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In this paper we analyze the convergence properties of two-level and W-cyclemultigrid solvers for the numerical solution of the linear system of equationsarising from hp-version symmetric interior penalty discontinuous Galerkindiscretizations of second-order elliptic partial differential equations onpolygonal/polyhedral meshes. We prove that the two-level method convergesuniformly with respect to the granularity of the grid and the polynomialapproximation degree p, provided that the number of smoothing steps, whichdepends on p, is chosen sufficiently large. An analogous result is obtained forthe W-cycle multigrid algorithm, which is proved to be uniformly convergentwith respect to the mesh size, the polynomial approximation degree, and thenumber of levels, provided the number of smoothing steps is chosen sufficientlylarge. Numerical experiments are presented which underpin the theoreticalpredictions; moreover, the proposed theoretical assumptions are not fullysatisfied.

Multigrid algorithms for $hp$-version Interior Penalty Discontinuous Galerkin methods on polygonal and polyhedral meshes