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H2-based equilibrated residual error estimates

  • Kexin Ding
  • Yuwen Li
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Abstract

We develop a framework for equilibrated residual a posteriori error analysis of finite element methods, based on decomposing vector- and tensor-valued Sobolev spaces into H2 and H1 spaces. To estimate the residual functional acting on H2 component, the method reconstructs symmetric and piecewise polynomial tensors on vertex patches in broken H(divdiv) spaces. We apply the proposed framework to conforming finite element discretizations of biharmonic equation, Poisson's equation, and linear elasticity in mixed form. As a result, we obtain several novel equilibrated residual a posteriori error estimates, providing guaranteed error control for energy error of biharmonic equation, L2 norm error of Poisson's equation, and stress error of mixed linear elasticity. They also serve as the first polynomial-degree-robust a posteriori lower bounds of discretization errors. Numerical experiments illustrate the effectiveness of the H2-based error estimators.

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Cite

BibTeXRIS
BibTeX
@misc{ding2026h2based,
  title = {{$H^2$-based equilibrated residual error estimates}},
  author = {Kexin Ding and Yuwen Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39210},
  eprint = {2610.39210},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39210},
}
RIS
TY  - UNPB
TI  - H^2-based equilibrated residual error estimates
AU  - Kexin Ding
AU  - Yuwen Li
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39210
UR  - https://arxiv.org/abs/2610.39210
AB  - We develop a framework for equilibrated residual a posteriori error analysis of finite element methods, based on decomposing vector- and tensor-valued Sobolev spaces into H^2 and H^1 spaces. To estimate the residual functional acting on H^2 component, the method reconstructs symmetric and piecewise polynomial tensors on vertex patches in broken H(divdiv) spaces. We apply the proposed framework to conforming finite element discretizations of biharmonic equation, Poisson's equation, and linear elasticity in mixed form. As a result, we obtain several novel equilibrated residual a posteriori error estimates, providing guaranteed error control for energy error of biharmonic equation, L^2 norm error of Poisson's equation, and stress error of mixed linear elasticity. They also serve as the first polynomial-degree-robust a posteriori lower bounds of discretization errors. Numerical experiments illustrate the effectiveness of the H^2-based error estimators.
ER  -