P-Multigrid Method for the Discontinuous Galerkin Discretization of Elliptic Problems

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 t...

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Veröffentlicht in:	Journal of scientific computing Jg. 105; H. 3; S. 76
Hauptverfasser:	Lei, Nuo, Zhang, Donghang, Zheng, Weiying
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.12.2025 Springer Nature B.V
Schlagworte:	Algorithms Approximation Basis functions Computational Mathematics and Numerical Analysis Convergence Discretization Galerkin method Iterative methods Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical functions Mathematics Mathematics and Statistics Partial differential equations Polynomials Theoretical multigrid algorithm discontinuous Galerkin method elliptic problems polynomial smoother 65N55 65F08 65N30
ISSN:	0885-7474, 1573-7691
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Abstract	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.
AbstractList	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. 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(p2) 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 C2-smooth interfaces.
ArticleNumber	76
Author	Lei, Nuo Zhang, Donghang Zheng, Weiying
Author_xml	– sequence: 1 givenname: Nuo surname: Lei fullname: Lei, Nuo organization: Hua Loo-Keng Center for Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences – sequence: 2 givenname: Donghang surname: Zhang fullname: Zhang, Donghang organization: Academy of Mathematics and Systems Science, Chinese Academy of Sciences – sequence: 3 givenname: Weiying surname: Zheng fullname: Zheng, Weiying email: zwy@lsec.cc.ac.cn organization: LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, School of Mathematical Science, University of Chinese Academy of Sciences
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Snippet	In this paper, we propose a W -cycle p -multigrid method for solving the p -version symmetric interior penalty discontinuous Galerkin (SIPDG) discretization of... In this paper, we propose a W-cycle p-multigrid method for solving the p-version symmetric interior penalty discontinuous Galerkin (SIPDG) discretization of...
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SubjectTerms	Algorithms Approximation Basis functions Computational Mathematics and Numerical Analysis Convergence Discretization Galerkin method Iterative methods Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical functions Mathematics Mathematics and Statistics Partial differential equations Polynomials Theoretical
Title	P-Multigrid Method for the Discontinuous Galerkin Discretization of Elliptic Problems
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