Error estimates with polynomial growth O(ε−1) for the HHO method on polygonal meshes of the Allen-Cahn model

A novel approach is presented to tackle the Allen-Cahn equation arising from phase separation in alloys, by utilizing the hybrid high-order (HHO) method on polygonal meshes. The primary challenge in this equation lies in employing a straightforward Gronwall inequality-type argument for error estimation with exponential growth factor $e^{(CT/{\epsilon }^2)}$ as ε approaches zero. The application of the discrete Lyapunov functional and the discrete HHO spectrum estimate of the linearized Allen-Cahn operator ${\lambda }_{AC}^{HHO}$ are used to overcome this exponential growth factor and achieve polynomial growth of order $O\left({\epsilon }^{-1}\right)$ for error bounds in error estimations. Rigorous convergence analyses are established for the fully implicit schemes, which are energy stable. However, due to the implicit processing of the nonlinear term, the computational cost significantly increases. To enhance computational efficiency, a static condensation process is hired by using the HHO method, resulting in optimal convergence rates in $L^2$ norm. Finally, various numerical experiments on diverse meshes are conducted to validate our theoretical findings.