3D Laplace Equation



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 $u$ 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 $u$ 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 $u;$ 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}$ .


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).


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


  Collocation BEM

  Galerkin BEM



[1] S. Nintcheu Fata.
     Explicit expressions for 3D boundary integrals in potential theory.
     Int. J. Num. Meth. Eng., 78(1):32-47, 2009.