Convergence and superconvergence analysis for nonlinear delay reaction–diffusion system with nonconforming finite element

Abstract

AbstractIn this article, we propose and analyze several numerical methods for the nonlinear delay reaction–diffusion system with smooth and nonsmooth solutions, by using Quasi‐Wilson nonconforming finite element methods in space and finite difference methods (including uniform and nonuniform L1 and L2‐1σ schemes) in time. The optimal convergence results in the senses of L2‐norm and broken H1‐norm, and H1‐norm superclose results are derived by virtue of two types of fractional Grönwall inequalities. Then, the interpolation postprocessing technique is used to establish the superconvergence results. Moreover, to improve computational efficiency, fast algorithms by using sum‐of‐exponential technique are built for above proposed numerical schemes. Finally, we present some numerical experiments to confirm the theoretical correctness and show the effectiveness of the fast algorithms.