A high order numerical method for analysis and simulation of 2D semilinear Sobolev model on polygonal meshes

Abstract

In this article, we design and analyze a hybrid high-order method for a semilinear Sobolev model on polygonal meshes. The method offers distinct advantages over traditional approaches, demonstrating its capability to achieve higher-order accuracy while reducing the number of unknown coefficients. We derive error estimates for the semi-discrete formulation of the method. Subsequently, these convergence rates are employed in full discretization with the Crank–Nicolson scheme. The method is demonstrated to converge optimally with orders of $O\left({\tau }^2+h^{k+1}\right)$ in the energy-type norm and $O\left({\tau }^2+h^{k+2}\right)$ in the $L^2$ norm. The reported method is supported by a series of computational tests encompassing linear, semilinear and Allen–Cahn models.