Numerical Methods for Geometric PDE on Manifolds with Arbitrary Topology

University Of California-San Diego, La Jolla CA

Investigators

Abstract

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, appear in a wide range of physical and mathematical problems; one of the primary motivations for this project is the Einstein, which are of central importance to gravitational wave science. 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 potentially complicated manifold rather than a simple shape in 3-space. Moreover, both the geometry and the topology of this manifold may evolve over time, depending on the particular model. The research results will have a broad impact on areas of mathematics such as geometric analysis, as well as in astrophysics and general relativity. The methods developed will contribute to the advancement of numerical methods for complex three-dimensional constrained nonlinear dynamical simulations. The simulation technology the PIs produce 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. The two graduate students involved in the project will be co-trained by both investigators; this will involve regular interaction between all four members of the team. The primary technical aim of this project is to develop a general approximation theory framework, 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 3-dimension. 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, the PIs will study 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. The PIs' approach is to develop an atlas-based discretization using techniques such as the multi-cube framework and the local simplex approximation techniques developed by the project team. To develop a corresponding error analysis framework, the PIs will exploit the variational crimes framework for methods in surfaces, such as methods based on finite element exterior calculus.