In many applications, piecewise continuous functions are commonly interpolated over meshes. However, accurate high-order manipulations of such functions can be challenging due to potential spurious oscillations known as the Gibbs phenomena. To address this challenge, we propose a novel approach, Robust Discontinuity Indicators (RDI), which can efficiently and reliably detect both C^{0} and C^{1} discontinuities for node-based and cell-averaged values. We present a detailed analysis focusing on its derivation and the dual-thresholding strategy. A key advantage of RDI is its ability to handle potential inaccuracies associated with detecting discontinuities on non-uniform meshes, thanks to its innovative discontinuity indicators. We also extend the applicability of RDI to handle general surfaces with boundaries, features, and ridge points, thereby enhancing its versatility and usefulness in various scenarios. To demonstrate the robustness of RDI, we conduct a series of experiments on non-uniform meshes and general surfaces, and compare its performance with some alternative methods. By addressing the challenges posed by the Gibbs phenomena and providing reliable detection of discontinuities, RDI opens up possibilities for improved approximation and analysis of piecewise continuous functions, such as in data remap.

翻译：在众多应用中，分段连续函数通常基于网格进行插值。然而，由于吉布斯现象可能引发的虚假振荡，这类函数的高精度高阶处理颇具挑战性。针对这一难题，我们提出了一种创新方法——鲁棒间断性指示器（RDI），可高效可靠地检测节点值及单元平均值中的C^{0}和C^{1}间断。我们详细分析了其推导过程及双阈值策略。RDI的核心优势在于，凭借其创新的间断性指示机制，能够有效处理非均匀网格上间断检测中可能存在的误差，从而确保检测精度。此外，我们将RDI的适用范围扩展至包含边界、特征点及脊点的通用曲面，显著提升了其在多种场景下的灵活性与实用性。为验证RDI的鲁棒性，我们在非均匀网格与通用曲面上开展了一系列实验，并将其性能与若干替代方法进行对比。通过克服吉布斯现象带来的挑战并提供可靠的间断性检测，RDI为改进分段连续函数（如数据重映射领域）的逼近与分析开辟了新途径。