Convergence and Superconvergence Analysis of a Nonconforming Finite Element Variable-Time-Step BDF2 Implicit Scheme for Linear Reaction-Diffusion Equations

In this paper, an effective fully-discrete implicit scheme for solving linear reaction-diffusion equations is constructed by using the variable-time-step two-step backward differentiation formula (VSBDF2) in time combining with the nonconforming finite element methods in space. By introducing a modified energy projection operator, a discrete Laplace operator, the discrete orthogonal convolution kernels, we obtain the optimal and sharp error estimates of order \(O(h^2+\tau ^2)\) $O(h^2+{\tau }^2)$ in \(L^2\) $L^2$ -norm and \(O(h+\tau ^2)\) $O(h+{\tau }^2)$ in \(H^1\) $H^1$ -norm under a mild restriction \(0<r_k< r_{\max }\approx 4.8645\) $0<r_k<r_{max}\approx 4.8645$ for the ratio of adjacent time steps \(r_k\) $r_k$ . Furthermore, with the help of a modified discrete Grönwall inequality and the combination technique of interpolation and projection operators, we achieved the superclose result between the interpolation function \(I_hu\) $I_{h}u$ and finite element solution \(u_h\) $u_h$ in \(H^1\) $H^1$ -norm of order \(O(h^2+\tau ^2)\) $O(h^2+{\tau }^2)$ , which together with the interpolation postprocessing operator \(\Pi _{2h}\) ${\mathrm{\Pi }}_{2h}$ leads to the global superconvergence result about \(u-\Pi _{2h}u_h\) $u-{\mathrm{\Pi }}_{2h}{u}_h$ in \(H^1\) $H^1$ -norm of order \(O(h^2+\tau ^2)\) $O(h^2+{\tau }^2)$ . Finally, numerical tests are provided to verify the theoretical analysis.