AI 中文总结
针对椭圆界面问题，提出基于平衡通量重建的后验误差估计器，考虑建模与数值误差，局部化在网格单元，能评估不同特征影响，可指示粗化与保留特征，证明了可靠性并经数值验证。
AI 中文摘要
我们提出了一种后验误差估计器，它依赖于平衡通量重建，以便在椭圆界面问题中做出分析感知的界面粗化决策。界面粗化包括简化分隔具有不同物理性质区域的内部界面，以促进网格划分过程并减少自由度总数。此过程扩展了去特征化的概念，这里的特征位于内部界面。重点是具有不连续扩散系数的拉普拉斯问题。该估计器考虑了界面粗化产生的建模误差和粗化界面问题解的离散近似产生的数值误差。它局部化在网格单元上，其常数明确跟踪扩散系数之间的对比度。不同特征的影响可单独评估，给出局部贡献，指示在给定网格尺寸下哪些特征应粗化，哪些应保留。使用平衡通量重建使我们能够严格界定整体数值误差源。我们证明了估计器的可靠性，并通过几个数值例子进行了验证，包括内部界面完全去除的情况。
英文摘要
We propose an a posteriori error estimator that relies on an equilibrated flux reconstruction to enable analysis-aware interface coarsening decisions in elliptic interface problems. Interface coarsening consists of simplifying an internal interface that separates regions characterized by different physical properties in order to facilitate the meshing process and reduce the total number of degrees of freedom. This process extends the concept of defeaturing, where small features on the boundary of a computational domain are removed. Here, the features lie on an interior interface instead. The focus is on a Laplace problem with a discontinuous diffusion coefficient. The estimator accounts for both the modeling error arising from the interface coarsening and the numerical error from the discrete approximation of the solution to the coarsened-interface problem. It is localized on the mesh elements, and its constants explicitly track the contrast between the diffusion coefficients. The impact of different features can be assessed individually, yielding local contributions that indicate which features to coarsen and which to retain for a given mesh size. The use of an equilibrated flux reconstruction allows us to sharply bound the bulk numerical source of error. We prove the reliability of the estimator and verify it across several numerical examples, including the case where an internal interface is fully removed.
