Matrix-Free Evaluation Strategies for Continuous and Discontinuous Galerkin Discretizations on Unstructured Tetrahedral Grids
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912584833171456 |
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| author | Still, Dominik Fehn, Niklas Wall, Wolfgang A. Kronbichler, Martin |
| author_facet | Still, Dominik Fehn, Niklas Wall, Wolfgang A. Kronbichler, Martin |
| contents | This study presents novel strategies for improving the node-level performance of matrix-free evaluation of continuous and discontinuous Galerkin spatial discretizations on unstructured tetrahedral grids. In our approach the underlying integrals of a generic finite-element operator are computed cell-by-cell through numerical quadrature using tabulated dense local matrices of shape functions, achieving high throughput for low to moderate-order polynomial degrees. By employing dense matrix-matrix products instead of matrix-vector products for the cell-wise interpolation, the method reaches over $60\%$ of peak performance. The optimization strategies exploit explicit data parallelism to enhance computational efficiency, complemented by a hierarchical mesh reordering algorithm that improves data locality. The matrix-free implementation achieves up to a $6\times$ speedup compared to a global sparse matrix-based approach at a polynomial degree of three. The effectiveness of the method is demonstrated through numerical experiments on the Poisson and Navier--Stokes equations. The Poisson operator is preconditioned by a hybrid multigrid scheme that combines auxiliary continuous finite-element spaces, polynomial and geometric coarsening where possible while employing algebraic multigrid on the coarse mesh. Within the preconditioner, the implementation transitions between the matrix-free and matrix-based strategies for optimal efficiency. Finally, we analyze the strong scaling behavior of the Poisson and Helmholtz operators, demonstrating the method's potential to solve large real-world problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_10226 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Matrix-Free Evaluation Strategies for Continuous and Discontinuous Galerkin Discretizations on Unstructured Tetrahedral Grids Still, Dominik Fehn, Niklas Wall, Wolfgang A. Kronbichler, Martin Numerical Analysis Mathematical Software Performance 65M60, 68W10, 76M10 This study presents novel strategies for improving the node-level performance of matrix-free evaluation of continuous and discontinuous Galerkin spatial discretizations on unstructured tetrahedral grids. In our approach the underlying integrals of a generic finite-element operator are computed cell-by-cell through numerical quadrature using tabulated dense local matrices of shape functions, achieving high throughput for low to moderate-order polynomial degrees. By employing dense matrix-matrix products instead of matrix-vector products for the cell-wise interpolation, the method reaches over $60\%$ of peak performance. The optimization strategies exploit explicit data parallelism to enhance computational efficiency, complemented by a hierarchical mesh reordering algorithm that improves data locality. The matrix-free implementation achieves up to a $6\times$ speedup compared to a global sparse matrix-based approach at a polynomial degree of three. The effectiveness of the method is demonstrated through numerical experiments on the Poisson and Navier--Stokes equations. The Poisson operator is preconditioned by a hybrid multigrid scheme that combines auxiliary continuous finite-element spaces, polynomial and geometric coarsening where possible while employing algebraic multigrid on the coarse mesh. Within the preconditioner, the implementation transitions between the matrix-free and matrix-based strategies for optimal efficiency. Finally, we analyze the strong scaling behavior of the Poisson and Helmholtz operators, demonstrating the method's potential to solve large real-world problems. |
| title | Matrix-Free Evaluation Strategies for Continuous and Discontinuous Galerkin Discretizations on Unstructured Tetrahedral Grids |
| topic | Numerical Analysis Mathematical Software Performance 65M60, 68W10, 76M10 |
| url | https://arxiv.org/abs/2509.10226 |