Unconditional error analysis of linearized BDF2 mixed virtual element method for semilinear parabolic problems on polygonal meshes

Abstract

In this paper, we construct, analyze, and numerically validate a class of H(div)-mixed virtual element method for the semilinear parabolic problem in mixed form, in which the parabolic problem is reformulated in terms of the velocity and the pressure of the time-dependent Darcy flow. The Newton linearized method for the nonlinear term is designed to cooperate with the second-order backward differentiation formula of the temporal discretization scheme. This allows each time step to only require the solution of a small and well-structured linear system rather than the solution of a nonlinear system. The linearization improves computational efficiency without decreasing convergence rates. Moreover, the “Fortin” operator and its approximation properties are applied to derive an optimal error estimates O(hk+1+τ2) for the virtual element solution of the velocity and the pressure. Finally, its remarkable performance is illustrated by several numerical examples that also validate the theoretical rates of convergence.