Unconditional superconvergence analysis of a novel energy dissipation nonconforming Crank-Nicolson FEM for Sobolev equations with high order Burgers' type nonlinearity

Abstract

A novel energy dissipation Crank-Nicolson (C-N) fully discrete scheme is established by low order nonconforming $E{Q}_1^{rot}$ element for solving the Sobolev equations with high order Burgers' type nonlinearity. Firstly, the boundedness of the discrete solution in the broken $H^1$-norm is achieved directly by the energy dissipation property without using the known time-space splitting technique in the existing literatures, and its well-posedness is demonstrated by the Brouwer fixed point theorem. Secondly, by utilizing the special characters of nonconforming $E{Q}_1^{rot}$ element, the unconditional superclose result of order $O(h^2+{\tau }^2)$ in the broken $H^1$-norm is gained strictly with no restrictions between the spatial partition parameter h and the time step τ. Moreover, the corresponding global superconvergent error estimate of order $O(h^2+{\tau }^2)$ is proved by applying an interpolation post-processing approach. Thirdly, an application to some different finite elements and nonlinear PDEs is discussed, which shows that the proposed scheme and the analysis presented herein can be considered as a general framework to cope with. Lastly, the theoretical results are validated by four numerical examples.