Unconditional superconvergence analysis for nonlinear hyperbolic equation with nonconforming finite element

Nonlinear hyperbolic equation is studied by developing a linearized Galerkin finite element method (FEM) with nonconforming EQ1rot element. A time-discrete system is established to split the error into two parts which are called the temporal error and the spatial error, respectively. The temporal error is proved skillfully which leads to the analysis for the regularity of the time-discrete system. The spatial error is derived τ-independently with order O(h2+hτ) in broken H1-norm. The final unconditional superclose result of u with order O(h2+τ2) is deduced based on the above achievements. The two typical characters of this nonconforming EQ1rot element (see Lemma 1 below) play an important role in the procedure of proof. At last, a numerical example is provided to support the theoretical analysis. Here, h is the subdivision parameter, and τ, the time step.