Unconditionally optimal error estimates of linearized Crank-Nicolson Virtual element methods for quasilinear parabolic problems on general polygonal meshes

Abstract

In this paper, we construct, analyze, and numerically validate a linearized Crank-Nicolson virtual element method (VEM) for solving quasilinear parabolic problems on general polygonal meshes. In particular, we consider the more general nonlinear term a(x,u), which does not require Lipschitz continuity or uniform ellipticity conditions. To ensure that the fully discrete solution remains bounded in L∞ norm, we construct two novel elliptic projections and apply a new error splitting technique. With the help of the boundedness of the numerical solution and delicate analysis of the nonlinear term, we derive the optimal error estimates for any k-order VEMs without any time-step restrictions. Numerical experiments on various polygonal meshes validate the accuracy of the theoretical analysis and the unconditional convergence of the proposed scheme.