IntETec > 3D Poisson Equation
3D Poisson Equation

Download CBEM_POI, a complete package for solving the 3D Poisson equation using a boundary-only discretization.

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 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., DOI:10.1016/j.apnum.2010.07.003, 2010.