ERC Consolidator Grant

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). PDEs are the foundation of almost every simulation in computational physics (from classical mechanics to geophysics, astrophysics, hydrodynamics, and micromagnetism) and even in computational finance and 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. 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. 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.