A Space-Time Multigrid Method for Space-Time Finite Element Discretizations of Parabolic and Hyperbolic PDEs
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929453673742336 |
|---|---|
| author | Margenberg, Nils Munch, Peter |
| author_facet | Margenberg, Nils Munch, Peter |
| contents | We present a space-time multigrid method based on tensor-product space-time finite element discretizations. The method is facilitated by the matrix-free capabilities of the {\ttfamily deal.II} library. It addresses both high-order continuous and discontinuous variational time discretizations with spatial finite element discretizations. The effectiveness of multigrid methods in large-scale stationary problems is well established. However, their application in the space-time context poses significant challenges, mainly due to the construction of suitable smoothers. To address these challenges, we develop a space-time cell-wise additive Schwarz smoother and demonstrate its effectiveness on the heat and acoustic wave equations. The matrix-free framework of the {\ttfamily deal.II} library supports various multigrid strategies, including $h$-, $p$-, and $hp$-refinement across spatial and temporal dimensions. Extensive empirical evidence, provided through scaling and convergence tests on high-performance computing platforms, demonstrate high performance on perturbed meshes and problems with heterogeneous and discontinuous coefficients. Throughputs of over a billion degrees of freedom per second are achieved on problems with more than a trillion global degrees of freedom. The results prove that the space-time multigrid method can effectively solve complex problems in high-fidelity simulations and show great potential for use in coupled problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04372 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Space-Time Multigrid Method for Space-Time Finite Element Discretizations of Parabolic and Hyperbolic PDEs Margenberg, Nils Munch, Peter Numerical Analysis Computational Physics 65M60, 65M55, 65F10, 65Y05 We present a space-time multigrid method based on tensor-product space-time finite element discretizations. The method is facilitated by the matrix-free capabilities of the {\ttfamily deal.II} library. It addresses both high-order continuous and discontinuous variational time discretizations with spatial finite element discretizations. The effectiveness of multigrid methods in large-scale stationary problems is well established. However, their application in the space-time context poses significant challenges, mainly due to the construction of suitable smoothers. To address these challenges, we develop a space-time cell-wise additive Schwarz smoother and demonstrate its effectiveness on the heat and acoustic wave equations. The matrix-free framework of the {\ttfamily deal.II} library supports various multigrid strategies, including $h$-, $p$-, and $hp$-refinement across spatial and temporal dimensions. Extensive empirical evidence, provided through scaling and convergence tests on high-performance computing platforms, demonstrate high performance on perturbed meshes and problems with heterogeneous and discontinuous coefficients. Throughputs of over a billion degrees of freedom per second are achieved on problems with more than a trillion global degrees of freedom. The results prove that the space-time multigrid method can effectively solve complex problems in high-fidelity simulations and show great potential for use in coupled problems. |
| title | A Space-Time Multigrid Method for Space-Time Finite Element Discretizations of Parabolic and Hyperbolic PDEs |
| topic | Numerical Analysis Computational Physics 65M60, 65M55, 65F10, 65Y05 |
| url | https://arxiv.org/abs/2408.04372 |