# Background

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

or   or

where represents a source function prescribed on .

A solution of the Poisson 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 function and the kernels and are given respectively by

Here is the usual Euclidean norm in defined as . In addition, it was established in [1] that the Newton potential admits a boundary representation as

where

with denoting an extension of the source function into any ball centered at and containing . In particular, a continuation of the source function can be specified as

This representation of the Newton potential in term of surface integral allows a numerical solution of the Poisson equation that does not require a volume-fitted mesh.

# Solution via a boundary element method

To approximately solve the Poisson 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 functions and are assumed to have a polynomial variation over each triangle (boundary element)

Galerkin BEM

# References

[1] S. Nintcheu Fata.
Treatment of domain integrals in boundary element methods.
Appl. Num. Math., 62(6):720-735, 2012.