In FD we approximate the continuous derivatives in the PDE by using suitable Taylor expansions (or, equivalently, Lagrangian polynomials).

In FE a variational formulation is typically adopted, that is the PDE is projected along suitable shape functions to minimize the error in a certain space. The discretization is therefore introduceb by means of the shape functions degree.

In my opinion Finite Element Method (FEM )is more accurate than Finite Difference Method (FDM)as the first approximate the problem of any differential equation by an element(like triangular, square,.....), but the second depends on number of nodes which may not be appropriate to solve the problem.

Nabel, the accuracy order is not the key in the difference between the two methods. You can also adopt compact FD of very high order of accuracy.

The key is more in the way the problem is solved, FD uses the pointiwise formulation, FEM adopts a variational (weak) formulation

[[ This is a difficult question to answer.

"The FDM is the oldest and is based upon the application of a local Taylor expansion to approximate the differential equations. The FDM uses a topologically square network of lines to construct the discretization of the PDE. This is a potential bottleneck of the method when handling complex geometries in multiple dimensions. This issue motivated the use of an integral form of the PDEs and subsequently the development of the finite element (FE)and finite volume(FV) techniques." (http://www2.imperial.ac.uk/ssherw/spectralhp/papers/HandBook.pdf)

Here are two references to review so you can get a better feel for these methods.

http://files.campus.edublogs.org/blog.nus.edu.sg/dist/4/1978/files/2012/01/CN4118R_Final_Report_U080118W_OliverYeo-1r6dfjw.pdf (see page 10 for a very nice comparison in the types of problems they were interested in - computational fluid dynamics)

There are some nice references for these methods at http://www2.imperial.ac.uk/ssherw/spectralhp/papers/HandBook.pdf (See section 7 for very nice references) ]].

Dear Qassim Ali ,

Look the link, please.

https://math.stackexchange.com/questions/424672/what-is-the-difference-between-finite-difference-methods-finite-element-methods

http://feaforall.com/difference-between-fem-and-fdm/

Regards, Shafagat

Hi Fillipo

Ok, I agree with you and precisely I agree with my colleague Dr. Isam. Please refer to the link and the following answer which may convenience:-

In my humble opinion, FEM is the most flexible one in terms of dealing with complex geometry and complicated boundary conditions. FEM also allows the adaptive/local procedure to get higher order local approximation or battling singularities. FEM's basis can be discontinuous and not well-defined pointwisely, which is a nice heritage from the Hilbert space framework. For computational fluid dynamics and electromagnetism, FEM is the way to incorporate the intrinsic geometrical properties of the solutions.

For FVM: partly you can refer to my answer here: How should a numerical solver treat conserved quantities? It is also worth noting that FVM can only have lower order of approximation.

In some recently development in FEM addresses the problem I mentioned in the answer above. For example, for convection-dominated pde, tradition continuous Galerkin framework for FEM doesn't work well, which introduces dissapation over time and oscillation over material-layers for the numerical solution. Now there are Discontinuous Galerkin FEM (higher order FVM) and hybrized DGFEM (see here: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems) to remedy these two effects.

FDM and FVM are easy to implement, but you get trade-off from this convenience of implementation for limited usage for different PDEs.

https://math.stackexchange.com/questions/424672/what-is-the-difference-between-finite-difference-methods-finite-element-methods

Yes. While I agree about the FD formulation, difficult to generalize on complex geometries and non-structured grids, the FVM shares the same features of FEM on complex grids and can be developed to get high order accuracy. On the other hand, FVM is nothing but a particular FEM, where the shape functions are piecewice constant. Numerical conservation in FVM is implied by definition owing to the unique definition of flux function while FEM does not automatically fulfills that. Furthermore, FVM can naturally manage non-regular solution.

Accuracy has little to do with the method used but how it is used and depends on the number of nodes and elements employed. However, FEM are more powerful in dealing with complex problems than FDM. Finite differences methods are easier to use than finite elements. FDM transforms differential equations into systems of algebraic equations while FEM transform a domain into connected elements. You need to take courses and/or read books to truly understand the two methods.

Please check reference below.. hope it will be helpful for you… Regards…

https://www.cs.auckland.ac.nz/courses/compsci369s1c/lectures/GG-notes/CS369-FDM-FEM.pdf