Parametric Polynomial Preserving Recovery on Manifolds
For researchers in numerical analysis and computational geometry, PPPR provides a more robust and simpler gradient recovery method for manifold data.
This paper introduces parametric polynomial preserving recovery (PPPR) for gradient recovery on discretized manifolds, which does not require tangent spaces or symmetry conditions, achieving superconvergence that is stable under high curvature. Numerical examples demonstrate its effectiveness and superiority over existing methods.
This paper investigates gradient recovery schemes for data defined on discretized manifolds. The proposed method, parametric polynomial preserving recovery (PPPR), does not require the tangent spaces of the exact manifolds, and they have been assumed for some significant gradient recovery methods in the literature. Another advantage of PPPR is that superconvergence is guaranteed without the symmetric condition which has been asked in the existing techniques. There is also numerical evidence that the superconvergence by PPPR is high curvature stable, which distinguishes itself from the others. As an application, we show its capability of constructing an asymptotically exact \textit{a posteriori} error estimator. Several numerical examples on two-dimensional surfaces are presented to support the theoretical results and comparisons with existing methods are documented.