## Green's identities

For the things that follow we will deduce a formula for integrating by parts being based on the well-known Divergence Theorem

Equation 1.1.

$\phantom{\rule{1.00em}{0ex}}\underset{\Omega }{\int }\mathrm{div}\mathbf{u}\phantom{\rule{0.278em}{0ex}}dx=\underset{\partial \Omega }{\int }〈\mathbf{u},\mathbf{n}〉\phantom{\rule{0.278em}{0ex}}d{s}_{x}$

for some differentiable function $\mathbf{u}$ . In Equation 1.1 $\partial \Omega$ is the sufficiently smooth boundary of the domain of interest $\Omega$ and $\mathbf{n}$ is its outer normal vector. With the definition

Equation 1.2.

$\phantom{\rule{1.00em}{0ex}}\mathbf{u}=\psi \mathbf{grad}\phi \phantom{\rule{1.00em}{0ex}}⟹\phantom{\rule{1.00em}{0ex}}\mathrm{div}\mathbf{u}=\psi \Delta \phi +〈\mathbf{grad}\psi ,\mathbf{grad}\phi 〉$

we end up with Green's first identity

Equation 1.3.

$\phantom{\rule{1.00em}{0ex}}\underset{\Omega }{\int }\psi \Delta \phi +〈\mathbf{grad}\psi ,\mathbf{grad}\phi 〉dx=\underset{\partial \Omega }{\int }\psi \genfrac{}{}{0.1ex}{}{\partial \phi }{\partial \mathbf{n}}d{s}_{x}$

where

Equation 1.4.

$\phantom{\rule{1.00em}{0ex}}\genfrac{}{}{0.1ex}{}{\partial \phi }{\partial \mathbf{n}}=\underset{\Omega \ni \stackrel{˜}{x}\to x\in \partial \Omega }{\mathrm{lim}}〈\mathbf{grad}\phi \left(\stackrel{˜}{x}\right),\mathbf{n}\left(x\right)〉$

denotes the normal derivative of the function $\phi$ . Green's first identity is perfectly suited to be used as starting point for the derivation of Finite Element Methods — at least for the Laplace equation. Next, we consider the function $\mathbf{u}$ from Equation 1.1 to be composed by the product of the gradient of $\psi$ times the function $\phi$ . The gives a similar equation like Equation 1.3 but with swapped differential operators now acting on $\psi$ rather than on $\phi$ . Subtracting this formula from the Green's first identity yields Green's second identity

Equation 1.5.

$\phantom{\rule{1.00em}{0ex}}\underset{\Omega }{\int }\psi \phantom{\rule{0.167em}{0ex}}\Delta \phi -\Delta \psi \phantom{\rule{0.167em}{0ex}}\phi \phantom{\rule{0.167em}{0ex}}dx=\underset{\partial \Omega }{\int }\psi \genfrac{}{}{0.1ex}{}{\partial \phi }{\partial \mathbf{n}}-\genfrac{}{}{0.1ex}{}{\partial \psi }{\partial \mathbf{n}}\phi \phantom{\rule{0.167em}{0ex}}d{s}_{x}$

where the symmetric parts of each formula have been cancelled out. Contrary to Green's first identity on which Finite Element Methods can be build Green's second identity forms somehow the basis for the construction of Boundary Element Methods as will be shown in the following sections.