  Adaptive multilevel finite element methods are developed and analyzed for
certain elliptic systems arising in geometric analysis and general relativity.
This class of nonlinear elliptic systems of tensor equations on manifolds is
first reviewed, and then adaptive multilevel finite element methods for
approximating solutions to this class of problems are considered in some
detail. Two a posteriori error indicators are derived, based on local residuals
and on global linearized adjoint or dual problems. The design of Manifold Code
(MC) is then discussed; MC is an adaptive multilevel finite element software
package for 2- and 3-manifolds. It employs a posteriori error estimation,
adaptive simplex subdivision, unstructured algebraic multilevel methods, global
inexact Newton methods, and numerical continuation methods for the numerical
solution of nonlinear covariant elliptic systems on 2- and 3-manifolds. Some of
the more interesting features of MC are described in detail, including some new
ideas for topology and geometry representation in simplex meshes, and an
unusual partition of unity-based method for exploiting parallel computers. A
short example is then given which involves the Hamiltonian and momentum
constraints in the Einstein equations, a representative nonlinear 4-component
covariant elliptic system on a Riemannian 3-manifold which arises in general
relativity. A number of operator properties and solvability results recently
established are first summarized, making possible two quasi-optimal a priori
error estimates for Galerkin approximations which are then derived. These two
results complete the theoretical framework for effective use of adaptive
multilevel finite element methods. A sample calculation using the MC software
is then presented.
