Derivation of the Boundary Integral Equations
In order to derive the boundary integral equations used in BEM, you take the limit of the volume potential operators as they approach the boundary. This limit process is nontrivial since on the boundary the integrals are evaluated for x ≠ y. The limit process is represented by a trace operator γ0. In addition, the normal derivative trace operator γ1 = γ0∂n is needed.
Operating with the trace operator γ0 on the (volume) representation for u gives
on ∂Ω
where γ0u = u on ∂Ω (the trace of u is u).
The boundary integral operator V
is called the single-layer boundary potential and has the same form as its corresponding volume potential:
on ∂Ω
Define the double-layer boundary potential operator as:
on ∂Ω
The relationship between the volume and boundary versions of the double-layer potential operator is nontrivial:
where the additional term comes from, roughly speaking, “cutting the singularity in half” on the boundary. This results in the following boundary integral equation for u:
on ∂Ω
where, in order to simplify the notation,
is used for the normal flux. In addition, operator notation is here used, for example, K(u) = Ku.
In the COMSOL Multiphysics implementation of BEM, the normal flux ψ is represented as a dependent variable alongside u.
To derive a boundary integral equation for ψ, operate with the trace operator γ1 = γ0∂n on the representation for u:
on ∂Ω
Define the adjoint double-layer boundary potential operator as
on ∂Ω
Again, the relationship between the volume and boundary potential operators is nontrivial as follows:
on ∂Ω
Define the hypersingular integral operator as
on ∂Ω
This results in the following boundary integral equation for ψ:
on ∂Ω