Geometric Finite Element Methods

Introduction

Partial differential equations (PDEs) describe important models in science and engineering. Many of these PDE-based models encode fundamental geometric and topological principles. For general relativity, gravity is described as the curvature of spacetime governed by the Einstein equations. For materials, defects and microstructures can be modeled as geometric quantities such as curvature.

Importance of Simulation

Since controlled experiments and analytical solutions are only available in very special cases, it is essential to simulate these equations on computers. Despite significant progress in the past decades, cutting-edge applications still call for reliable numerical methods.

Challenges in Numerical Methods

In numerical relativity, codes may break down or significantly lose precision in long-term simulation of black holes due to the violation of geometric constraints. For continuum with microstructures, convergence may degenerate as multiple length scales are present. The common challenge behind these examples is to find an intrinsic way to discretize high-order tensors in geometry with certain symmetries.

Research Objectives

My research will address the fundamental problem of discretizing high-order tensors by bringing together geometry, algebra, PDEs, and numerical analysis.

1. I will develop an algebraic framework and a systematic construction of tensorial finite elements with symmetries.
2. By clarifying mathematical structures at both continuous and discrete levels, I will investigate reliable methods for discretizing the Einstein equations and continuum models with microstructures.

Expected Contributions

The new framework will also inspire the development of fundamental concepts and models, and establish novel connections between:

• Numerical schemes
• Discrete geometry
• Measure-valued solutions of PDEs
• Discrete physics, e.g., quantum gravity and lattice gauge theory.