Long-time error estimation for semi-linear parabolic equations

The long-time error estimation approach of Sun and Ewing (Dyn. Contin. Discrete Impuls. Systems Ser. B Appl. Algorithms, 9 (2002) 115–129) is applied here for the error analysis and estimation of linear and semi-linear parabolic partial differential equations. The analysis is carried out using the stability–smoothing indicator, the smoothing assumption, the moving attractor, the exact error propagation and the two-level error propagation analysis introduced by Sun and Ewing (Dyn. Contin. Discrete Impuls. Systems Ser. B Appl. Algorithms, 9 (2002) 115–129). Moreover, an inverse elliptic projection is employed here as a key technique in dealing with the spatial discretization error. The error estimates obtained are uniform in time. The results are substantiated by a complete mathematical analysis and numerical experiments.