Superconvergnce analysis of an energy-stable implicit scheme with variable time steps and anisotropic spatial nonconforming finite elements for the nonlinear Sobolev equations

Abstract

A fully discrete implicit scheme is presented and analyzed for the nonlinear Sobolev equations, which combines an anisotropic spatial nonconforming FEM with the variable-time-step BDF2 such that nonuniform meshes can be adopted in both time and space simultaneously. We prove that the fully discrete scheme is uniquely solvable, possesses the modified discrete energy dissipation law, and achieves second-order accuracy in both temporal and spatial directions under mild meshes conditions (adjacent time-step ratio condition $0<r_n:=\frac{{\tau }_n}{{\tau }_{n-1}}<r_{\max }\approx 4.8645$ and anisotropic space meshes). The analysis approach involves a priori boundedness of the finite element solution, anisotropic properties of the element, energy projection error, DOC kernels and a modified discrete Grönwall inequality. Theoretical results reveal that the error in $H^1$-norm is sharp in time and optimal or even superconvergent in space. Abundant numerical experiments verify the theoretical results, and demonstrate the efficiency and accuracy of the proposed fully discrete scheme.