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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Bibliographic Details
Published in:Journal of scientific computing Vol. 105; no. 3; p. 76
Main Authors: Lei, Nuo, Zhang, Donghang, Zheng, Weiying
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2025
Springer Nature B.V
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-025-03105-7