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

Abstract

AbstractThis 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⋅1011 (more than one hundred billion) unknowns on more than 32,000 processes with a matrix-free full multigrid solver demonstrates its extreme-scalability.Keywords: Matrix-free finite elementshybrid tetrahedral gridsblock-structured gridsparallel data structures AcknowledgmentsThe authors gratefully acknowledge the scientific support and HPC resources provided by the Erlangen National High Performance Computing Center (NHR@FAU) of the Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU).Disclosure statementNo potential conflict of interest was reported by the author(s).Notes1 https://i10git.cs.fau.de/hyteg/hyteg2 https://gauss-allianz.de/en/project/title/CoMPS3 https://www.nhr-verein.deAdditional informationFundingThe authors gratefully acknowledge funding through the joint BMBF project CoMPSFootnote2 (grant 16ME0647K). The authors would like to thank the NHR-Verein e.V.Footnote3 for supporting this work/project within the NHR Graduate School of National High Performance Computing (NHR). The Gauss Centre for Supercomputing e.V. funded this project by providing computing time on the GCS Supercomputer HPE Apollo Hawk at the High Performance Computing Center Stuttgart (grant TN17/44103). NHR funding is provided by federal and Bavarian state authorities. NHR@FAU hardware is partially funded by the German Research Foundation (DFG) – 440719683.