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Singularity matrix

 

Until now we have discussed singularities depending only on test functions which vanish. Suppose we have two singularities S1 and S2, depending respectively on test functions f1 and f2. Namely, assume that f1 vanishes at both S1 and S2, while f2 vanishes at only S2. 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 [11]). This matrix is a compact way to describe the relation between the singularities and all test functions.

Suppose we are interested in ns singularities and nt test functions which are needed to detect and locate the singularities. Then let S be the ns×nt matrix, such that:

Sij = ì
ï
í
ï
î
0
singularity i: test function j must vanish,
1
singularity i: test function j must not vanish,
otherwise
singularity i: ignore test function j.
(30)