Superconvergence of differential structure for finite element methods on perturbed surface meshes

TL;DR

This paper investigates superconvergence phenomena for differential structures on perturbed surface meshes, introducing geometric supercloseness and an algorithmic framework for gradient recovery without exact geometric data, supported by numerical validation.

Contribution

It introduces geometric supercloseness and a new gradient recovery framework applicable to perturbed surfaces, advancing superconvergence analysis in finite element methods.

Findings

Proves superconvergence of gradient recovery on deviated surfaces.

Develops an algorithmic framework for gradient recovery without exact geometry.

Numerical examples validate the theoretical superconvergence results.

Abstract

Superconvergence of differential structure on discretized surfaces is studied in this paper. The newly introduced geometric supercloseness provides us with a fundamental tool to prove the superconvergence of gradient recovery on deviated surfaces. An algorithmic framework for gradient recovery without exact geometric information is introduced. Several numerical examples are documented to validate the theoretical results.