## 3D Poisson Equation

# 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

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)

## Approaches:

Galerkin BEM

# References

- [1] S. Nintcheu Fata.

Treatment of domain integrals in boundary element methods.

*Appl. Num. Math.*, 62(6):720-735, 2012.