3D Laplace Equation

Details

Background

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

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

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

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

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 potential and the kernels and $\boldsymbol{H}$ are given respectively by

$\displaystyle G(\boldsymbol{x},\boldsymbol{y})=\frac1{4\pi}\frac1{\Vert\boldsym... ...mbol{x},\boldsymbol{y}\!\in\!\mathbb{R}^3,\quad\boldsymbol{x}\ne\boldsymbol{y}.$

Here $\Vert\boldsymbol{x}\Vert$ is the usual Euclidean norm in $\mathbb{R}^3$ defined as $\Vert\boldsymbol{x}\Vert=\sqrt{x_1^2+x_2^2+x_3^2}$ .

Solution via a boundary element method

To approximately solve the Laplace 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).