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In many applications, piecewise continuous functions are commonlyinterpolated over meshes. However, accurate high-order manipulations of suchfunctions can be challenging due to potential spurious oscillations known asthe Gibbs phenomena. To address this challenge, we propose a novel approach,Robust Discontinuity Indicators (RDI), which can efficiently and reliablydetect both C^{0} and C^{1} discontinuities for node-based and cell-averagedvalues. We present a detailed analysis focusing on its derivation and thedual-thresholding strategy. A key advantage of RDI is its ability to handlepotential inaccuracies associated with detecting discontinuities on non-uniformmeshes, thanks to its innovative discontinuity indicators. We also extend theapplicability of RDI to handle general surfaces with boundaries, features, andridge points, thereby enhancing its versatility and usefulness in variousscenarios. To demonstrate the robustness of RDI, we conduct a series ofexperiments on non-uniform meshes and general surfaces, and compare itsperformance with some alternative methods. By addressing the challenges posedby the Gibbs phenomena and providing reliable detection of discontinuities, RDIopens up possibilities for improved approximation and analysis of piecewisecontinuous functions, such as in data remap.

Robust Discontinuity Indicators for High-Order Reconstruction of Piecewise Smooth Functions