hp-discontinuous Galerkin method for the generalized Burgers-Huxley equation with weakly singular kernels
<|reference_start|>hp-discontinuous Galerkin method for the generalized Burgers-Huxley equation with weakly singular kernels: In this work, we investigate the $hp$-discontinuous Galerkin (DG) time-stepping method for the generalized Burgers-Huxley equation with memory, a non-linear advection-diffusion-reaction problem featuring weakly singular kernels. We derive a priori error estimates for the semi-discrete 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 $hp$-finite element method (conforming), followed by the $hp$-discontinuous Galerkin 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.<|reference_end|>
@article{mahajan2024hp-discontinuous,
    title={hp-discontinuous Galerkin method for the generalized Burgers-Huxley
  equation with weakly singular kernels},
    author={Sumit Mahajan, Arbaz Khan},
    journal={arXiv preprint arXiv:2409.00818},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.00818},
    primaryClass={math.NA cs.NA}
}