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

In this paper, we extend our previous work on the W-cycle p -multigrid method to efficiently solve elliptic problems with discontinuous coefficients, using the symmetric weighted interior penalty discontinuous Galerkin (SWIPDG) method for high-order discretization. We present a rigorous convergence analysis for the p -multigrid method in the presence of discontinuous coefficients and hierarchical Legendre polynomial basis, under the assumption of piecewise H 2 -regularity of the solution. Our analysis proves that the convergence rate is uniform with respect to the mesh size, the polynomial degree p , and the ratio of discontinuous coefficients. Theoretical results show that our method is both efficient and robust in handling elliptic problems with large coefficient jumps. Notably, our method achieves competitive performance by with only O ( p ) smoothing steps. Extensive numerical experiments are conducted to validate the theoretical findings, demonstrating the effectiveness and robustness of the proposed approach.