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Crouzeix--Raviart--Marini realisation of computable a priori L2-error bounds on anisotropic meshes

  • Hiroki Ishizaka
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Abstract

We consider conforming Pone and lowest-order Crouzeix--Raviart (CR) approximations of the Dirichlet Poisson problem in two and three dimensions. We obtain computable a priori L2-error bounds on anisotropic simplicial meshes without a global H2-regularity assumption. For piecewise constant data, the Marini relation gives an equilibrated lowest-order Raviart--Thomas (RT) flux from the scalar CR solution. The square of the resulting Marini constant is the largest generalised eigenvalue of a problem involving only scalar conforming and CR matrices. Combined with the elementwise Poincaré constant, it controls both the conforming and CR errors. The CR L2 estimate uses positivity of the discrete CR--conforming gap operator rather than a nonconforming Aubin--Nitsche argument. We also derive an exact decomposition of the Marini defect into the CR--conforming energy gap and an explicit geometric term with no aspect-ratio factor. On the algebraically graded L-shaped mesh family considered here, q>3/2 yields κM,h=O(h) and computable O(h2) L2-operator bounds.

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Cite

BibTeXRIS
BibTeX
@misc{ishizaka2026crouzeixraviartmarini,
  title = {{Crouzeix--Raviart--Marini realisation of computable a priori $L^2$-error bounds on anisotropic meshes}},
  author = {Hiroki Ishizaka},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06057},
  eprint = {2609.06057},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06057},
}
RIS
TY  - UNPB
TI  - Crouzeix--Raviart--Marini realisation of computable a priori L^2-error bounds on anisotropic meshes
AU  - Hiroki Ishizaka
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.06057
UR  - https://arxiv.org/abs/2609.06057
AB  - We consider conforming Pone and lowest-order Crouzeix--Raviart (CR) approximations of the Dirichlet Poisson problem in two and three dimensions. We obtain computable a priori L^2-error bounds on anisotropic simplicial meshes without a global H^2-regularity assumption. For piecewise constant data, the Marini relation gives an equilibrated lowest-order Raviart--Thomas (RT) flux from the scalar CR solution. The square of the resulting Marini constant is the largest generalised eigenvalue of a problem involving only scalar conforming and CR matrices. Combined with the elementwise Poincaré constant, it controls both the conforming and CR errors. The CR L^2 estimate uses positivity of the discrete CR--conforming gap operator rather than a nonconforming Aubin--Nitsche argument. We also derive an exact decomposition of the Marini defect into the CR--conforming energy gap and an explicit geometric term with no aspect-ratio factor. On the algebraically graded L-shaped mesh family considered here, q>3/2 yields κ_M,h=O(h) and computable O(h^2) L^2-operator bounds.
ER  -