Analytical regularization

In physics and applied mathematics, analytical regularization is a technique used to convert boundary value problems which can be written as Fredholm integral equations of the first kind involving singular operators into equivalent Fredholm integral equations of the second kind. The latter may be easier to solve analytically and can be studied with discretization schemes like the finite element method or the finite difference method because they are pointwise convergent. In computational electromagnetics, it is known as the method of analytical regularization. It was first used in mathematics during the development of operator theory before acquiring a name.^[1]

Method

Analytical regularization proceeds as follows. First, the boundary value problem is formulated as an integral equation. Written as an operator equation, this will take the form

${\displaystyle GX=Y}$

with ${\displaystyle Y}$ representing boundary conditions and inhomogeneities, ${\displaystyle X}$ representing the field of interest, and ${\displaystyle G}$ the integral operator describing how Y is given from X based on the physics of the problem. Next, ${\displaystyle G}$ is split into ${\displaystyle G_{1}+G_{2}}$, where ${\displaystyle G_{1}}$ is invertible and contains all the singularities of ${\displaystyle G}$ and ${\displaystyle G_{2}}$ is regular. After splitting the operator and multiplying by the inverse of ${\displaystyle G_{1}}$, the equation becomes

${\displaystyle X+G_{1}^{-1}G_{2}X=G_{1}^{-1}Y}$

or

${\displaystyle X+AX=B}$

which is now a Fredholm equation of the second type because by construction ${\displaystyle A}$ is compact on the Hilbert space of which ${\displaystyle B}$ is a member.

In general, several choices for ${\displaystyle \mathbf {G} _{1}}$ will be possible for each problem.^[1]