p-multigrid method for the discontinuous Galerkin discretization of elliptic problems
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| Format: | Preprint |
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2025
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| _version_ | 1866908543649579008 |
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| author | Lei, Nuo Zhang, Donghang Zheng, Weiying |
| author_facet | Lei, Nuo Zhang, Donghang Zheng, Weiying |
| contents | In this paper, we propose a $W$-cycle $p$-multigrid method for solving the $p$-version symmetric interior penalty discontinuous Galerkin (SIPDG) discretization of elliptic problems. This SIPDG discretization employs hierarchical Legendre polynomial basis functions. Inspired by the uniform convergence theory of the $W$-cycle $hp$-multigrid method in [P. F. Antonietti, et al., SIAM J. Numer. Anal., 53 (2015)], we provide a rigorous convergence analysis for the proposed $p$-multigrid method, considering both inherited and non-inherited bilinear forms of SIPDG discretization. Our theoretical results show significant improvement over [P. F. Antonietti, et al., SIAM J. Numer. Anal., 53 (2015)], reducing the required number of smoothing steps from $O(p^2)$ to $O(p)$, where $p$ is the polynomial degree of the discrete broken polynomial space. Moreover, the convergence rate remains independent of the mesh size. Several numerical experiments are presented to verify our theoretical findings. Finally, we numerically verify the effectiveness of the $p$-multigrid method for unfitted finite element discretization in solving elliptic interface problems on general $C^{2} $-smooth interfaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_13669 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | p-multigrid method for the discontinuous Galerkin discretization of elliptic problems Lei, Nuo Zhang, Donghang Zheng, Weiying Numerical Analysis 65M60, 65N30, 65F08 In this paper, we propose a $W$-cycle $p$-multigrid method for solving the $p$-version symmetric interior penalty discontinuous Galerkin (SIPDG) discretization of elliptic problems. This SIPDG discretization employs hierarchical Legendre polynomial basis functions. Inspired by the uniform convergence theory of the $W$-cycle $hp$-multigrid method in [P. F. Antonietti, et al., SIAM J. Numer. Anal., 53 (2015)], we provide a rigorous convergence analysis for the proposed $p$-multigrid method, considering both inherited and non-inherited bilinear forms of SIPDG discretization. Our theoretical results show significant improvement over [P. F. Antonietti, et al., SIAM J. Numer. Anal., 53 (2015)], reducing the required number of smoothing steps from $O(p^2)$ to $O(p)$, where $p$ is the polynomial degree of the discrete broken polynomial space. Moreover, the convergence rate remains independent of the mesh size. Several numerical experiments are presented to verify our theoretical findings. Finally, we numerically verify the effectiveness of the $p$-multigrid method for unfitted finite element discretization in solving elliptic interface problems on general $C^{2} $-smooth interfaces. |
| title | p-multigrid method for the discontinuous Galerkin discretization of elliptic problems |
| topic | Numerical Analysis 65M60, 65N30, 65F08 |
| url | https://arxiv.org/abs/2509.13669 |