A robust $C^0$ interior penalty method for a gradient-elastic Kirchhoff plate model

TL;DR

This paper introduces a robust $C^0$ interior penalty finite element method for the gradient-elastic Kirchhoff plate model, achieving optimal error estimates and convergence without extra regularity assumptions, verified through numerical experiments.

Contribution

It develops a novel $C^0$ interior penalty method combining Hermite elements for the GEKP model, with proven robustness and optimal error estimates.

Findings

Achieves optimal error estimates for the GEKP model.

Proves convergence without additional regularity assumptions.

Numerical experiments confirm theoretical results.

Abstract

This paper is devoted to proposing and analyzing a robust $C^0$ interior penalty method for a gradient-elastic Kirchhoff plate (GEKP) model over a convex polygon. The numerical method is obtained by combining the triangular Hermite element and a $C^0$ interior penalty method, which can avoid the use of higher order shape functions or macroelements. Next, a robust regularity estimate is established for the GEKP model based on our earlier result for a triharmonic equation on a convex polygon. Furthermore, some local lower bound estimates of the a posteriori error analysis are established. These together with an enriching operator and its error estimates lead to a C\'{e}a-like lemma. Thereby, the optimal error estimates are achieved, which are also robust with respect to the small size parameter. In addition, it is proved that this numerical method is convergent without any additional regularity assumption for the exact solution. Some numerical experiments are performed to verify the theoretical findings.