Optimal error estimations and superconvergence analysis of anisotropic FEMs with variable time steps for reaction–diffusion equations

Abstract

By combining variable-time-step two-step backward differentiation formula (VSBDF2) with anisotropic finite element methods (FEMs), a fully discrete scheme with non-uniform meshes both in time and space is constructed for the reaction–diffusion equations. Two approaches are provided to prove the optimal error estimates in $L^2$-norm and global superconvergence result in $H^1$-norm under a mild adjacent time-step ratio restriction $0<r_k<r_{max}\approx 4.8645$. The first approach is based on the use of anisotropic properties of the interpolation operators, but needs a higher regularity of the solution and is only valid for some special elements. The second approach is based on the combination technique of interpolation and energy projection operators, and needs a lower regularity of the solution, which can be regarded as a unified framework of convergence analysis. Numerical experiments are provided to demonstrate our theoretical analysis.