Consequently, solutions to non self-adjoint fluid dynamic problems by GFEM are often corrupted by spurious node to node oscillations. This problem has motivated the development of alternatives to the GFEM which preclude oscillations without requiring mesh or time-step refinement.

Least squares finite element method (LSFEM) recently developed by  Bonan Jiang et al. seems to overcome these difficulties by providing a symmetric positive definite system of equations and freedom from meeting LBB condition (A difficult condition which the Galerkin form of finite element formulations must satisfy). However, higher continuity requirements for second-order terms in the governing equations force the introduction of additional unknowns by use of an equivalent first-order system of equations or the use of C1 continuous basis functions. These additional unknowns lead to increased memory and computing time requirements that have limited the application of LSFEM to large-scale practical problems, such as three-dimensional compressible viscous flows.

I worked towards development of a novel and simple finite element method as the dissertation topic in my PhD, that employs a least-squares method for first-order derivatives and a Galerkin method for second order derivatives in the governing Navier-Stokes equations, thereby avoiding the need for additional unknowns required by a pure LSFEM approach. This method doesn’t need to satisfy the LBB condition. Galerkin part of the method exploits the benefit of integration by parts. When the unsteady form of the governing equations is used, a streamline upwinding term is introduced naturally by the least-squares method. Resulting system matrix is always symmetric and positive definite and can be solved by iterative solvers like pre-conditioned conjugate gradient method. The method is stable for convection-dominated flows and allows for equal-order basis functions for both pressure and velocity. The stability and accuracy of the method are demonstrated with preliminary results of several incompressible and compressible Fluid Dynamics benchmark problems solved using low-order C0 continuous elements.

The papers can be downloaded from the publication list on my profile.

FEM methods ARE very popular for fluid simulations, although for the reasons explained in the previous answer their basic version does not extend immediately to problems based on hyperbolic equations. One fundamental advance that contributed to solving some of the problems of Galerkin discretizations for hyperbolic problems was given by

http://www.sciencedirect.com/science/article/pii/0045782582900718

Also the advances in Discontinuous Galerkin methods (which are FEM as well) should be considered to have a clearer view of the available options.

"Popular" is a subjective term.  But you are correct, almost all commercial and popular (OpenFOAM) CFD codes are based on FV methods.

The key is the following idea (especially the second half of it).

To discretize a PDE you must approximate the solution (velocity and pressure in your case)  AND you must satisfy the equations (mass and momentum in your case) in an approximate way as well.  This latter part is how FV and FE differ.

Classic FE methods (Galerkin FE methods) use the same mathematical approximation for both the solutions and the equations.  FV methods use a different approximation for them. The FE choice (same approximation) works well when the PDE has a similar type of symmetry (called self-adjoint PDE).  But the advection term of Navier-Stokes is NOT self-adjoint.  It is actually anti-self-adjoint.   So the basic approximation assumptions of the Galerkin FE method struggles with the advection term.  Because it imposes a symmetry on the discrete version that does not exist in real life (in the PDE). 

There are plenty of FE fixes to this issue.   For example, SUPG and its many variants make the basis functions and test functions (solution approximation and equation approximation) different.  Upwinding test functions can do the same thing. Least squares is a brute force mathematical way to make any numerical method (for anything) work - but never work well.  And Discontinuous Galerkin FE methods are for all intents, essentially FV methods.  They are named something else by the FE community perhaps to hide the fact that DG FE methods hat work well for non-self-adjoint problems (like Navier-Stokes) are just FV methods.  

FV comes directly from control volume analysis. Which is familiar and intuitive to engineers.  It is therefore appealing to engineers.  Engineers write commercial CFD codes.

FE comes directly from Hilbert function spaces.  These are familiar and intuitive to mathematicians.  It is therefore appealing to mathematicians.

You can always find a version of FE that is identical to FV. You can always find a version of FD that is identical to FE. So the differences are more attitude and approach than differences in achievable results. 

There are number reasons and depend upon problem to problem of fluid dynamics/mechanics. FEM is having more data book keeping, type of elements, discretization technique, order of accuracy.

Fluid motion is governed by the Navier–Stokes equations, a set of coupled and nonlinear partial differential equations derived from the basic laws of conservation of mass, momentum and energy.  The unknowns are usually the flow velocity, the pressure and density and temperature. The analytical solution of this equation is impossible hence scientists resort to laboratory experiments in such situations.  Computational fluid dynamics (CFD) is an additional tool in the arsenal of scientists. The central process in CFD is the process of discretization, i.e. the process of taking differential equations with an infinite number of degrees of freedom, and reducing it to a system of finite degrees of freedom.Various discretization schemes have been developed to cope with a variety of these issues. The most notable for our purposes are finite difference methods, finite volume methods, finite element methods, and spectral methods. Both finite element and finite difference methods are low order methods, however, finite volume methods are primarily used in aerodynamics applications where strong shocks and discontinuities in the solution occur. Finite volume method solves an integral form of the governing equations so that local continuity property do not have to hold.

Hai Nithin

FVM is more accurate in CFD analysis of flow domain having irregular boundaries . Errors can be reduced while meshing. FEM is best for stress analysis where mesh refinement is not much important and have less memory constraint

As per my survey, Finite difference method (FDM) tops (46%) and finite volume method (FVM) and finite element method (FEM) share their portion of 39% and 15%, respectively among the mathematical models.

check the nice explanation from Comsol: https://br.comsol.com/blogs/fem-vs-fvm