# Boundary Element Method

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:
${\displaystyle \nabla ^{2}u=0}$
• Poisson equation:
${\displaystyle \nabla ^{2}u=-f}$
• Helmholtz equation or Diffusion equation in frequency domain:
${\displaystyle \nabla ^{2}u-k^{2}u=-f}$

## 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:

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

where

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