This talk is planned to consist of two parts. In the first part, I want to discuss some recent results 
on the existence of solutions with arbitrarily specifiable extrinsic mean curvature for the standard 
conformal formulation of the Einstein constraint equations, which is a semilinear elliptic system that 
arises in general relativity. In the second part, after giving some background on convergence and complexity 
analysis of adaptive finite element methods, I intend to talk about designing provably convergent (and optimal 
complexity) adaptive finite element methods for solving the Einstein constraint equations. The first part is 
based on a joint work with M. Holst and G. Nagy, and the second part is on a joint work with M. Holst.