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 EQ1rot element for solving the Sobolev equations with high order Burgers' type nonlinearity. Firstly, the boundedness of the discrete solution in the broken H1-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 EQ1rot element, the unconditional superclose result of order O(h2+τ2) in the broken H1-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(h2+τ2) 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.