3D Poisson Equation



Let $\Omega \!\subset\!\mathbb{R}^3$ be a bounded, simply or multiply connected domain in with a Lipschitz boundary The Poisson equation for a scalar function $u$ in is given by

$\displaystyle \Delta u(\boldsymbol{x})+b(\boldsymbol{x})=0$   or$\displaystyle \quad\nabla^2u(\boldsymbol{x})+b(\boldsymbol{x})=0$   or$\displaystyle \quad \frac{\partial^2u}{\partial x_1^2} + \frac{\partial^2u}{\partial x_2^2}+ \frac{\partial^2u}{\partial x_3^2} + b(x_1,x_2,x_3)= 0$   

where $ b$ represents a source function prescribed on $\overline{\Omega}\!=\!\Omega\cup\Gamma$.

 A solution $u$ of the Poisson equation admits an integral representation, known as the Green's representation formula, expressed as

$\displaystyle u(\boldsymbol{x}) = \int_{\Gamma }G(\boldsymbol{x},\boldsymbol{y}...
...symbol{y}) {\rm d}\Omega _{\boldsymbol{y}}, \quad\boldsymbol{x}\!\in\!\Omega ,$   

where $ \boldsymbol{n}$ is the unit normal to $\Gamma$ directed towards the exterior of $ \Omega ;$ $ t=\partial u/\partial n$ is the flux associated with the function$u;$ and the kernels $G$ and $ \boldsymbol{H}$are given respectively by

$\displaystyle G(\boldsymbol{x},\boldsymbol{y})=\frac1{4\pi}\frac1{\Vert\boldsym...

Here $ \Vert\boldsymbol{x}\Vert$ is the usual Euclidean norm in $ \mathbb{R}^3$ defined as $ \Vert\boldsymbol{x}\Vert=\sqrt{x_1^2+x_2^2+x_3^2}$. In addition, it was established in [1] that the Newton potential admits a boundary representation as

$\displaystyle \int_{\Omega }G(\boldsymbol{x},\boldsymbol{y}) b(\boldsymbol{y})...
...{y}) {\rm d}\Gamma _{\!\boldsymbol{y}},\quad\boldsymbol{x}\!\in\!\mathbb{R}^3,$   


$\displaystyle \mathcal{F}(\boldsymbol{x},\boldsymbol{y})=\int_{0}^{1}z h(\bold...
...oldsymbol{y}-\boldsymbol{x})) dz,\quad \boldsymbol{y}\!\in\!\overline{\Omega }$   

with$ h$ denoting an extension of the source function $ b$ into any ball centered at $ \boldsymbol{x}$ and containing $ \overline{\Omega }$ . In particular, a continuation $ h$ of the source function $b$ can be specified as

$\displaystyle h(\boldsymbol{x}) = \left\{ \begin{array}{rl} \!\!\!b(\boldsymbol...
...ymbol{x}\!\in\! \mathbb{R}^3\!\setminus\!\overline{\Omega } \end{array} \right.$   

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 $\Gamma $ is usually discretized into flat triangles $\Gamma _j$ using a mesh generation software (e.g. CUBIT).


With reference to [1], the functions $u$ and $t$ are assumed to have a polynomial variation over each triangle (boundary element)$\Gamma _j.$


  Collocation BEM

  Galerkin BEM



[1] S. Nintcheu Fata.
     Treatment of domain integrals in boundary element methods.
     Appl. Num. Math., 62(6):720-735, 2012.