Unconditional superconvergence analysis of a modified nonconforming energy stable BDF2 FEM for Sobolev equations with Burgers’ type nonlinearity

Abstract

The aim of this paper is to develop a modified energy stable 2-step backward differentiation formula (BDF2) fully discrete scheme and study its superconvergence behavior with the nonconforming quadrilateral quasi-Wilson element for the Sobolev equations with Burgers’ type nonlinearity. The existence and uniqueness of the numerical solution are proved by Brouwer fixed point theorem. Then, different from the so-called time–space splitting technique, some typical properties of this element especially the consistency error can reach order O(h2), one order higher than its interpolation error, and the interpolation post-processing approach are used to derive the unconditional supercloseness and superconvergence results on quadrilateral meshes without the restriction between the mesh size h and the time step τ. At last, a numerical example is carried out and a comparison with the Wilson element is also given to confirm the theoretical analysis.