Fundamental data structures for matrix-free finite elements on hybrid tetrahedral grids

• Published in: International journal of parallel, emergent and distributed systems Vol. 39; no. 1; pp. 51 - 74
• Main Authors: Kohl, Nils, Bauer, Daniel, Böhm, Fabian, Rüde, Ulrich
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 02.01.2024; Taylor & Francis Ltd
• Subjects: block-structured grids; Data structures; Discretization; Finite element method; hybrid tetrahedral grids; Mathematical analysis; Matrix-free finite elements; Multigrid methods; parallel data structures
• ISSN: 1744-5760; 1744-5779
• DOI: 10.1080/17445760.2023.2266875

This paper presents efficient data structures for the implementation of matrix-free finite element methods on block-structured, hybrid tetrahedral grids. It provides a complete categorization of all geometric sub-objects that emerge from the regular refinement of the unstructured, tetrahedral coarse grid and describes efficient iteration patterns and analytical linearization functions for the mapping of coefficients to memory addresses. This foundation enables the implementation of fast, extreme-scalable, matrix-free, iterative solvers, and in particular geometric multigrid methods by design. Their application to the variable-coefficient Stokes system subject to an enriched Galerkin discretization and to the curl-curl problem discretized with Nédélec edge elements showcases the flexibility of the implementation. Finally, the solution of a curl-curl problem with $ 1.6\cdot 10^{11} $ 1.6 ⋅ 10 11 (more than one hundred billion) unknowns on more than 32,000 processes with a matrix-free full multigrid solver demonstrates its extreme-scalability.