New Frontiers in Optimal Adaptivity

Introduction

The ultimate goal of any numerical method is to achieve maximal accuracy with minimal computational cost. This is also the driving motivation behind adaptive mesh refinement algorithms to approximate partial differential equations (PDEs).

Importance of PDEs

PDEs are the foundation of almost every simulation in computational physics, including:

1. Classical mechanics
2. Geophysics
3. Astrophysics
4. Hydrodynamics
5. Micromagnetism
6. Computational finance
7. Machine learning

Without adaptive mesh refinement, such simulations fail to reach significant accuracy even on the strongest computers before running out of memory or time. The goal of adaptivity is to achieve a mathematically guaranteed optimal accuracy vs. work ratio for such problems.

Challenges in Adaptive Mesh Refinement

However, adaptive mesh refinement for time-dependent PDEs is mathematically not understood, and no optimal adaptive algorithms for such problems are known. The reason is that several key ideas from elliptic PDEs do not work in the non-stationary setting, and the established theory breaks down.

Project Objectives

This ERC project aims to overcome these longstanding open problems by developing and analyzing provably optimal adaptive mesh refinement algorithms for time-dependent problems with relevant applications in computational physics.

Methodology

This will be achieved by exploiting a new mathematical insight that, for the first time in the history of mesh refinement, opens a viable path to understand adaptive algorithms for time-dependent problems. The approaches bridge several mathematical disciplines, such as:

• Finite element analysis
• Matrix theory
• Non-linear PDEs
• Error estimation

Thus, this project breaks new ground in the mathematics and application of computational PDEs.