**1. Introduction**

The finite element method (FEM) consider is the most of flexibility common technique used for dealing with various kinds of application in many fields, for instance, in engineering, in chemistry and in biology. The derivation of a posteriori error estimates for linear and nonlinear parabolic problems are gaining increasing interest and there is a significant implementation of the method now are understandable and available in the literature [1–9]. However, There is less progress has been made comparatively in the proving of a posteriori error bounds for semilinear parabolic problems [10–13]. These estimations play a crucial rule in designing adaptive mesh refinement algorithms and consequently leading to a good accuracy while reducing the computational cost of the scheme.

The key technique used in the proofs is the elliptic reconstruction idea, introduced by Makridakis and Nochetto for spatially discrete conforming FEM [2] and extended to fully discrete conforming FEM by Lakkis and Makridakis [3] These ideas have been carried forward also to fully discrete schemes involving spatially non-conforming/dG methods in [14]. The choice of this technique for deriving a posteriori error for parabolic problem is motivated by the following factors.

First, elliptic reconstruction allows us to utilise the readily available elliptic a posteriori estimates [2] to bound the main part of the spatial error. Second, this technique combines the energy approach and appropriate pointwise representation of the error in order to arrive to optimal order a posteriori estimators in the *L*<sup>∞</sup> *L*<sup>2</sup> � � norm. As a result, this approach will lead to optimal order in both *L*<sup>2</sup> *H*<sup>1</sup> � � and *L*<sup>∞</sup> *L*<sup>2</sup> � �-type norms, while the results obtained by the standard energy methods are only optimal order in *L*<sup>2</sup> *H*<sup>1</sup> � �-type norms.

The aim of this Chapter is to derive a posteriori error bounds for the fully discrete in two cases Lipschitz and non Lipschitz. Continuation Argument will be used to deal with nonlinear forcing terms.
