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hp-discontinuous Galerkin method for the generalized Burgers-Huxley equation with weakly singular kernels

Sumit Mahajan, Arbaz Khan

Abstract

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.

hp-discontinuous Galerkin method for the generalized Burgers-Huxley equation with weakly singular kernels

Abstract

In this work, we investigate the -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 -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 -finite element method (conforming), followed by the -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.
Paper Structure (16 sections, 37 theorems, 160 equations, 1 figure, 13 tables)

This paper contains 16 sections, 37 theorems, 160 equations, 1 figure, 13 tables.

Table of Contents

  1. Introduction
  2. $hp$-DG time-stepping
  3. Notations and Weak formulation
  4. Time discretization
  5. Projection operators
  6. Error Estimates
  7. Fully-discrete scheme
  8. $hp$-FEM in space
  9. Error estimates
  10. $hp$-Discontinuous Galerkin FEM in space
  11. Error estimates
  12. Numerical studies
  13. Accuracy verification
  14. Application to Fractional differential equations
  15. Prey-predator system
  16. ...and 1 more sections

Key Result

Theorem 2.1

For $1 \leq n \leq N$, let $u \in \mathrm{C}\mathopen{}\mathclose{\left(\mathopen{}\mathclose{\left[t_{n-1}, t_n\right] ; \mathrm{H}_0^1\right)$. Then we have the following: (i) If $u$ is analytic on $\mathopen{}\mathclose{\left[t_{n-1}}}, t_n\right]$ with values in $\mathrm{H}_0^1$, there holds (ii) For any $0 \leq q_n \leq p_n$ and $\mathopen{}\mathclose{\left.u\right|_{\mathrm{J}_n}}} \in \mat

Figures (1)

  • Figure 1: The solution plot for Prey(1st row) and predator (2nd row) density for different memory coefficient.

Theorems & Definitions (72)

  • Remark 1
  • Remark 2
  • Theorem 2.1
  • proof
  • Lemma 1
  • Lemma 2
  • Lemma 3: SVL, lemma 6.3
  • Lemma 4: SVL, lemma 6.4
  • Lemma 5
  • proof
  • ...and 62 more