Matrix-free GPU-accelerated saddle-point solvers for high-order problems in $H(\mathrm{div})$
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| Format: | Preprint |
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2023
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| author | Pazner, Will Kolev, Tzanio Vassilevski, Panayot |
| author_facet | Pazner, Will Kolev, Tzanio Vassilevski, Panayot |
| contents | This work describes the development of matrix-free GPU-accelerated solvers for high-order finite element problems in $H(\mathrm{div})$. The solvers are applicable to grad-div and Darcy problems in saddle-point formulation, and have applications in radiation diffusion and porous media flow problems, among others. Using the interpolation-histopolation basis (cf. SIAM J. Sci. Comput., 45 (2023), A675-A702, arXiv:2203.02465), efficient matrix-free preconditioners can be constructed for the $(1,1)$-block and Schur complement of the block system. With these approximations, block-preconditioned MINRES converges in a number of iterations that is independent of the mesh size and polynomial degree. The approximate Schur complement takes the form of an M-matrix graph Laplacian, and therefore can be well-preconditioned by highly scalable algebraic multigrid methods. High-performance GPU-accelerated algorithms for all components of the solution algorithm are developed, discussed, and benchmarked. Numerical results are presented on a number of challenging test cases, including the "crooked pipe" grad-div problem, the SPE10 reservoir modeling benchmark problem, and a nonlinear radiation diffusion test case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_12387 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Matrix-free GPU-accelerated saddle-point solvers for high-order problems in $H(\mathrm{div})$ Pazner, Will Kolev, Tzanio Vassilevski, Panayot Numerical Analysis This work describes the development of matrix-free GPU-accelerated solvers for high-order finite element problems in $H(\mathrm{div})$. The solvers are applicable to grad-div and Darcy problems in saddle-point formulation, and have applications in radiation diffusion and porous media flow problems, among others. Using the interpolation-histopolation basis (cf. SIAM J. Sci. Comput., 45 (2023), A675-A702, arXiv:2203.02465), efficient matrix-free preconditioners can be constructed for the $(1,1)$-block and Schur complement of the block system. With these approximations, block-preconditioned MINRES converges in a number of iterations that is independent of the mesh size and polynomial degree. The approximate Schur complement takes the form of an M-matrix graph Laplacian, and therefore can be well-preconditioned by highly scalable algebraic multigrid methods. High-performance GPU-accelerated algorithms for all components of the solution algorithm are developed, discussed, and benchmarked. Numerical results are presented on a number of challenging test cases, including the "crooked pipe" grad-div problem, the SPE10 reservoir modeling benchmark problem, and a nonlinear radiation diffusion test case. |
| title | Matrix-free GPU-accelerated saddle-point solvers for high-order problems in $H(\mathrm{div})$ |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2304.12387 |