A new approach to finite element simulations of general relativity

Abstract

In order to study gravitational waves, we introduce a new approach to finite element simulation of general relativity. This approach is based on approximating the Weyl curvature directly through new stable mixed finite elements for the Einstein-Bianchi system. We design and analyze these novel finite elements by adapting the recently developed Finite Element Exterior Calculus (FEEC) framework to abstract Hodge wave equations. This framework enables us to borrow key ideas from Reissner-Mindlin plate bending and elasticity with weakly imposed symmetries to maintain stability of the method. The stability of a discretization often relies on deep connections between fundamental branches of mathematics: the FEEC mimics these connections for the numerical method to achieve similar stability to that of the original equations. The recent development of FEEC has had a transformative impact on electromagnetism and related computational problems, and we are expanding it to general relativity.