I am developing a BEM code based on a deal.ii tutorial. Consider the Poisson equation
$$ \Delta u=-f\,, $$ and its Green's function $G\left(\mathbf{x},\mathbf{x}'\right)$ with the property $$ \Delta G=-\delta\left(\mathbf{x}-\mathbf{x}'\right)\,. $$ Applying Green's 2nd identity leads to the well known boundary integral equation (BIE) $$ \int\limits_{\Omega}u\left(\mathbf{x}\right)\delta\left(\mathbf{x}-\mathbf{x}'\right)\mathrm{d}V =\int\limits_{\Omega}f\left(\mathbf{x}\right)G\left(\mathbf{x},\mathbf{x}'\right)\mathrm{d}V +\oint\limits_{\partial\Omega}\mathbf{n}\cdot\left[\cdots\right]\mathrm{d}\Gamma $$ I am considering the integral on left hand side of the equation. It is well known that $$ \int\limits_{\Omega}u\left(\mathbf{x}\right)\delta\left(\mathbf{x}-\mathbf{x}'\right)\mathrm{d}V=\alpha\left(\mathbf{x}'\right)u\left(\mathbf{x}'\right)\,, $$ where as the factor $\alpha$, i.e. the fraction of the solid angle, depends on the location of the point $\mathbf{x}'$. Typically it is given by $$ \alpha\left(\mathbf{x}\right)=\begin{cases} 0 & \mathbf{x}\notin\Omega\\ 1 & \mathbf{x}\in\Omega\\ \frac{1}{2} & \mathbf{x}\in\Gamma & \text{smooth boundary}\\ \frac{\theta}{4\pi} & \mathbf{x}\in\Gamma & \text{3D corner} \end{cases} $$ You get these expressions by simple analytic evaluations of the integral. If the BIE is discretized, these analytic expressions may be not be appropiate any more. Therefore, the author of the tutorial states that the fraction of solid angle may be calculate by $$ \alpha\left(\mathbf{x}'\right)=1+\oint\mathbf{n}\cdot\nabla G\left(\mathbf{x},\mathbf{x}'\right)\mathrm{d}\Gamma\,, $$ and in fact uses this term in the discretization.
Finally my question: How can be the equation above can be derived?
examples/step-34/doc/intro.dox
so that others who read it have the benefit of having the information there? $\endgroup$ – Wolfgang Bangerth Jan 14 '16 at 19:22deal.ii
's git repository and modified the fileexamples/step-34/doc/intro.dox
. I am allowed to make a commit and push it to the online repository? $\endgroup$ – sebastian_g Jan 19 '16 at 10:24git help format-patch
to see how to do this. (In the future, the way to do it is to "fork" the repository on your own github account, "clone" your fork onto your own harddrive, make modifications there, push to your fork, and create a pull request from your fork to the main repository.) $\endgroup$ – Wolfgang Bangerth Jan 19 '16 at 14:02