Robust Multigrid for Cartesian Interior Penalty DG Formulations of the Poisson Equation in 3D
Provides a scalable and robust solver for high-order discontinuous Galerkin discretizations of elliptic problems, addressing computational bottlenecks in large-scale simulations.
A polynomial multigrid method with a weighted overlapping Schwarz smoother achieves residual reductions of about two orders of magnitude per V(1,1) cycle for the Poisson equation on 3D Cartesian grids, with linear runtime scaling and robustness up to polynomial order 32 and element aspect ratios up to 48.
We present a polynomial multigrid method for the nodal interior penalty formulation of the Poisson equation on three-dimensional Cartesian grids. Its key ingredient is a weighted overlapping Schwarz smoother operating on element-centered subdomains. The MG method reaches superior convergence rates corresponding to residual reductions of about two orders of magnitude within a single V(1,1) cycle. It is robust with respect to the mesh size and the ansatz order, at least up to ${P=32}$. Rigorous exploitation of tensor-product factorization 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, the method proved feasible for anisotropic grids with element aspect ratios up to 48.