Robust multigrid for Cartesian interior penalty DG formulations of the Poisson equation in 3D. (English) Zbl 1382.65454
Bittencourt, Marco L. (ed.) et al., Spectral and high order methods for partial differential equations, ICOSAHOM 2016. Selected papers from the ICOSAHOM conference, June 27 – July 1, 2016, Rio de Janeiro, Brazil. Cham: Springer (ISBN 978-3-319-65869-8/hbk; 978-3-319-65870-4/ebook). Lecture Notes in Computational Science and Engineering 119, 189-201 (2017).
Summary: We present a polynomial multigrid (MG) 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.
For the entire collection see [Zbl 1383.65001].
For the entire collection see [Zbl 1383.65001].
MSC:
65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
65Y20 | Complexity and performance of numerical algorithms |
65F10 | Iterative numerical methods for linear systems |