A posteriori error estimates for the Generalized Burgers-Huxley equation with weakly singular kernels

Abstract: 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.