At this point we need to comment on trace operators which are needed when deriving the respective boundary integral operators. Actually, one trace operator, namely the Neumann trace, has been already introduced in Equation 1.4. Nevertheless, to give a more common notion of those trace operators we will refer them by ${\gamma}_{0}$ an ${\gamma}_{1}$ from now on. For the Laplace equation they are defined as

**Equation 1.10. **

$\phantom{\rule{1.00em}{0ex}}\left({\gamma}_{0}\phi \right)\left(x\right)={\displaystyle \underset{\Omega \ni \tilde{x}\to x\in \partial \Omega}{\mathrm{lim}}}\phi \left(\tilde{x}\right)\phantom{\rule{0.167em}{0ex}},\phantom{\rule{2.00em}{0ex}}\left({\gamma}_{1}\phi \right)\left(x\right)={\gamma}_{0}\langle \mathbf{grad}\phi \left(\tilde{x}\right),\mathbf{n}\left(x\right)\rangle \phantom{\rule{0.278em}{0ex}}.$

With those trace operators the representation formula becomes

**Equation 1.11. **

$\phantom{\rule{1.00em}{0ex}}\phi \left(\tilde{x}\right)={\displaystyle \underset{\partial \Omega}{\int}}\left({\gamma}_{1}\phi \right)\left(y\right)\phantom{\rule{0.167em}{0ex}}\left({\gamma}_{0}U\right)(y-\tilde{x})-\left({\gamma}_{0}\phi \right)\left(y\right)\phantom{\rule{0.167em}{0ex}}\left({\gamma}_{1}U\right)(y-\tilde{x})\phantom{\rule{0.167em}{0ex}}d{s}_{y}\phantom{\rule{1.00em}{0ex}}\forall \tilde{x}\in {\mathbb{R}}^{3}\setminus \partial \Omega ,\phantom{\rule{0.167em}{0ex}}y\in \partial \Omega \phantom{\rule{0.278em}{0ex}}.$

Finally, the boundary integral operators are obtained by the application
of the Dirichlet trace
${\gamma}_{0}$
and the Neumann trace
${\gamma}_{1}$
to the representation formula. If the domain
$\Omega $
where the analysis takes place is an interior domain we end up with the
two following boundary integral equations

**Equation 1.12. **

$\phantom{\rule{1.00em}{0ex}}{\gamma}_{0}\phi =V{\gamma}_{1}\phi +(\genfrac{}{}{0.1ex}{}{1}{2}I-K){\gamma}_{0}\phi $

**Equation 1.13. **

$\phantom{\rule{1.00em}{0ex}}{\gamma}_{1}\phi =(\genfrac{}{}{0.1ex}{}{1}{2}I+{K}^{\prime}){\gamma}_{1}\phi +D{\gamma}_{0}\phi \phantom{\rule{0.278em}{0ex}}.$

The above equations are usually denoted as the first and the second
boundary integral equation, respectively. In case of an exterior
boundary value problem those two equations take a similar form as above
but with some sign changes in front of the operators. The operator
$I$
simply denotes the identity operator. It represents the jump in normal
direction across the boundary. Since we will only address Galerkin
Boundary Element schemes the factor in front of the identity can be
safely set to 0.5. All the remaining operators in Equation 1.12 and in Equation 1.13 are the most
common boundary integral operators. They are usually named as the single
layer operator, the double layer operator, the adjoint double layer
operator, and as the hypersingular integral operator. Their definitions
are

**Equation 1.14. **

$\phantom{\rule{1.00em}{0ex}}\begin{array}{cc}\hfill & \left(Vw\right)\left(x\right)={\displaystyle \underset{\partial \Omega}{\int}}U(y-x)\phantom{\rule{0.167em}{0ex}}w\left(y\right)\phantom{\rule{0.167em}{0ex}}d{s}_{y}\hfill \\ & \left(Kw\right)\left(x\right)={\displaystyle \underset{\u03f5\to 0}{\mathrm{lim}}}{\displaystyle \underset{\partial \Omega :|y-x|\ge \u03f5}{\int}}\left({\gamma}_{1,y}U\right)(y-x)\phantom{\rule{0.167em}{0ex}}w\left(y\right)\phantom{\rule{0.167em}{0ex}}d{s}_{y}\hfill \\ & \left({K}^{\prime}w\right)\left(x\right)={\displaystyle \underset{\u03f5\to 0}{\mathrm{lim}}}{\displaystyle \underset{\partial \Omega :|y-x|\ge \u03f5}{\int}}\left({\gamma}_{1,x}U\right)(y-x)\phantom{\rule{0.167em}{0ex}}w\left(y\right)\phantom{\rule{0.167em}{0ex}}d{s}_{y}\hfill \\ & \left(Dw\right)\left(x\right)=-{\gamma}_{1,x}{\displaystyle \underset{\u03f5\to 0}{\mathrm{lim}}}{\displaystyle \underset{\partial \Omega :|y-x|\ge \u03f5}{\int}}\left({\gamma}_{1,y}U\right)(y-x)\phantom{\rule{0.167em}{0ex}}w\left(y\right)\phantom{\rule{0.167em}{0ex}}d{s}_{y}\phantom{\rule{0.278em}{0ex}}.\hfill \end{array}$

Special attention has to be paid to the fact that the fundamental solution is singular at the
origin. Usually, this singularity is common for boundary integral
operators. Hence, the definitions above contain some special limiting
processes already indicating the different kind of singularities those
operators consist of. Since the single layer potential is weakly
singular the integral exists in an improper sense. These singularities
are quite easy to tackle. The situation changes for the double layer
potential as well as for the adjoint double layer potential. In general,
the occurring integrals have to be understand as strongly singular
integrals. They are also referred to as *Cauchy principal
values* when a limiting process is applied which approaches
the singularity uniformly from every direction. As the name already
induces the hypersingular integral operator is somehow the
*worst case scenario*. These kind of singularities
are commonly denoted as *finite part
integrals*. Within a computational scheme integrals of this
kind are hardly computable in general. Fortunately there exists a remedy
for this drawback. Within Galerkin schemes integration by parts can be
used in order to express the associated hypersingular bilinear form via
the single layer operator. Examples will be given within the tutorials.

Note that the integral operators do not necessarily need to act on
some Dirichlet data
${\gamma}_{0}\phi $
or Neumann data
${\gamma}_{1}\phi $
. In fact, the definition in Equation 1.14 is given for some more or less arbitrary density
function
$w$
. Hence, one could also think of Boundary Element Methods which are not
build upon the representation formula. In fact, those methods exist and
they are the so-called *indirect Boundary Element
Methods*. Contrary, schemes that utilise the boundary integral
equations Equation 1.12 and/or Equation 1.13 are widely known as *direct Boundary
Element Methods*.

In the tutorial part of this documentation we will embed the deduced boundary integral operators into Galerkin schemes. In contrast to other possible discretisation schemes like, e.g., Collocation or Nyström methods, the Galerkin methods have the advantage of preserving essential properties of the underlying boundary integral operators also on a discrete level.