  This paper explores the residual based a posteriori error estimations for the
generalized Burgers-Huxley equation (GBHE) featuring weakly singular kernels.
Initially, we present a reliable and efficient error estimator for both the
stationary GBHE and the semi-discrete GBHE with memory, utilizing the
discontinuous Galerkin finite element method (DGFEM) in spatial dimensions.
Additionally, employing backward Euler and Crank Nicolson discretization in the
temporal domain and DGFEM in spatial dimensions, we introduce an estimator for
the fully discrete GBHE, taking into account the influence of past history. The
paper also establishes optimal $L^2$ error estimates for both the stationary
GBHE and GBHE. Ultimately, we validate the effectiveness of the proposed error
estimator through numerical results, demonstrating its efficacy in an adaptive
refinement strategy.
