Multigrid Algorithms for $hp$-Discontinuous Galerkin Discretizations of Elliptic Problems

Abstract

We present W-cycle $h$-, $p$-, and $hp$-multigrid algorithms for the solution of the linear system of equations arising from a wide class of $hp$-version discontinuous Galerkin discretizations of elliptic problems. Starting from a classical framework in geometric multigrid analysis, we define a smoothing and an approximation property, which are used to prove uniform convergence of the W-cycle scheme with respect to the discretization parameters and the number of levels, provided the number of smoothing steps is chosen of order $p^{2+\mu}$, where $p$ is the polynomial approximation degree and $\mu=0,1$. A discussion on the effects of employing inherited or noninherited sublevel solvers is also presented. Numerical experiments confirm the theoretical results.