<i>hp</i> -discontinuous Galerkin method for the generalized Burgers–Huxley equation with weakly singular kernels

Abstract In this work we investigate the $hp$-discontinuous Galerkin (DG) time-stepping method for the generalized Burgers–Huxley equation with memory, a nonlinear advection–diffusion–reaction problem featuring weakly singular kernels. We derive a priori error estimates for the semidiscrete scheme using $hp$-DG time-stepping, with explicit dependence on the local mesh size, polynomial degree and solution regularity, achieving optimal convergence in the energy norm. For the fully discrete scheme we initially implement the $hp$-conforming finite element method, followed by the $hp$-DG method. We establish the well-posedness and stability of the fully discrete scheme and provide corresponding a priori estimates. The effectiveness of the proposed method is demonstrated through numerical validation on a series of test problems.