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
![$\displaystyle \Delta u(\boldsymbol{x})+b(\boldsymbol{x})=0$](/Imgs/ImgPoisson/poi_img6.png)
![$\displaystyle \quad\nabla^2u(\boldsymbol{x})+b(\boldsymbol{x})=0$](/Imgs/ImgPoisson/poi_img7.png)
![$\displaystyle \quad \frac{\partial^2u}{\partial x_1^2} + \frac{\partial^2u}{\partial x_2^2}+ \frac{\partial^2u}{\partial x_3^2} + b(x_1,x_2,x_3)= 0$](/Imgs/ImgPoisson/poi_img8.png)
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
![$\displaystyle u(\boldsymbol{x}) = \int_{\Gamma }G(\boldsymbol{x},\boldsymbol{y}...
...symbol{y}) {\rm d}\Omega _{\boldsymbol{y}}, \quad\boldsymbol{x}\!\in\!\Omega ,$](/Imgs/ImgPoisson/poi_img11.png)
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
![$\displaystyle G(\boldsymbol{x},\boldsymbol{y})=\frac1{4\pi}\frac1{\Vert\boldsym...
...mbol{x},\boldsymbol{y}\!\in\!\mathbb{R}^3,\quad\boldsymbol{x}\ne\boldsymbol{y}.$](/Imgs/ImgLaplace/img17.png)
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
![$\displaystyle \int_{\Omega }G(\boldsymbol{x},\boldsymbol{y}) b(\boldsymbol{y})...
...{y}) {\rm d}\Gamma _{\!\boldsymbol{y}},\quad\boldsymbol{x}\!\in\!\mathbb{R}^3,$](/Imgs/ImgPoisson/poi_img21.png)
where
![$\displaystyle \mathcal{F}(\boldsymbol{x},\boldsymbol{y})=\int_{0}^{1}z h(\bold...
...oldsymbol{y}-\boldsymbol{x})) dz,\quad \boldsymbol{y}\!\in\!\overline{\Omega }$](/Imgs/ImgPoisson/poi_img22.png)
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
![$\displaystyle h(\boldsymbol{x}) = \left\{ \begin{array}{rl} \!\!\!b(\boldsymbol...
...ymbol{x}\!\in\! \mathbb{R}^3\!\setminus\!\overline{\Omega } \end{array} \right.$](/Imgs/ImgPoisson/poi_img26.png)
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).
![\includegraphics[height=1.5in]{ct896g}](/Imgs/ImgLaplace/img21.png)
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.