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PreprintCC BY-NC-ND 4.0arXiv

Adaptive Multilevel Discontinuous Galerkin Methods on GPUs

  • Michał Wichrowski
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Abstract

I present a matrix-free symmetric interior penalty discontinuous Galerkin method for adaptively refined Cartesian meshes on GPUs. At non-matching interfaces, auxiliary \emph{shadow cells} represent the adjacent coarse polynomial on the fine level. This approach converts non-matching interfaces into matching faces, permitting uniform face evaluation throughout the mesh. I prove that this construction is equivalent to the standard non-matching formulation. Furthermore, this representation yields a local geometric multigrid method in which frozen shadows provide inter-level boundary data and carry residual contributions to coarser levels. A primal--dual pairing eliminates shadow assembly from the Krylov iteration. Numerical experiments with cubic elements (p=3) on an NVIDIA A100 show stable multigrid convergence under increasing refinement depth and efficient GPU execution.

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Cite

BibTeXRIS
BibTeX
@misc{wichrowski2026adaptive,
  title = {{Adaptive Multilevel Discontinuous Galerkin Methods on GPUs}},
  author = {Michał Wichrowski},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04534},
  eprint = {2609.04534},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04534},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Adaptive Multilevel Discontinuous Galerkin Methods on GPUs
AU  - Michał Wichrowski
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.04534
UR  - https://arxiv.org/abs/2609.04534
AB  - I present a matrix-free symmetric interior penalty discontinuous Galerkin method for adaptively refined Cartesian meshes on GPUs. At non-matching interfaces, auxiliary \emph{shadow cells} represent the adjacent coarse polynomial on the fine level. This approach converts non-matching interfaces into matching faces, permitting uniform face evaluation throughout the mesh. I prove that this construction is equivalent to the standard non-matching formulation. Furthermore, this representation yields a local geometric multigrid method in which frozen shadows provide inter-level boundary data and carry residual contributions to coarser levels. A primal--dual pairing eliminates shadow assembly from the Krylov iteration. Numerical experiments with cubic elements (p=3) on an NVIDIA A100 show stable multigrid convergence under increasing refinement depth and efficient GPU execution.
ER  -