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)+T-2) in L-2-norm and O(h + T-2) in H1 -norm under a mild restriction 0 < r(k) < r(max) approximate to 4.8645 for the ratio of adjacent time steps r(k.) Furthermore, with the help of a modified discrete Gronwall inequality and the combination technique of interpolation and projection operators, we achieved the superclose result between the interpolation function I(h)u and finite element solution u(h) in H-1 norm of order O(h + T-2) which together with the interpolation postprocessing operator Pi(2h) leads to the global superconvergence result about u - Pi(2h)u(h) in H-1 -norm of order O(h + T-2). Finally, numerical tests are provided to verify the theoretical analysis.