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We present a polynomial multigrid method for the nodal interior penaltyformulation of the Poisson equation on three-dimensional Cartesian grids. Itskey ingredient is a weighted overlapping Schwarz smoother operating onelement-centered subdomains. The MG method reaches superior convergence ratescorresponding to residual reductions of about two orders of magnitude within asingle V(1,1) cycle. It is robust with respect to the mesh size and the ansatzorder, at least up to ${P=32}$. Rigorous exploitation of tensor-productfactorization yields a computational complexity of $O(PN)$ for $N$ unknowns,whereas numerical experiments indicate even linear runtime scaling. Moreover,by allowing adjustable subdomain overlaps and adding Krylov acceleration, themethod proved feasible for anisotropic grids with element aspect ratios up to48.

Robust Multigrid for Cartesian Interior Penalty DG Formulations of the Poisson Equation in 3D