Optimal convergence analysis of the lowest-order nonconforming virtual element method for general nonlinear parabolic equations

Abstract

A family of compact and linearly implicit Galerkin method is proposed for the general nonlinear parabolic equation based on second-order weighted implicit–explicit schemes in time and the lowest-order nonconforming virtual element discretization in space. The proposed method achieves second-order global accuracy in the temporal directions, and no additional initial iterations are required. To address consistency errors arising from nonconforming virtual element space, we construct two novel elliptic projection operators and rigorously prove the convergence and boundedness of the projection solutions. With the help of a novel elliptic projection operator and the temporal–spatial error splitting technique, we establish the $L^{\infty }$ boundedness and unconditional optimal error estimate of the fully discrete solution. Several numerical experiments are presented to validate our theoretical discoveries.