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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal{O}(p^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal{O}(p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C^{2} $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-smooth interfaces.

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

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发表日期：

2025年11月4日

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Nuo Lei,
Donghang Zhang,
Weiying Zheng

摘要

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 $\mathcal{O}(p^2)$ $O (p^{2})$ to $\mathcal{O}(p)$ $O (p)$ , 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} $ $C^{2}$ -smooth interfaces.