For the things that follow we will deduce a formula for integrating by parts being based on the well-known Divergence Theorem
for some differentiable function
. In Equation 1.1
is the sufficiently smooth boundary of the domain of interest
and
is its outer normal vector. With the definition
we end up with Green's first
identity
where
denotes the normal derivative of the function
. 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
from Equation 1.1 to be composed
by the product of the gradient of
times the function
. The gives a similar equation like Equation 1.3 but
with swapped differential operators now acting on
rather than on
. Subtracting this formula from the Green's first identity yields
Green's second
identity
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.