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Symmetric two-level preconditioning for the method of reflections

  • Jongho Park
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Abstract

We develop and analyze a symmetric two-level preconditioner for the method of reflections for Laplace problems in perforated domains. Using constant boundary modes, we define a coarse correction through a symmetric coercive dual formulation. Combining this coarse correction with parallel reflection corrections gives a symmetric positive definite preconditioner for solving the dual system by the conjugate gradient method. Under a geometric separation condition, we prove a condition number bound independent of the number and sizes of the holes and the diameter of the ambient domain. We also derive geometry-dependent convergence estimates for the one-level methods that explain the need for a coarse correction.

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Cite

BibTeXRIS
BibTeX
@misc{park2026symmetric,
  title = {{Symmetric two-level preconditioning for the method of reflections}},
  author = {Jongho Park},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14447},
  eprint = {2610.14447},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14447},
}
RIS
TY  - UNPB
TI  - Symmetric two-level preconditioning for the method of reflections
AU  - Jongho Park
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.14447
UR  - https://arxiv.org/abs/2610.14447
AB  - We develop and analyze a symmetric two-level preconditioner for the method of reflections for Laplace problems in perforated domains. Using constant boundary modes, we define a coarse correction through a symmetric coercive dual formulation. Combining this coarse correction with parallel reflection corrections gives a symmetric positive definite preconditioner for solving the dual system by the conjugate gradient method. Under a geometric separation condition, we prove a condition number bound independent of the number and sizes of the holes and the diameter of the ambient domain. We also derive geometry-dependent convergence estimates for the one-level methods that explain the need for a coarse correction.
ER  -