SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2439-2458, December 2024.
Abstract. 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 a 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 us 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.

中文翻译：

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平衡通量：特征去除问题的后验误差估计器


SIAM 数值分析杂志，第 62 卷，第 6 期，第 2439-2458 页，2024 年 12 月。

抽象。该文提出一种基于平衡磁通量重构的后验误差估计器，用于有限元离散化背景下的特征去除问题。特征去除包括通过删除被认为与给定偏微分方程的解的近似无关的特征来简化几何结构。在这项工作中，重点是特征边界上具有诺依曼边界条件的泊松方程。估计器既考虑了所谓的去特征误差，也考虑了通过在去特征域上近似解而产生的数值误差。与之前为特征去除问题提出的其他估计器不同，使用平衡磁通量重建使我们能够获得误差的数值分量的尖锐界限。此外，它不需要计算特征边界上数值通量的法向轨迹：这使得估计器非常适合有限元离散化，其中数值通量的法向轨迹通常在单元之间是不连续的。估计器的可靠性在几个数值示例中得到了证明和验证。它还显示了识别最相关特征的能力，以预测未来对自适应策略的应用。