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Singularity matrix

Until now we have discussed singularities depending only on test functions which vanish. Suppose we have two singularities and , depending respectively on test functions and . Namely, assume that vanishes at both and , while vanishes at only . 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 singularities and test functions which are needed to detect and locate the singularities. Then let be the matrix, such that:

| (30)