Have a look on:

https://www.researchgate.net/post/whats_the_differences_between_MoM_FEM_FDTD_FIT

http://www.edaboard.com/thread116998.html

super thanks!

In general, time-domain methods are superior for simulating extremely large devices or devices incorporating nonlinear properties.  While there does exist time-domain BIE and FEM algorithms, I suspect you are talking about frequency-domain versions of them which are much more common.  Frequency-domain methods are usually much faster for simulations that are not very large and for simulating devices that are highly resonant.

From here we need to compare BIE to FEM.  The FEM uses a volume mesh while BIE only uses surface meshes.  If you are simulating something with a lot of geometric complexity, FEM will likely be the best method.  If you are simulating something with simpler geometry, like scattering from a sphere, the BIE will likely be your most efficient method.  Many modern commercial software hybridize the two so you get the best of both worlds.

Last, I am not sure if you are looking at developing your own codes or not, but this is also a consideration.  FDTD will be the easiest to implement because it is more intuitive and there are many resources for you.  For example, see

http://emlab.utep.edu/ee5390fdtd.htm

The most difficult to implement for you will be the BIE.  There is much less information and resources available about this method.  FEM will take more work to develop than FDTD, but there is still a lot of information available and even example codes.

I created a short summary of many different numerical methods in electromagnetics in Lecture 1 at the following link.  While the comparison is not complete, it is the most complete comparison I have seen on the internet and I am continually evolving it.  

http://emlab.utep.edu/ee5390cem.htm

Hope this helps!!

Very nice thanks

dear Professor Raymond Rumpf  your answer is very useful.

i have another question. 

which one of numerical method is used for close space and which one for open space?  how can we use the MOM method for close space? how can we change the space from open to close in MOM method?

the mesh at FEM is volume mesh and for MOM is both surface and volume?

thank you

I am not completely sure what you mean by open and close space.  A technique like the method of moments is only required to mesh the objects being simulated.  It does not require space to be truncated or handled specially at the boundaries of your space.  Maybe this is what you mean by open space?  If so, method of moments and boundary element methods are very good.  I think, however, the best numerical method depends more on the device you are simulating and the physics you want to account for than it does anything else.