There is currently tremendous interest in geometric PDEs, due in part to the geometric flow program used recently to solve the Poincare conjecture. Geometric PDEs also play an expanding role in many other applications, such as understanding the gravitational wave models of Einstein. The need to validate these models has led to the construction of gravitational wave detectors in the last several years, such as the NSF-funded LIGO project. In this lecture, we consider the coupled nonlinear elliptic constraints in the Einstein equations, a geometric flow which describes the propagation of gravitational waves generated by collisions of massive objects such as black holes. The constraint equations must be solved numerically to produce initial data for gravitational wave simulations and to enforce the constraints during dynamical simulations. In the first part of the lecture, we consider a thirty-year-old open question involving existence of solutions to the constraint equations on space-like hyper-surfaces with arbitrarily prescribed mean extrinsic curvature, and we give a partial answer using a priori estimates and a new type of topological fixed-point argument.

In the second part of this lecture, we develop some adaptive numerical methods for which we can prove a number of useful results on convergence, optimality, and scalability. Based on the a priori estimates developed in the first part of the talk, we first establish some critical discrete estimates. We then derive error estimates for Galerkin approximations and describe a class of nonlinear approximation algorithms based on adaptive finite element methods (AFEM). We establish some new AFEM convergence and optimality results for geometric PDE problems with non-monotone nonlinearities such as the Einstein constraints.

We finish by illustrating the algorithms with some examples using the Finite Element ToolKit (FETK).