Trace Operators and Boundary Integral Operators

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.

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

With those trace operators the representation formula becomes

Equation 1.11.

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

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.

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

and

Equation 1.13.

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

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.

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

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.