Robust Multigrid for Cartesian Interior Penalty DG Formulations of the Poisson Equation in 3D
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.
Funding by German Research Foundation (DFG) in frame of the project STI 157/4-1 is gratefully acknowledged.
- 2.P. Bastian, M. Blatt, R. Scheichl Algebraic multigrid for discontinuous Galerkin discretizations of heterogeneous elliptic problems. Numer. Linear Algebra Appl. 19, 367–388 (2012)Google Scholar
- 3.J. Bramble, Multigrid Methods. Pitman Research Notes Mathematical Series, vol. 294 (Longman Scientific & Technical, Harlow, 1995)Google Scholar
- 5.K.J. Fidkowski, T.A. Oliver, J. Lu, D.L. Darmofal, p-multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. J. Comput. Phys. 207, 92–113 (2005)Google Scholar
- 9.B.T. Helenbrook, H.L. Atkins, D.J. Mavriplis, Analysis of p-multigrid for continuous and discontinuous finite element discretizations. AIAA Paper 2003–3989 (2003)Google Scholar
- 17.B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. Frontiers in Mathematics, vol. 35 (SIAM, Philadelphia, PA, 2008)Google Scholar