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

A novel energy dissipation Crank-Nicolson (C-N) fully discrete scheme is established by low order nonconforming E Q 1 r o t 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 r o t element, the unconditional superclose result of order O ( h 2 + τ 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 + τ 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.