A Posteriori $L_{\infty}(H^{1})$ Error Bound in Finite Element Approximation of Semdiscrete Semilinear Parabolic Problems

This work aims to construct a posteriori error bounds for semidiscrete semilinear parabolic problems in terms of $L$∞ (H1) norm. The curtail idea is to adapt the elliptic reconstruction technique introduced by Makridakis and Nochetto [8], this allows us to use error estimators derived for elliptic problems in order to obtain parabolic estimators that are of optimal order in space and time for non-Lipschitz nonlinearities, using relevant Sobolev Imbedding through continuation argument. These error bounds are subsequently used to reduce the computational of the scheme.