An a posteriori error estimator based on an equilibrated flux reconstruction is proposed for defeaturing problems in the context of finite element discretizations. Defeaturing consists in the simplification of a geometry by removing features that are considered not relevant for the approximation of the solution of a given PDE. In this work, the focus is on Poisson equation with Neumann boundary conditions on the feature boundary. The estimator accounts both for the so-called defeaturing error and for the numerical error committed by approximating the solution on the defeatured domain. Unlike other estimators that were previously proposed for defeaturing problems, the use of the equilibrated flux reconstruction allows to obtain a sharp bound for the numerical component of the error. Furthermore, it does not require the evaluation of the normal trace of the numerical flux on the feature boundary: this makes the estimator well-suited for finite element discretizations, in which the normal trace of the numerical flux is typically discontinuous across elements. The reliability of the estimator is proven and verified on several numerical examples. Its capability to identify the most relevant features is also shown, in anticipation of a future application to an adaptive strategy.

翻译：针对有限元离散框架下的去特征问题，提出了一种基于平衡通量重构的后验误差估计器。去特征技术通过移除对给定偏微分方程解近似影响较小的几何特征来简化几何结构。本研究聚焦于特征边界上具有诺伊曼边界条件的泊松方程。该估计器同时考虑了所谓的去特征误差以及在简化几何域上近似解时产生的数值误差。与先前提出的去特征问题估计器不同，平衡通量重构方法能够为数值误差分量提供精确上界。此外，该方法无需评估特征边界上数值通量的法向迹线——这使该估计器特别适用于数值通量法向迹线在单元间通常不连续的有限元离散方案。通过多个数值算例验证了估计器的可靠性，并展示了其识别关键特征的能力，为后续应用于自适应策略奠定了基础。