In this lecture, we consider nonlinear approximation techniques for treating singularities in geometric analysis and general relativity. We first review approximation theory for Petrov-Galerkin and Galerkin techniques for nonlinear variational problems. We then examine the use of a posteriori error estimation for adaptive construction of discrete (finite element, wavelet, spectral) spaces for deriving nonlinear approximation techniques; these techniques attempt to meet a target approximation quality using discrete spaces of minimal dimension, and are of increasing importance in modeling and computational science.
We then turn to nonlinear elliptic problems in geometric analysis, and focus on the constraints in Einstein flow. We look briefly at weak solution theory on manifolds with boundary, and various Lp and Sobolev estimates for the constraints which are required to develop approximation theory. We then derive a priori and a posteriori error estimates for Petrov-Galerkin approximations to the constraints, and develop some nonlinear approximation algorithms based on adaptive multilevel finite element methods. We illustrate some of the approximation techniques using the Finite Element ToolKit (FEtk).
If time permits, we will describe the use of the nonlinear approximation techniques to enforce constraints during numerical integration of evolution systems such as the Yang-Mills and Einstein equations, by the use of variational techniques. These techniques yield discrete solutions which exactly satisfy the (discrete) constraints at each discrete moment in time, yet a very simple argument shows that the solutions retain the accuracy of standard time integration methods which do not enforce constraints.