Superconvergence analysis for nonlinear reaction-diffusion equation with BDF-FEM

An implicit backward differential formula (BDF) scheme is constructed for nonlinear reaction-diffusion equation, and superconvergence results are studied with the Galerkin finite element method (FEM). The existence and uniqueness of the numerical solution are given by using of the function's monotonicity. Splitting technique is utilized to get rid of the ratio between the time step tau and the subdivision parameter h. Temporal error estimates in H-2-norm are derived, which implies the boundedness of the solutions about the time-discrete equations in H-2-norm. Unconditional spatial error estimates in L-2-norm are deduced, which help bound the numerical solutions in L-infinity-norm. The unconditional superconvergent property of u(n) in H-1-norm with order O(h(2) + tau(2)) is obtained by the relationship between the classic Ritz projection operator and the the corresponding interpolation operator. The global superconvergent property is deduced through the above results. Two numerical examples show the validity of the theoretical analysis.