Finite difference computations that involve spatial adaptation commonly employ an
equidistribution principle. In these cases, a new mesh is constructed such that a
given monitor function is equidistributed in some sense. Typical choices of the
monitor function involve the solution or one of its many derivatives. This straightforward
concept has proven to be extremely effective and practical. However, selections of core
monitoring functions are often challenging
and crucial to the computational success. This paper concerns six different
designs of the monitoring function that targets a highly nonlinear partial
differential equation that exhibits both quenching-type and degeneracy singularities.
While the first four monitoring strategies are within the so-called
primitive regime, the rest belong to a later category of the
modified type, which requires the priori knowledge of certain
important quenching solution characteristics. Simulated examples are given
to illustrate our study and conclusions.