Superconvergence analysis for nonlinear Ginzburg-Landau equation with backward differential formula finite element method

Superconvergence analysis for nonlinear Ginzburg-Landau equation with 2-step backward differential formula (BDF) finite element method (FEM) is proposed. Auxiliary equation known as a time-discrete system is suggested to split the error into temporal error and spatial error. Prior estimates for solutions of the finite element approximation equation and the auxiliary equation in $H^{1}$-norm are obtained by Hölder inequality, Gagliardo-Nirenberg inequality and the other techniques. The uniqueness for numerical solution, the temporal error in $H^{2}$-norm and the unconditional spatial error in $L^{2}$-norm are derived with the help of the prior estimates. As a result, the numerical solution in $L^{\infty}$-norm is bounded. Based on the deduced achievements, unconditional superclose result of order $O(h^{2}+\tau ^{2})$ in $H^{1}$-norm is gained by taking difference between two adjacent time levels of the error equation. At last, global superconvergence result is shown through the known interpolated postprocessing technique. Two numerical examples show the validity of the theoretical analysis. Here, τ is the time step and h is the subdivision parameter.