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Adaptive multilevel finite element methods are developed and analyzed forcertain elliptic systems arising in geometric analysis and general relativity.This class of nonlinear elliptic systems of tensor equations on manifolds isfirst reviewed, and then adaptive multilevel finite element methods forapproximating solutions to this class of problems are considered in somedetail. Two a posteriori error indicators are derived, based on local residualsand on global linearized adjoint or dual problems. The design of Manifold Code(MC) is then discussed; MC is an adaptive multilevel finite element softwarepackage for 2- and 3-manifolds. It employs a posteriori error estimation,adaptive simplex subdivision, unstructured algebraic multilevel methods, globalinexact Newton methods, and numerical continuation methods for the numericalsolution of nonlinear covariant elliptic systems on 2- and 3-manifolds. Some ofthe more interesting features of MC are described in detail, including some newideas for topology and geometry representation in simplex meshes, and anunusual partition of unity-based method for exploiting parallel computers. Ashort example is then given which involves the Hamiltonian and momentumconstraints in the Einstein equations, a representative nonlinear 4-componentcovariant elliptic system on a Riemannian 3-manifold which arises in generalrelativity. A number of operator properties and solvability results recentlyestablished are first summarized, making possible two quasi-optimal a priorierror estimates for Galerkin approximations which are then derived. These tworesults complete the theoretical framework for effective use of adaptivemultilevel finite element methods. A sample calculation using the MC softwareis then presented.

Adaptive Numerical Treatment of Elliptic Systems on Manifolds