# Background

Let be a bounded, simply or multiply connected domain in with a Lipschitz boundary The Laplace equation for a scalar function in is given by

or   or

A solution of the Laplace equation admits an integral representation, known as the Green's representation formula, expressed as

where is the unit normal to directed towards the exterior of is the flux associated with the potential and the kernels and are given respectively by

Here is the usual Euclidean norm in defined as .

# Solution via a boundary element method

To approximately solve the Laplace equation via a Boundary Element Method (BEM), the surface is usually discretized into flat triangles using a mesh generation software (e.g. CUBIT).

With reference to [1], the potential and flux are assumed to have a polynomial variation over each triangle (boundary element)

# References

[1] S. Nintcheu Fata.
Explicit expressions for 3D boundary integrals in potential theory.
Int. J. Num. Meth. Eng., 78(1):32-47, 2009.