  High-order surface reconstruction is an important technique for CAD-free,
mesh-based geometric and physical modeling, and for high-order numerical
methods for solving partial differential equations (PDEs) in engineering
applications. In this paper, we introduce a novel method for accurate and
robust reconstructions of piecewise smooth surfaces from a triangulated
surface. Our proposed method extends the Continuous Moving Frames (CMF) and the
Weighted Averaging of Local Fittings (WALF) methods (Engrg. Comput. 28 (2012))
in two main aspects. First, it utilizes a Hermite-style least squares
approximation to achieve fourth and higher-order accuracy with compact support,
even if the input mesh is relatively coarse. Second, it introduces an iterative
feature-aware parameterization to ensure high-order accurate, G0 continuous
reconstructions near sharp features. We present the theoretical framework of
the method and compare it against CMF and WALF in terms of accuracy and
stability. We also demonstrate that the use of the Hermite-style reconstruction
in the solutions of PDEs using finite element methods (FEM), and show that
quartic and sextic FEMs using the high-order reconstructed surfaces produce
nearly identical results as using exact geometry while providing additional
flexibility.
