Computational and Applied Mathematics seminar

Abstract: Matrix-free finite element methods have gained a lot of attention in the last decade due to their suitability for modern CPU- and GPU-based systems in contrast to the matrix-based alternatives. By now, a number of open-source libraries, like deal.II, DUNE, MFEM, Firedrake, and libCEED, offer matrix-free support. Current research topics include performance optimizations, hardware portability, preconditioning development, and adaptation to new challenging applications. In my presentation, I will give a brief introduction into matrix-free finite-element computations, present a possible realization from the deal.II finite-element library [1], and discuss challenging engineering applications [2, 3]. Finally, I will discuss concepts for constructing preconditioners for matrix-free methods. For the setup of preconditioners, one normally requires a matrix-representation of the linear operator, whose construction might be inherently expensive. I will talk about data-local preconditioned conjugate gradient methods [4], geometric and polynomial multigrid methods [5, 6] (which are applicable to the solution of stabilized incompressible Navier-Stokes equations, locally refined meshes, hp-FEM), domain decomposition methods [7], and block preconditioners [3, 8].

[1] M. Kronbichler and K. Kormann, 2012.
[2] P. Munch, K. Kormann, and M. Kronbichler, 2021.
[3] P. Munch, V. Ivannikov, C. Cyron, and M. Kronbichler, 2024
[4] M. Kronbichler, D. Sashko, and P. Munch, 2022.
[5] P. Munch, T. Heister, L. Prieto Saavedra, and M. Kronbichler, 2022.
[6] P. Munch, K. Ljungkvist, and M. Kronbichler, 2022.
[7] P. Munch and M. Kronbichler, 2023.
[8] P. Munch, I. Dravins, M. Kronbichler, and M. Neytcheva, 2023.