A posteriori error estimation for coarse or distorted meshes
Abstract
Finite element simulations carried out on meshes issued from real-world geometries — patient-specific anatomies, scanned industrial parts, fractured media — almost never satisfy the shape-regularity assumptions under which classical a posteriori error analysis is written. Practitioners are therefore left without a quantitative answer to a very legitimate question: how wrong is my simulation when my mesh is coarse and contains badly shaped elements? In this work, we study to which extent equilibrated flux estimators can provide such an answer. Indeed, they deliver a guaranteed and fully computable upper bound of the energy-norm discretization error, with a constant equal to one; and, crucially for the present study, its derivation uses no shape-regularity, quasi-uniformity or angle condition. The only geometric ingredient is the Poincaré constant on a single cell, which Payne and Weinberger showed to be at most hK /π on any convex cell K, however distorted (hK being the cell size). Mesh quality affects the local efficiency — the sharpness of the estimator — but never its reliability, so that the failure mode under distortion is pessimism rather than false confidence. We study the case of conforming Lagrange finite element methods: the analysis is written for general cells in two and three space dimensions, and the numerical experiments are carried out with simplicial elements in two dimensions. We detail two flux reconstructions — a global H(div) mixed one and the patchwise one (in the spirit of works from Braess and Schöberl or Ern and Vohralík) — and we map the geometric quality measures used by the meshing community onto the constants that do, and do not, appear in the analysis. The estimator is then assessed numerically on families of coarse, locally damaged, anisotropic and globally distorted meshes, against the residual, Zienkiewicz–Zhu-type recovery and Bank–Weiser estimators, and a duality argument extends the guarantee to a quantity of interest. Across the 49 computations reported here — including meshes of a letter glyph produced by two independent meshers, one of which ships with a zero-area cell — the guaranteed bound never failed, while the recovery estimator reported less error than was committed in every run on a mesh of the unit square. The Bank–Weiser estimator is sharper on every well-shaped mesh, but a single needle cell out of two thousand costs it a factor of 2.6, and it falls below the true error on the L-shaped domain, where nothing in its output signals the failure. The two reconstructions of the guaranteed bound are found to degrade under different distortions: the global one under global anisotropy, the patchwise one under an accumulation of flattened cells within a single patch.
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BibTeX
@misc{chouly2026posteriori,
title = {{A posteriori error estimation for coarse or distorted meshes}},
author = {Franz Chouly and Michel Duprez and Claudio Lobos and Zuqi Tang},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12088},
eprint = {2610.12088},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05786448v1},
note = {CC BY-NC 4.0},
}
RIS
TY - UNPB
TI - A posteriori error estimation for coarse or distorted meshes
AU - Franz Chouly
AU - Michel Duprez
AU - Claudio Lobos
AU - Zuqi Tang
PY - 2026
DA - 2026/10/08
PB - HAL
DO - 10.48550/arxiv.2610.12088
UR - https://hal.science/hal-05786448v1
AB - Finite element simulations carried out on meshes issued from real-world geometries — patient-specific anatomies, scanned industrial parts, fractured media — almost never satisfy the shape-regularity assumptions under which classical a posteriori error analysis is written. Practitioners are therefore left without a quantitative answer to a very legitimate question: how wrong is my simulation when my mesh is coarse and contains badly shaped elements? In this work, we study to which extent equilibrated flux estimators can provide such an answer. Indeed, they deliver a guaranteed and fully computable upper bound of the energy-norm discretization error, with a constant equal to one; and, crucially for the present study, its derivation uses no shape-regularity, quasi-uniformity or angle condition. The only geometric ingredient is the Poincaré constant on a single cell, which Payne and Weinberger showed to be at most hK /π on any convex cell K, however distorted (hK being the cell size). Mesh quality affects the local efficiency — the sharpness of the estimator — but never its reliability, so that the failure mode under distortion is pessimism rather than false confidence. We study the case of conforming Lagrange finite element methods: the analysis is written for general cells in two and three space dimensions, and the numerical experiments are carried out with simplicial elements in two dimensions. We detail two flux reconstructions — a global H(div) mixed one and the patchwise one (in the spirit of works from Braess and Schöberl or Ern and Vohralík) — and we map the geometric quality measures used by the meshing community onto the constants that do, and do not, appear in the analysis. The estimator is then assessed numerically on families of coarse, locally damaged, anisotropic and globally distorted meshes, against the residual, Zienkiewicz–Zhu-type recovery and Bank–Weiser estimators, and a duality argument extends the guarantee to a quantity of interest. Across the 49 computations reported here — including meshes of a letter glyph produced by two independent meshers, one of which ships with a zero-area cell — the guaranteed bound never failed, while the recovery estimator reported less error than was committed in every run on a mesh of the unit square. The Bank–Weiser estimator is sharper on every well-shaped mesh, but a single needle cell out of two thousand costs it a factor of 2.6, and it falls below the true error on the L-shaped domain, where nothing in its output signals the failure. The two reconstructions of the guaranteed bound are found to degrade under different distortions: the global one under global anisotropy, the patchwise one under an accumulation of flattened cells within a single patch.
ER -