# Boundary Element Method

(Redirected from BEM)

Boundary Element Method (BEM) is a numerical method of solving partial differential equations (PDEs) presented in the integral form known as Boundary Integral Equation (BIE).

## Partial Differential Equations (PDEs)

The BEMLAB library has been implemented to support following PDEs:

• Laplace equation:
$\nabla ^{2}u=0$ • Poisson equation:
$\nabla ^{2}u=-f$ • Helmholtz equation or Diffusion equation in frequency domain:
$\nabla ^{2}u-k^{2}u=-f$ ## Boundary Integral Equation (BIE)

Boundary Integral Equation (BIE) is the basic equation which is solved by BEM. PDEs have to be presented in the form of the following BIE:

$c_{i}(\mathbf {r} _{i})u_{i}(\mathbf {r} _{i})+\int \limits _{\Gamma }u(\mathbf {r} ){\frac {\partial G(\mathbf {r} _{i},\mathbf {r} )}{\partial n}}d\Gamma (\mathbf {r} )=\int \limits _{\Gamma }q(\mathbf {r} )G(\mathbf {r} _{i},\mathbf {r} )d\Gamma (\mathbf {r} )+\int \limits _{\Omega }f(\mathbf {r} _{\Omega })G(\mathbf {r} _{i},\mathbf {r} _{\Omega })d\Omega (\mathbf {r} _{\Omega })$ where

• $\displaystyle \Omega$ - domain of the problem, where PDE is being solved,
• $\Gamma$ - boundary of the investigated domain $\Omega$ ,
• $u$ - potential,
• $q$ - normal derivative of potential,
• $G$ - Green function, also known as fundamental solution,
• $f$ - domain function