Adaptive multigrid for high-order discontinuous Galerkin methods based on the full approximation scheme
Abstract
We propose an adaptive multigrid (MG) method for discontinuous Galerkin formulations of elliptic problems using Brandt's full approximation scheme (FAS). Unlike common approaches, this method achieves local hp-refinement of hexahedral meshes without the need for hanging nodes. The core component of the FAS-MG method is an overlapping Schwarz smoother, which is optionally accelerated by a Krylov method. This smoother is designed for unstructured curvilinear meshes but maintains a tensor-product structure for fast diagonalization. Numerical experiments demonstrate the exceptional efficiency of the FAS-MG method. Dedicated studies confirm its robustness against high aspect ratios, element deformation, and irregular mesh topology. We also verify its capability for dynamic parallel mesh adaptation using the wave-front benchmark of Červený, Dobrev, and Kolev (SIAM J. Sci. Comp. 41, 2019). Finally, we present preliminary results of extending the method to incompressible Navier-Stokes problems.
Related
- Adaptive Multilevel Discontinuous Galerkin Methods on GPUs
- Adaptive local representations for Helmholtz Trefftz discontinuous Galerkin methods
- Discrete hypocoercive estimates for discontinuous Galerkin methods: application to the Vlasov-Poisson-Fokker-Planck system
- A discontinuous Galerkin approximation of the Dean--Kawasaki equation
- Hybridizable Discontinuous Galerkin methods for the Keller-Segel model
BibTeX
@misc{stiller2026adaptive,
title = {{Adaptive multigrid for high-order discontinuous Galerkin methods based on the full approximation scheme}},
author = {Jörg Stiller},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.14123},
eprint = {2608.14123},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.14123},
}
RIS
TY - UNPB
TI - Adaptive multigrid for high-order discontinuous Galerkin methods based on the full approximation scheme
AU - Jörg Stiller
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.14123
UR - https://arxiv.org/abs/2608.14123
AB - We propose an adaptive multigrid (MG) method for discontinuous Galerkin formulations of elliptic problems using Brandt's full approximation scheme (FAS). Unlike common approaches, this method achieves local hp-refinement of hexahedral meshes without the need for hanging nodes. The core component of the FAS-MG method is an overlapping Schwarz smoother, which is optionally accelerated by a Krylov method. This smoother is designed for unstructured curvilinear meshes but maintains a tensor-product structure for fast diagonalization. Numerical experiments demonstrate the exceptional efficiency of the FAS-MG method. Dedicated studies confirm its robustness against high aspect ratios, element deformation, and irregular mesh topology. We also verify its capability for dynamic parallel mesh adaptation using the wave-front benchmark of Červený, Dobrev, and Kolev (SIAM J. Sci. Comp. 41, 2019). Finally, we present preliminary results of extending the method to incompressible Navier-Stokes problems.
ER -