  The conformal formulation of the Einstein constraint equations is first
reviewed, and we then consider the design, analysis, and implementation of
adaptive multilevel finite element-type numerical methods for the resulting
coupled nonlinear elliptic system. We derive weak formulations of the coupled
constraints, and review some new developments in the solution theory for the
constraints in the cases of constant mean extrinsic curvature (CMC) data,
near-CMC data, and arbitrarily prescribed mean extrinsic curvature data. We
then outline some recent results on a priori and a posteriori error estimates
for a broad class of Galerkin-type approximation methods for this system which
includes techniques such as finite element, wavelet, and spectral methods. We
then use these estimates to construct an adaptive finite element method (AFEM)
for solving this system numerically, and outline some new convergence and
optimality results. We then describe in some detail an implementation of the
methods using the FETK software package, which is an adaptive multilevel finite
element code designed to solve nonlinear elliptic and parabolic systems on
Riemannian manifolds. We finish by describing a simplex mesh generation
algorithm for compact binary objects, and then look at a detailed example
showing the use of FETK for numerical solution of the constraints.
