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, 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 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.

Surface Crouzeix–Raviart element for the Laplace–Beltrami equation

“…As proved in [24], the definition of the tangent gradient (5.1) is invariant under different chosen of regular isomorphic parametrization function r.…”

“…Superconvergent post-processing. In this section, we generalize the parametric polynomial preserving recovery [24] to the surface Crouzeix-Raviart element.…”

“…Then, we reconstruct the local approximation surface S i over Ω i . As in [24], the approximate surface S i can be approximated by graph of a quadratic function on Ω i . That is S i = rh,i (Ω i ) = ∪ ξ∈Ωi (ξ, s i (ξ)), where…”

“…However, there are two barriers to the surface Crouzeix-Raviart element: first, it requires the exact normal vectors; second, it requires the edge centers located on the exact surface. Those two difficulties can be alleviated by going back to the original definition of the covariant derivative as in [24]. In specific, we firstly adopt a least-squares procedure to recover the local parametric map and then employ another least-squares fitting on the parameter domain.…”

Superconvergent gradient recovery for virtual element methods

“…Remark 4.4. As observed in [24], the least-squares fitting procedure will not improve the accuracy of the solution approximation. We can remove one degree of freedom in the least-squares fitting procedure by assuming…”