  A method is developed within an adaptive framework to solve quasilinear
diffusion problems with internal and possibly boundary layers starting from a
coarse mesh. The solution process is assumed to start on a mesh where the
problem is badly resolved, and approximation properties of the exact problem
and its corresponding finite element solution do not hold. A sequence of
stabilized and inexact partial solves allow the mesh to be refined to capture
internal layers while an approximate solution is built eventually leading to an
accurate approximation of both the problem and its solution. The innovations in
the current work include a closed form definition for the numerical dissipation
and inexact scaling parameters on each mesh refinement, as well as a
convergence result for the residual of the discrete problem. Numerical
experiments demonstrate the method on a range of problems featuring steep
internal layers and high solution dependent frequencies of the diffusion
coefficients.
