Dissipative nonconforming virtual element method for the fourth order nonlinear extended Fisher-Kolmogorov equation

Abstract

A unified framework of nonconforming virtual element methods (VEMs), including C0 and fully nonconforming virtual elements, are built for the fourth order extended Fisher-Kolmogorov (EFK) equation. For the constructed semi-discrete and fully discrete schemes, the unique solvability is obtained. Then the numerical schemes are proved to be dissipative in the senses of discrete energy, which can be further used to derive the priori bounds of the discrete solutions. In addition, a new analytical technique is used to derive the unconditional convergence of the numerical schemes without any restriction on the grid ratio. Finally, numerical experiments are provided to confirm the theoretical results.