Parametric Polynomial Preserving Recovery on Manifolds

TL;DR

This paper introduces PPPR, a gradient recovery method on discretized manifolds that guarantees superconvergence without tangent space knowledge and demonstrates high curvature stability through numerical experiments.

Contribution

The paper presents PPPR, a novel gradient recovery scheme on manifolds that does not require tangent space information and guarantees superconvergence without symmetry conditions.

Findings

PPPR achieves superconvergence on discretized manifolds.

PPPR is stable under high curvature conditions.

Numerical examples validate theoretical results and outperform existing methods.

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, 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.