Convergence analysis of weak Galerkin finite element variable-time-step BDF2 implicit scheme for parabolic equations

Abstract

In this paper, we propose a fully discrete implicit method for parabolic problem. The variable-time-step BDF2 method is applied in time combining with the weak Galerkin finite element method in space. Optimal error estimates of $O(h^r+{\tau }^2)$ in $H^1$-norm and $O(h^{r+1}+{\tau }^2)$ in $L^2$-norm are derived under the time-step ratio $0<r_k⩽4.8645$. Numerical experiments confirm the theoretical findings. Furthermore, an adaptive scheme is introduced and validated to enhance the computational performance.