p-multigrid method for the discontinuous Galerkin discretization of elliptic problems

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.