An hp Multigrid Approach for Tensor-Product Space-Time Finite Element Discretizations of the Stokes Equations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918240220872704 |
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| author | Margenberg, Nils Bause, Markus Munch, Peter |
| author_facet | Margenberg, Nils Bause, Markus Munch, Peter |
| contents | We present a monolithic $hp$ space-time multigrid method for tensor-product space-time finite element discretizations of the Stokes equations. Geometric and polynomial coarsening of the space-time mesh is performed, and the entire algorithm is expressed through rigorous mathematical mappings. For the discretization, we use inf-sup stable pairs $\mathbb Q_{r+1}/\mathbb P_{r}^{\text{disc}}$ of elements in space and a discontinuous Galerkin (DG$(k)$) discretization in time with piecewise polynomials of order $k$. The key novelty of this work is the application of $hp$ multigrid techniques in space and time, facilitated and accelerated by the matrix-free capabilities of the deal$.$II library. While multigrid methods are well-established for stationary problems, their application in space-time formulations encounter unique challenges, particularly in constructing suitable smoothers. To overcome these challenges, we employ space-time cell and vertex star patch based Vanka smoothers. Extensive tests on high-performance computing platforms demonstrate the efficiency of our \( hp \) multigrid approach on problem sizes exceeding a trillion degrees of freedom (dofs), sustaining throughputs of hundreds of millions of dofs per second. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09159 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An hp Multigrid Approach for Tensor-Product Space-Time Finite Element Discretizations of the Stokes Equations Margenberg, Nils Bause, Markus Munch, Peter Numerical Analysis Computational Physics 65M60, 65M55, 65F10, 65Y05 We present a monolithic $hp$ space-time multigrid method for tensor-product space-time finite element discretizations of the Stokes equations. Geometric and polynomial coarsening of the space-time mesh is performed, and the entire algorithm is expressed through rigorous mathematical mappings. For the discretization, we use inf-sup stable pairs $\mathbb Q_{r+1}/\mathbb P_{r}^{\text{disc}}$ of elements in space and a discontinuous Galerkin (DG$(k)$) discretization in time with piecewise polynomials of order $k$. The key novelty of this work is the application of $hp$ multigrid techniques in space and time, facilitated and accelerated by the matrix-free capabilities of the deal$.$II library. While multigrid methods are well-established for stationary problems, their application in space-time formulations encounter unique challenges, particularly in constructing suitable smoothers. To overcome these challenges, we employ space-time cell and vertex star patch based Vanka smoothers. Extensive tests on high-performance computing platforms demonstrate the efficiency of our \( hp \) multigrid approach on problem sizes exceeding a trillion degrees of freedom (dofs), sustaining throughputs of hundreds of millions of dofs per second. |
| title | An hp Multigrid Approach for Tensor-Product Space-Time Finite Element Discretizations of the Stokes Equations |
| topic | Numerical Analysis Computational Physics 65M60, 65M55, 65F10, 65Y05 |
| url | https://arxiv.org/abs/2502.09159 |