Collaborative Research: Finite Element Methods for Discretizing Geometric PDEs with Nonlinear Constraints and Gauge Freedom

Colorado State University, Fort Collins CO

Investigators

Abstract

This project is concerned with the approximate solution of certain systems of 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; 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 with an infinite-dimensional symmetry group. The Cauchy formulation for such systems yields a constrained evolution system which has to be augmented with gauge-fixing conditions in order to get a unique evolution vector field. The project will involve constructing finite element discretizations for solving such geometric PDE systems; various techniques for dealing with evolution systems with constraints will be analyzed, including constraint-projection using variational techniques (where the numerical solution is projected back to the constraint manifold after some number of time steps), the use of special finite elements which automatically solve the linearized constraints and thereby remain on a piecewise-linear approximation to the constraint manifold, and finally least-squares approaches which only control the constraints rather than enforce them. In particular, stability results guaranteeing convergence of the numerical solution to the continuum solution will be derived, at least for the linearized equations. A posteriori error estimates will be derived for studying properties of the discretizations, and for building adaptive methods. This project involves the design, development, and implementation of new mathematical and computational techniques for solving a large class of important, challenging, and pressing mathematical problems in multiscale and multiphysics modeling and simulation. The techniques developed will lead to the aquisition of new knowledge in areas of science such as relatistic astrophysics, by making possible more reliable and accurate simulations of phenomena such as gravitational collapse, nonlinear stability of deformed rotating black holes, binary black hole collision, and the production and emission of gravitational radiation. Most of these problems are currently of great interest due to the recent construction of gravitational wave detectors such as the NSF-funded LIGO devices in Lousiana and Washington. The results from this project will have a 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, and the technology produced will provide powerful tools for the exploration of models in astrophysics and relativity as well as in some areas of pure mathematics such as geometric analysis.