3D Poisson Equation

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

Background

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

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

where

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}......dsymbol{y}) {\rm d}\Gamma _{\!\boldsymbol$

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......mbol{x},\boldsymbol{y}\!\in\!\mathbb{R}^3$

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

where

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

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 a mesh generation software (e.g. CUBIT).

$\includegraphics[height=1.5in]{ct896g}$

With reference to [1], the functions $u$

and

are assumed to have a polynomial variation over each triangle (boundary element) $\Gamma _j.$

Approaches:

Collocation BEM

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.