Speaker
Description
In this talk, we introduce a new hybrid approach for building geometrically informed algebraic multigrid preconditioners. Classical geometric multigrid methods are well known to be optimal preconditioners for linear systems arising from elliptic partial differential equations (PDEs). However, for very fine or complex geometries, the generation of a hierarchy might be not feasible. This issue is usually solved by employing algebraic multigrid (AMG) frameworks, which require only the system matrix to be given, without exploiting any geometrical information, but using matrix-entries only.
Our approach builds on a domain-decomposition setting. The core idea is to enrich the Nicolaides coarse space [1] with higher order basis functions, in order to extend AMG methodologies to high order elements. To this aim, we devise transfer operators defined through an efficient agglomeration algorithm based on the R-tree spatial data structures [2]. This coarsening strategy allows to inject geometrical information into the transfer operators, which are used as a tool to build coarser spaces.
We present a comprehensive set of numerical experiments, both in two and three dimensions on both structured and unstructured meshes, thereby confirming the effectiveness and efficiency of our approach as a multigrid preconditioner for continuous finite element discretizations.
[1] V. Dolean, P. Jolivet, F. Nataf. An Introduction to Domain Decomposition Methods: Algorithms, Theory and Parallel Implementation. SIAM, 2015
[2] M. Feder, A. Cangiani, L. Heltai. R3MG: R-tree based agglomeration of polytopal grids with applications to multilevel methods. Journal of Computational Physics, 526:113773, 2025