Download CBEM_POI, a complete package for solving the 3D Poisson equation using a boundary-only discretization.
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
represents a source function prescribed on
of the Poisson equation admits an integral representation, known as the Green's representation formula, expressed as
is the unit normal to
directed towards the exterior
is the flux associated with the function
and the kernels
are given respectively by
is the usual Euclidean norm in
. In addition, it was established in  that the Newton potential admits a boundary representation as
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.
To approximately solve the Poisson equation via a Boundary Element Method (BEM), the surface
is usually discretized into flat
using a mesh a mesh generation software (e.g. CUBIT
With reference to , the functions
are assumed to have a polynomial variation over each triangle (boundary element)
- S. Nintcheu Fata.
Treatment of domain integrals in boundary element methods.
Appl. Num. Math., DOI:10.1016/j.apnum.2010.07.003, 2010.