Geometric Multigrid Preconditioners for conforming Finite Element Methods on Arbitrarily Agglomerated Meshes
Abstract
Geometric multigrid methods for conforming finite element discretisations classically require a nested mesh hierarchy, forcing coarse meshes to be conforming, which is unsuitable for general domains. In order to eliminate this geometric constraint, we introduce and analyse a family of multilevel auxiliary space preconditioners. At the first level, a conforming Lagrangian discretisation on a fine simplicial mesh is preconditioned by a non-conforming polytopal discretisation posed on an arbitrarily agglomerated coarse mesh. At the subsequent levels, the coarse problem is solved by a geometric multigrid method for that polytopal discretisation on further agglomerated meshes, so that the whole hierarchy is generated from the fine mesh by agglomeration alone. We consider two polytopal discretisations, a symmetric interior penalty discontinuous Galerkin method and a hybrid high-order method. The prolongation operator at the first level is an element-wise interpolation onto the fine mesh followed by nodal averaging, preceded in the hybrid high-order case by a local potential reconstruction. For point-Jacobi and vertex-star patch additive Schwarz smoothers, we prove that the condition number of the preconditioned system is bounded by O((H/h)3), where H/h is the coarsening ratio, and by O((H/h)2) for the DG variant with penalty proportional to H/h. Numerical experiments in two and three dimensions, on structured and unstructured meshes with agglomerates generated by automatic clustering algorithms, confirm the robustness of the preconditioners.
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BibTeX
@misc{badia2026geometric,
title = {{Geometric Multigrid Preconditioners for conforming Finite Element Methods on Arbitrarily Agglomerated Meshes}},
author = {Santiago Badia and Jordi Manyer},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06012},
eprint = {2611.06012},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06012},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geometric Multigrid Preconditioners for conforming Finite Element Methods on Arbitrarily Agglomerated Meshes
AU - Santiago Badia
AU - Jordi Manyer
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06012
UR - https://arxiv.org/abs/2611.06012
AB - Geometric multigrid methods for conforming finite element discretisations classically require a nested mesh hierarchy, forcing coarse meshes to be conforming, which is unsuitable for general domains. In order to eliminate this geometric constraint, we introduce and analyse a family of multilevel auxiliary space preconditioners. At the first level, a conforming Lagrangian discretisation on a fine simplicial mesh is preconditioned by a non-conforming polytopal discretisation posed on an arbitrarily agglomerated coarse mesh. At the subsequent levels, the coarse problem is solved by a geometric multigrid method for that polytopal discretisation on further agglomerated meshes, so that the whole hierarchy is generated from the fine mesh by agglomeration alone. We consider two polytopal discretisations, a symmetric interior penalty discontinuous Galerkin method and a hybrid high-order method. The prolongation operator at the first level is an element-wise interpolation onto the fine mesh followed by nodal averaging, preceded in the hybrid high-order case by a local potential reconstruction. For point-Jacobi and vertex-star patch additive Schwarz smoothers, we prove that the condition number of the preconditioned system is bounded by O((H/h)^3), where H/h is the coarsening ratio, and by O((H/h)^2) for the DG variant with penalty proportional to H/h. Numerical experiments in two and three dimensions, on structured and unstructured meshes with agglomerates generated by automatic clustering algorithms, confirm the robustness of the preconditioners.
ER -