# Difference between MoM and FEM

Method of Moments and Finite Element Methods are two of the most used methods in computational electromagnetics to solve electromagnetic equations. As it is known, in FEM sparse matrixes are used while MoM uses the full-matrixes.

My question is, why MoM uses full-matrix while a sparse matrix is obtained in FEM? Is there any advantage or disadvantage between these types of solutions?

• I know very little about MoM, but if it is generating a fully coupled matrix system, then the basis functions used in the MoM formulation must be very non-local, making all degrees of freedom coupled to each other. In contrast, FEM is designed with assumptions weak enough to allow for basis functions that are non zero only in small regions clustered around each of the nodal points in the mesh. – Paul Apr 30 '20 at 1:13

## 1 Answer

Method of Moments is the name given to the Boundary Element Method (BEM) in the Electromagnetism community. Since you are using Green functions for BEM you get fully populated and asymmetric matrices.

Some advantages that are commonly mentioned for BEM are:

• It reduces the dimensionality of the mesh. This might look to lead to smaller matrices but, in general, they are more expensive compared to sparse matrices for FEM. This might be an advantage when dealing with remeshing such as fracture mechanics and shape optimization.

• You can deal naturally with unbounded domains, such as exterior acoustics, antenna radiation, or seismic wave propagation.

You can add Fast Multipole Method to improve BEM performance. And, I would say that is the way to go instead of just using plain BEM nowadays.

### References

Liu, Y. (2009). Fast multipole boundary element method: theory and applications in engineering. Cambridge university press.

• +1 for the community specific reference. To be fair, fast integration like FMM/PME is almost always used for large integral equations like this. However, this is less about the method and more about numerics. – Spencer Bryngelson May 2 '20 at 4:05