  We describe and analyze preconditioned steepest descent (PSD) solvers for
fourth and sixth-order nonlinear elliptic equations that include p-Laplacian
terms on periodic domains in 2 and 3 dimensions. The highest and lowest order
terms of the equations are constant-coefficient, positive linear operators,
which suggests a natural preconditioning strategy. Such nonlinear elliptic
equations often arise from time discretization of parabolic equations that
model various biological and physical phenomena, in particular, liquid
crystals, thin film epitaxial growth and phase transformations. The analyses of
the schemes involve the characterization of the strictly convex energies
associated with the equations. We first give a general framework for PSD in
generic Hilbert spaces. Based on certain reasonable assumptions of the linear
pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD
iteration. We then apply the general the theory to the fourth and sixth-order
problems of interest, making use of Sobolev embedding and regularity results to
confirm the appropriateness of our pre-conditioners for the regularized
p-Lapacian problems. Our results include a sharper theoretical convergence
result for p-Laplacian systems compared to what may be found in existing works.
We demonstrate rigorously how to apply the theory in the finite dimensional
setting using finite difference discretization methods. Numerical simulations
for some important physical application problems -- including thin film epitaxy
with slope selection and the square phase field crystal model -- are carried
out to verify the efficiency of the scheme.
