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

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...

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Published inInternational 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
LanguageEnglish
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
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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 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.
AbstractList 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 ...
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.
Author Böhm, Fabian
Bauer, Daniel
Kohl, Nils
Rüde, Ulrich
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Snippet This paper presents efficient data structures for the implementation of matrix-free finite element methods on block-structured, hybrid tetrahedral grids. It...
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SubjectTerms block-structured grids
Data structures
Discretization
Finite element method
hybrid tetrahedral grids
Mathematical analysis
Matrix-free finite elements
Multigrid methods
parallel data structures
Title Fundamental data structures for matrix-free finite elements on hybrid tetrahedral grids
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