Collocation BEM for 3D Poisson Equation

Download CBEM_POI, a package for solving the 3D Poisson equation based on a piecewise constant Collocation Boundary Element Method, and employing a boundary-only discretization technique.


Piecewise constant collocation method

The functions $u$ and $t$ are assumed to be constant over each boundary element. It can be shown (see cbem_poiGuide.pdf provided in the package CBEM_POI) that the potentials

$\displaystyle g_j(\boldsymbol{x}) = \int_{\Gamma _{\!j}}G(\boldsymbol{x},\boldsymbol{y}) {\rm d}\Gamma _{\!\boldsymbol{y}},$   and$\displaystyle \quad h_j(\boldsymbol{x}) = \int_{\Gamma _{\!j}}\boldsymbol{H}(\b...
...{y}) {\rm d}\Gamma _{\!\boldsymbol{y}},\quad\boldsymbol{x}\!\in\!\mathbb{R}^3,$   

due to a uniform source distribution over a flat triangle $\Gamma _j,$ can be employed to successfully compute $u$ and $t$ on $\Gamma =\bigcup \overline{\Gamma }_{\!j}$. In addition, these potentials can be utilized to effectively calculate $u$ at interior points $\boldsymbol{x}\!\in\!\Omega .$ The analytic expressions for$g_j(\boldsymbol{x})$ and$h_j(\boldsymbol{x})$ over a flat triangle are given in [1]. Moreover, due to its boundary representation, the Newton potential can also be effectively evaluated without the need of a volume-fitted mesh [2]. A brief description of the boundary-only discretization of the Newton potential can be found in the user guide cbem_poiGuide.pdf provided in the package CBEM_POI.



[1] S. Nintcheu Fata.
     Explicit expressions for 3D boundary integrals in potential theory.
     Int. J. Num. Meth. Eng., 78(1):32-47, 2009.
[2] S. Nintcheu Fata.
     Treatment of domain integrals in boundary element methods.
     Appl. Num. Math., 62(6):720-735, 2012.