Electromagnetic fields, like many other physical processes, are governed by
partial differential equations
PDEs . Hence the
numerical methods for solving such problems can be classed with other methods
of solving PDEs, such as the
finite difference method ,
finite element method ,
and the
boundary element method .
A further important method method is the transmission line
method or transmission line matrix method
( TLM). This method
involves decribing the distributed electromagnetics field
as a network of circuits.
Most fundamentally, we consider the electromagnetic
field as a time-dependent or transient problem in which
case the FD-TD method and TLM methods can be applied.
Often, we can make simplifications. If the electromagentic
field is periodic then we can write the electromagnetic
properties as the sum of sinusoidal functions and the
maxwell equations
can be simplified to Helmholtz problems.
In this case a variety of methods can be applied, in particular
the finite element method, the finite difference method and
the boundary element method.
If the electric field is static (electrostatics) or the current is steady (magnetostatics) then the field is governed by the Laplace equation and again the finite element method, the finite difference method and the boundary element method are appropriate as methods of solution.
In general, in AC or DC problems, the finite element and finite difference methods are the better choice if there are many or continuous changes in material. If the material domian is easily defined then the boundary element method is appropriate.
In general electromagnetics, the finite difference method takes a special form, called the finite-difference time-domain method or (FDTD). The method involves dividing the domain of interest into an interlocking lattice of cubes (in 3D) for with the electric and magnetic field components taking values at particular points on each cube.
The Boundary Element Method is also known as the Method of Moments. It is particularly applicable to the Helmholtz and Laplace equations that arise in AC and DC electromagnetics. The web site Boundary Element Method contains manuals and methods for the solution of the Laplace and Helmholtz Equations by the BEM and hence they can be used for electromagnetic problems.
The Finite Element Method can be applied to the Helmholz or Laplace equations that arise in AC and DC electromagnetics. See the Books by Volakis (1998) and Jin (1993) for introductions to the method at science-books.net .