  Solid fuel ignition models, for which the dynamics of the temperature is
independent of the single-species mass fraction, attempt to follow the dynamics
of an explosive event. Such models may take the form of singular, degenerate,
reaction-diffusion equations of the quenching type, that is, the temporal
derivative blows up in finite time while the solution remains bounded.
Theoretical and numerical investigations have proved difficult for even the
simplest of geometries and mathematical degeneracies. Quenching domains are
known to exist for piecewise smooth boundaries, but often lack theoretical
estimates. Rectangular geometries have been primarily studied. Here, this
acquired knowledge is utilized to determine new theoretical estimates to
quenching domains for arbitrary piecewise, smooth, connected geometries.
Elliptical domains are of primary interest and a Peaceman-Rachford splitting
algorithm is then developed that employs temporal adaptation and nonuniform
grids. Rigorous numerical analysis ensures numerical solution monotonicity,
positivity, and linear stability of the proposed algorithm. Simulation
experiments are provided to illustrate the accomplishments.
