Fundamental Data Structures for Matrix-Free Finite Elements on Hybrid Tetrahedral Grids

TL;DR

This paper introduces specialized data structures for matrix-free finite element methods on hybrid tetrahedral grids, enabling scalable solvers for extremely large problems with over a hundred billion unknowns.

Contribution

It provides a comprehensive categorization of geometric sub-objects and efficient iteration patterns, facilitating the implementation of scalable matrix-free solvers for complex PDEs.

Findings

Successfully solved a curl-curl problem with over 10^11 unknowns

Demonstrated extreme scalability on 32,000 processes

Enabled efficient matrix-free multigrid methods for large-scale systems

Abstract

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 N\'ed\'elec edge elements showcases the flexibility of the implementation. Eventually, the solution of a curl-curl problem with $1.6 \cdot 10^{11}$ (more than one hundred billion) unknowns on more than $32000$ processes with a matrix-free full multigrid solver demonstrates its extreme-scalability.