This paper focuses on proposing and analyzing a robust \(C^0\) interior penalty method for a gradient-elastic Kirchhoff plate (GEKP) model over polygons, which is a sixth-order singularly perturbed problem. The numerical method is obtained by combining the triangular Hermite element and a \(C^0\) interior penalty method, avoiding the use of higher order shape functions or macroelements. Next, a robust regularity estimate is established for the GEKP model borrowing a regularity result for a triharmonic equation. 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 Céa-like inequality. Thereby, the optimal error estimates are achieved, which are robust with respect to the small size parameter as well. The numerical method is also proved to be convergent without any additional regularity assumption for the exact solution. Some numerical experiments are performed to verify the theoretical findings.
