Unified analysis of conforming and nonconforming virtual element methods for nonlinear Sobolev equations

In this paper, we establish a unified analysis of both the conforming and nonconforming virtual element methods for a class of nonlinear Sobolev equations in which the nonlinear diffusion coefficients, a(u) and b(u), may not satisfy global Lipschitz continuity or elliptical conditions. The discretization for the spatial variables is the virtual element method on general polygonal meshes, whereas the temporal discretization is accomplished with a linearized Crank–Nicolson scheme. By employing a temporal-spatial error splitting argument and introducing a novel projection operator, we prove that the numerical solution is bounded in the \(L^{\infty }\) $L^{\infty }$ -norm without any time step-size conditions. Moreover, for both conforming and nonconforming ones, optimal error estimates in \(L^2\) $L^2$ -norm and \(H^{1}\) $H^1$ -seminorm for the fully scheme are obtained unconditionally. Numerical results are provided on different polygonal meshes to confirm the theoretical analysis.