Generating Initial Data in General Relativity using Adaptive Finite Element Methods

TL;DR

This paper develops and analyzes adaptive finite element methods for solving the Einstein constraint equations in general relativity, providing new error estimates, convergence results, and implementation details using the FETK software.

Contribution

It introduces a novel adaptive finite element approach for the Einstein constraints, including new error estimates, convergence proofs, and practical implementation on Riemannian manifolds.

Findings

Derivation of weak formulations for coupled nonlinear elliptic systems

Development of adaptive finite element method with proven convergence and optimality

Successful implementation and demonstration using the FETK software package

Abstract

The conformal formulation of the Einstein constraint equations is first reviewed, and we then consider the design, analysis, and implementation of adaptive multilevel finite element-type numerical methods for the resulting coupled nonlinear elliptic system. We derive weak formulations of the coupled constraints, and review some new developments in the solution theory for the constraints in the cases of constant mean extrinsic curvature (CMC) data, near-CMC data, and arbitrarily prescribed mean extrinsic curvature data. We then outline some recent results on a priori and a posteriori error estimates for a broad class of Galerkin-type approximation methods for this system which includes techniques such as finite element, wavelet, and spectral methods. We then use these estimates to construct an adaptive finite element method (AFEM) for solving this system numerically, and outline some new convergence and optimality results. We then describe in some detail an implementation of the methods using the FETK software package, which is an adaptive multilevel finite element code designed to solve nonlinear elliptic and parabolic systems on Riemannian manifolds. We finish by describing a simplex mesh generation algorithm for compact binary objects, and then look at a detailed example showing the use of FETK for numerical solution of the constraints.