Singularity matrix

Until now we have discussed singularities depending only on test functions which vanish. Suppose we have two singularities $S_1$ and $S_2$, depending respectively on test functions $\phi_1$ and $\phi_2$. Namely, assume that $\phi_1$ vanishes at both $S_1$ and $S_2$, while $\phi_2$ vanishes at only $S_2$. Therefore we need a possibility to represent singularities using non-vanishing test functions.

To represent all singularities we will introduce a singularity matrix (as in [26]). This matrix is a compact way to describe the relation between the singularities and all test functions.

Suppose we are interested in $n_s$ singularities and $n_t$ test functions which are needed to detect and locate the singularities. Then let $S$ be the $n_s\times n_t$ matrix, such that:

\begin{displaymath}
S_{ij} = \left\{ \begin{array}{cl}
0 & \mbox{singularity ...
...ularity $i$: ignore test function $j$}.
\end{array} \right.
\end{displaymath} (30)