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PreprintarXiv licencearXiv

Improving accuracy of finite element solutions by smoothing

  • Yuwen Li
  • Han Shui
  • Ludmil Zikatanov
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Abstract

This paper develops a smoothing-based postprocessing method for improving the accuracy of finite element solutions. The method applies a few smoothing iterations, such as damped Jacobi, symmetrized Gauss-Seidel, or preconditioned conjugate gradient, with the initial guess being the current finite element solution embedded in an augmented finite element space. The resulting procedure is algebraic, easy to implement, and applicable to high-order and three-dimensional discretizations. For symmetric and positive-definite problems, we show that the accuracy of the smoothed solutions is improved by a smoothing factor under additive and multiplicative smoothers. This smoothing factor decays exponentially during the first few smoothing steps. The effectiveness of the proposed method is numerically demonstrated using the Poisson, Maxwell, biharmonic and Helmholtz equations.

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Cite

BibTeXRIS
BibTeX
@misc{li2026improving,
  title = {{Improving accuracy of finite element solutions by smoothing}},
  author = {Yuwen Li and Han Shui and Ludmil Zikatanov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.04786},
  eprint = {2605.04786},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.04786},
}
RIS
TY  - UNPB
TI  - Improving accuracy of finite element solutions by smoothing
AU  - Yuwen Li
AU  - Han Shui
AU  - Ludmil Zikatanov
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2605.04786
UR  - https://arxiv.org/abs/2605.04786
AB  - This paper develops a smoothing-based postprocessing method for improving the accuracy of finite element solutions. The method applies a few smoothing iterations, such as damped Jacobi, symmetrized Gauss-Seidel, or preconditioned conjugate gradient, with the initial guess being the current finite element solution embedded in an augmented finite element space. The resulting procedure is algebraic, easy to implement, and applicable to high-order and three-dimensional discretizations. For symmetric and positive-definite problems, we show that the accuracy of the smoothed solutions is improved by a smoothing factor under additive and multiplicative smoothers. This smoothing factor decays exponentially during the first few smoothing steps. The effectiveness of the proposed method is numerically demonstrated using the Poisson, Maxwell, biharmonic and Helmholtz equations.
ER  -