  We consider the design and analysis of numerical methods for approximating
positive solutions to nonlinear geometric elliptic partial differential
equations containing critical exponents. This class of problems includes the
Yamabe problem and the Einstein constraint equations, which simultaneously
contain several challenging features: high spatial dimension n >= 3, varying
(potentially non-smooth) coefficients, critical (even super-critical)
nonlinearity, non-monotone nonlinearity (arising from a non-convex energy), and
spatial domains that are typically Riemannian manifolds rather than simply open
sets in Rn. These problems may exhibit multiple solutions, although only
positive solutions typically have meaning. This creates additional complexities
in both the theory and numerical treatment of such problems, as this feature
introduces both non-uniqueness as well as the need to incorporate an inequality
constraint into the formulation. In this work, we consider numerical methods
based on Galerkin-type discretization, covering any standard bases construction
(finite element, spectral, or wavelet), and the combination of a barrier method
for nonconvex optimization and global inexact Newton-type methods for dealing
with nonconvexity and the presence of inequality constraints. We first give an
overview of barrier methods in non-convex optimization, and then develop and
analyze both a primal barrier energy method for this class of problems. We then
consider a sequence of numerical experiments using this type of barrier method,
based on a particular Galerkin method, namely the piecewise linear finite
element method, leverage the FETK modeling package. We illustrate the behavior
of the primal barrier energy method for several examples, including the Yamabe
problem and the Hamiltonian constraint.
