  An adaptive regularization strategy for stabilizing Newton-like iterations on
a coarse mesh is developed in the context of adaptive finite element methods
for nonlinear PDE. Existence, uniqueness and approximation properties are known
for finite element solutions of quasilinear problems assuming the initial mesh
is fine enough. Here, an adaptive method is started on a coarse mesh where the
finite element discretization and quadrature error produce a sequence of
approximate problems with indefinite and ill-conditioned Jacobians. The methods
of Tikhonov regularization and pseudo-transient continuation are related and
used to define a regularized iteration using a positive semidefinite penalty
term. The regularization matrix is adapted with the mesh refinements and its
scaling is adapted with the iterations to find an approximate sequence of
coarse mesh solutions leading to an efficient approximation of the PDE
solution. Local q-linear convergence is shown for the error and the residual in
the asymptotic regime and numerical examples of a model problem illustrate
distinct phases of the solution process and support the convergence theory.
