## 3D Laplace Equation

- Details

# 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

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)

## Approaches:

# References

- [1] S. Nintcheu Fata.

Explicit expressions for 3D boundary integrals in potential theory.

*Int. J. Num. Meth. Eng.*, 78(1):32-47, 2009.