hp-discontinuous Galerkin method for the generalized Burgers-Huxley equation with weakly singular kernels
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914932150239232 |
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| author | Mahajan, Sumit Khan, Arbaz |
| author_facet | Mahajan, Sumit Khan, Arbaz |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00818 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | hp-discontinuous Galerkin method for the generalized Burgers-Huxley equation with weakly singular kernels Mahajan, Sumit Khan, Arbaz Numerical Analysis 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. |
| title | hp-discontinuous Galerkin method for the generalized Burgers-Huxley equation with weakly singular kernels |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2409.00818 |