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.

翻译：本文研究了带弱奇异核的广义Burgers-Huxley方程（GBHE）基于残差的后验误差估计。首先，我们利用空间维度的间断伽辽金有限元方法（DGFEM），为稳态GBHE和带记忆的半离散GBHE建立了可靠且高效的误差估计子。此外，结合时间域的向后欧拉和Crank-Nicolson离散格式以及空间维度的DGFEM，我们引入了一个考虑历史影响的全离散GBHE估计子。本文还确立了稳态GBHE和GBHE的最优$L^2$误差估计。最后，通过数值结果验证了所提出的误差估计子的有效性，展示了其在自适应细化策略中的效能。