Next-Generation High-Order Methods: Sparse Discretizations and Scalable Multigrid Solvers

Per-Olof Persson is a Professor of Mathematics at the University of California, Berkeley, and a Faculty Senior Scientist at the Berkeley Lab. He received his Ph.D. from the Massachusetts Institute of Technology in 2005, where he also developed the widely used DistMesh algorithm for unstructured mesh generation in implicit and deforming geometries. He has also worked for several years on the development of commercial numerical software, such as the finite element package COMSOL Multiphysics. His research focuses on high-order discontinuous Galerkin methods for computational fluid and solid mechanics, spanning efficient discretizations, scalable solvers, and adjoint-based optimization. A second key area of his research is mesh generation, where he has developed methods for space-time and curved meshes, as well as new approaches based on machine learning.

Next-Generation High-Order Methods: Sparse Discretizations and Scalable Multigrid Solvers

High-order accurate numerical methods, such as the discontinuous Galerkin (DG) method, offer significant advantages over traditional low-order schemes for simulating complex phenomena like turbulent flows, multiphysics interactions, and wave propagation. However, their broad adoption is often limited by high computational costs and sensitivity to under-resolved features, such as shocks. This talk presents recent algorithmic advances in numerical discretizations and algebraic solvers designed to overcome these computational bottlenecks.

We begin by discussing the importance of unstructured curved meshes, highlighting the DistMesh algorithm as well as our recent work on Deep Reinforcement Learning for mesh generation. Next, I will introduce a suite of novel discretization schemes inspired by the DG method, including the naturally sparse Line-DG, half-closed DG, and FUSE schemes. These approaches yield system matrices that are orders of magnitude smaller than those generated by standard DG schemes. To solve these highly sparse systems, we couple our discretizations with efficient parallel solvers based on multigrid and optimally ordered incomplete factorizations. I will outline new developments in this area, including agglomeration-based multigrid, adaptive smoothed aggregation multigrid, and a novel construction that enables full static condensation for DG discretizations.

Finally, time permitting, I will discuss the use of adjoint methods for gradient-based optimization, with applications in optimal designs for flapping flight, and how these techniques can be used for high-order implicit shock tracking.