Title: Parametric Polynomial Preserving Recovery on Manifolds,
Abstract: 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
which have been assumed for some significant gradient recovery methods in the
literature. Another advantage is that superconvergence is guaranteed for PPPR
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 other methods. As
an application, we show that 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
make comparisons with state of the art methods.