Superconvergence analysis of an energy stable scheme with three step backward differential formula-finite element method for nonlinear reaction-diffusion equation

A three step backward differential formula scheme is proposed for nonlinear reaction-diffusion equation and superconvergence results are studied with Galerkin finite element method unconditionally. Energy stability is testified for the constructed scheme with an artificial term. Splitting technique is utilized to get rid of the ratio between the time step size tau and the subdivision parameter h. Temporal error estimate in H-2-norm is derived, which leads to the boundedness of the solutions of the time-discrete equations. Unconditional spatial error estimate in L-2-norm is deduced which help bound the numerical solutions in L-infinity-norm. Superconvergent property of un in H-1-norm with order O(h2+tau 3) is obtained by taking difference between two time levels of the error equations unconditionally. The global superconvergent property is deduced through the above results. Two numerical examples show the validity of the theoretical analysis.