This project was 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 gauge degrees of freedom, appear in a wide range of physical and mathematical problems; particularly relevant for this project were Maxwell's equations and Einstein's field equations.  The Cauchy formulation for such systems yields a constrained evolution system which has to be augmented with side conditions (constraints) to yield a unique evolution.  These constraints 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 Riemannian manifold with potentially non-trivial topology.  Moreover, both the geometry and the topology may evolve over time, depending on the particular model.

Intellectual Merit: The primary technical aim of this project was to develop general approximation theory results, together with reliable and provably convergent adaptive methods, for the intrinsic discretization of a general class of nonlinear geometric elliptic and evolution PDE on Riemannian 2- and 3-manifolds.  While the solution theory for this class of PDE has been intensively studied over the last thirty years, progress on the development of robust numerical methods with a corresponding approximation theory has been a more recent development.  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 R3.  For applications such as general relativity, a more general approach is needed that does not rely on the existence of such an embedding.  In this project, we studied the development of truly intrinsic discretizations that use no extrinsic information to produce a discretization, to allow for the development of numerical methods on Riemannian 2- and 3-manifolds with arbitrary topology.  Our approach was to build on the existing Finite Element Exterior Calculus (FEEC) Framework, extending it in the direction of evolution problems, problems on surfaces, and nonlinear problems, and at the same time allowing for adaptive algorithms.  The results from our project were disseminated in numerous lectures given at both national and international venues, and through the publication of a number of articles that appeared in major general and field-specific mathematics journals, including: The Bulletin of the American Mathematical Society, Communications in Mathematical Physics, the Journal of Computational Mathematics, and ACM Transactions on Mathematical Software.

Broader Impacts: The results from this project have had an impact on the study, design, and analysis of numerical methods for PDE, and have the further potential to have broad impact on related areas of mathematics and physics such as geometric analysis, astrophysics, and general relativity.  The methods we developed will contribute to the overall advancement of numerical methods for complex three-dimensional constrained nonlinear dynamical simulations, and the simulation technology we produced provides general tools for the exploration of models in astrophysics and relativity as well as in some areas of pure mathematics such as geometric analysis.  The graduate students and postdoctoral fellows involved in the project were co-trained by the investigators, and in addition to their research projects they were exposed to both organizing research workshops and writing funding proposals.  All of the postdocs involved in the project have now successfully transitioned to tenure-track faculty positions at research universities.

Last Modified: 06/26/2021
Modified by: Michael J Holst