Locally computable error estimators for conforming approximations of interface problems cannot be robust
Abstract
We prove an impossibility result for finite-range locally computable a posteriori error estimators for the conforming finite element discretization of an elliptic interface problem. For a class of interface problems with a checkerboard cross-point and {1,M}-valued coefficients, we construct two problem instances on the same interface-fitted mesh. The two instances share a piecewise-constant load for which the finite element solution and the data oscillation both vanish. The ratio of their exact energy errors grows at least proportionally to M1/4. Locality together with efficiency forces identical estimator values for these instances. The product of reliability and efficiency constants is therefore bounded below by a constant multiple of M1/4. Consequently, any locally computable error estimator for interface problems, whether of residual, equilibrated, or recovery type, cannot be simultaneously reliable and efficient with contrast-independent constants.
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BibTeX
@misc{li2026locally,
title = {{Locally computable error estimators for conforming approximations of interface problems cannot be robust}},
author = {Yuwen Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.27817},
eprint = {2609.27817},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27817},
}
RIS
TY - UNPB
TI - Locally computable error estimators for conforming approximations of interface problems cannot be robust
AU - Yuwen Li
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2609.27817
UR - https://arxiv.org/abs/2609.27817
AB - We prove an impossibility result for finite-range locally computable a posteriori error estimators for the conforming finite element discretization of an elliptic interface problem. For a class of interface problems with a checkerboard cross-point and {1,M}-valued coefficients, we construct two problem instances on the same interface-fitted mesh. The two instances share a piecewise-constant load for which the finite element solution and the data oscillation both vanish. The ratio of their exact energy errors grows at least proportionally to M^1/4. Locality together with efficiency forces identical estimator values for these instances. The product of reliability and efficiency constants is therefore bounded below by a constant multiple of M^1/4. Consequently, any locally computable error estimator for interface problems, whether of residual, equilibrated, or recovery type, cannot be simultaneously reliable and efficient with contrast-independent constants.
ER -