  Monotone finite difference methods provide stable convergent discretizations
of a class of degenerate elliptic and parabolic Partial Differential Equations
(PDEs). These methods are best suited to regular rectangular grids, which leads
to low accuracy near curved boundaries or singularities of solutions. In this
article we combine monotone finite difference methods with an adaptive grid
refinement technique to produce a PDE discretization and solver which is
applied to a broad class of equations, in curved or unbounded domains which
include free boundaries. The grid refinement is flexible and adaptive. The
discretization is combined with a fast solution method, which incorporates
asynchronous time stepping adapted to the spatial scale. The framework is
validated on linear problems in curved and unbounded domains. Key applications
include the obstacle problem and the one-phase Stefan free boundary problem.
