Numerical Methods for Geometric Partial Differential Equations with Applications in Numerical Relativity

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This project is concerned with the approximate solution of systems of stationary and evolution partial differential equations (PDE) arising at the intersection of mathematical physics and geometric analysis. Such systems of equations, known as Geometric PDE, with both constraints and extra degrees of freedom, appear in a wide range of physical and mathematical problems; examples include Maxwell's equations (or more generally the Yang-Mills equations on a curved background), and Einstein's field equations and other Hamiltonian systems. The initial-value formulation for such systems yields a constrained evolution system which has to be augmented with side conditions in order to get a unique evolution. The non-dynamical geometric PDE (as constraints or otherwise) are of great interest in their own right; examples include the Yamabe problem, the Hamiltonian and momentum constraints in the Einstein equations, and the Monge-Ampere equations, among others. One of the most challenging features of this class of problems, for both mathematical analysis and computational simulation, is the underlying spatial domain which has the structure of a manifold with potentially complicated topology. Moreover, both the geometry and the topology may evolve over time, depending on the particular model. The results of this project have the potential for broad impact on areas of mathematics such as geometric analysis, as well as in astrophysics and general relativity. The methods developed here will contribute to the advancement of numerical methods for complex three-dimensional constrained nonlinear dynamical simulations. The simulation technology produced will provide powerful tools for the exploration of mathematical and computational models in astrophysics and relativity, as well as in some areas of pure mathematics such as geometric analysis. This project provides research training opportunities for graduate students. The primary technical aims of this project are to develop new discretization techniques for a class of geometric PDE that includes the Einstein equations. The emphasis is on modeling cases that present particular challenges for current state-of-the-art methods and software currently used for the Einstein equations, such as the case of extreme mass ration binary black hole systems. The tools will be the development of approximation theory, together with reliable and provably convergent adaptive methods, for the intrinsic discretization of the class of nonlinear geometric PDE on Riemannian 2- and 3- manifolds. Most of the approaches to date, such as surface finite element methods for two-dimensional problems, are based on exploiting the embedding of the surface into three space, and then on use of method-of-lines discretization for separating the space and time discetizations. For applications such as general relativity, a more general approach is needed that does not rely on the existence of such an embedding, and does not on an a priori spatial slicing. This project studies the development of truly intrinsic discretizations that use no extrinsic information to produce a discretization, to allow for the development of numerical methods for evolution PDE on Riemannian 2- and 3-manifolds with arbitrary topology and without imposing an a priori discrete spatial slicing. The approach is to develop atlas-based discretization techniques and space-time discretizations based on explicit tent-pitching methods or fully implicit space-time discetizations. For the design of such methods and their analysis, researchers will exploit variational crimes frameworks developed by their team and collaborators for analyzing numerical methods posed on surfaces, and through use of the finite element exterior calculus framework.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.