A

numerical technique useful for solving

integral equations, the

unknown function being the

variable of

integration.

Typically the unknown function is expanded into a series of N basis functions which conform to the region of solution. These basis functions are generally chosen so that they satisfy the particulars of the physics under study (be it a study in fluid dynamics or electromagnetics), and also the boundary conditions such as the divergence and curl of the unknown quantity.

Once this basis function is inserted into the integral equation, typically you are left with one equation in N unknowns. To convert it into a linear system of N equations in N unknowns, the resulting equation is tested using a testing function over each of the N expansion regions.

The resulting matrix system is generally of the form Ax=B, and can be solved by efficient matrix solution algortihms on a computer.

The drawback of the Method of Moments is that when the
number of unknowns N grows to be large, the matrix A becomes very large because it is size N x N. Solving this matrix system can be an extremely time consuming operation. Matrix solvers that use factorization require CPU time on the order of N^{3}, which grows to astronomical proportions when N gets large. When N grows to be tens or hundreds of thousands (or even millions), you are suddenly in need of lots of gigabytes of memory and a hefty number of processors for this task. (This requirement has been reduced to some degree by highly clever MoM techniques such as the Fast Multipole Method)

Another drawback in the MoM is in the generation of the matrix system. These generally involve numerical integrations which can sometimes be very challenging. In Computational Electromagnetics, these integrations typically contain Green's Functions as the system kernel, which are singular when the testing points are close to the source points. This can make the implementation of these expressions in software a pain in the ass (see also fuck me harder).

A classic book on applications of the MoM to Electromagnetics is : R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan Company, 1968.