Smoother-type a posteriori error estimates for finite element methods
Abstract
This work develops user-friendly a posteriori error estimates of finite element methods, based on smoothers of linear iterative solvers. The proposed method employs simple smoothers, such as Jacobi or Gauss-Seidel iteration, on an auxiliary finer mesh to process the finite element residual for a posteriori error control. The implementation has linear complexity and requires only a coarse-to-fine prolongation operator. For symmetric problems, we prove the reliability and efficiency of smoother-type error estimators under a saturation assumption. Numerical experiments for various PDEs demonstrate that the proposed smoother-type error estimators outperform residual-type estimators in accuracy and exhibit robustness with respect to parameters and polynomial degrees.
Cite
@article{arxiv.2510.07677,
title = {Smoother-type a posteriori error estimates for finite element methods},
author = {Yuwen Li and Han Shui},
journal= {arXiv preprint arXiv:2510.07677},
year = {2026}
}