Issue |
ESAIM: M2AN
Volume 58, Number 5, September-October 2024
|
|
---|---|---|
Page(s) | 2035 - 2059 | |
DOI | https://doi.org/10.1051/m2an/2024064 | |
Published online | 21 October 2024 |
Numerical analysis of a class of penalty discontinuous Galerkin methods for nonlocal diffusion problems
1
Department of Applied Physics and Applied Mathematics and Data Science Institute, Columbia University, New York, NY 10027, USA
2
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA
3
School of Mathematics, South China University of Technology, Guangzhou, Canton 510641, P.R. China
4
Department of Mathematics, University of California San Diego, La Jolla, CA 92093, USA
* Corresponding author: jflu@scut.edu.cn
Received:
19
February
2024
Accepted:
5
August
2024
In this paper, we consider a class of discontinuous Galerkin (DG) methods for one-dimensional nonlocal diffusion (ND) problems. The nonlocal models, which are integral equations, are widely used in describing many physical phenomena with long-range interactions. The ND problem is the nonlocal analog of the classic diffusion problem, and as the interaction radius (horizon) vanishes, then the nonlocality disappears and the ND problem converges to the classic diffusion problem. Under certain conditions, the exact solution to the ND problem may exhibit discontinuities, setting it apart from the classic diffusion problem. Since the DG method shows its great advantages in resolving problems with discontinuities in computational fluid dynamics over the past several decades, it is natural to adopt the DG method to compute the ND problems. Based on [Q. Du, L. Ju, J. Lu and X. Tian, Commun. Appl. Math. Comput. 2 (2020) 31–55], we develop the DG methods with different penalty terms, ensuring that the proposed DG methods have local counterparts as the horizon vanishes. This indicates the proposed methods will converge to the existing DG schemes as the horizon vanishes, which is crucial for achieving asymptotic compatibility. Rigorous proofs are provided to demonstrate the stability, error estimates, and asymptotic compatibility of the proposed DG schemes. To observe the effect of the nonlocal diffusion, we also consider the time-dependent convection–diffusion problems with nonlocal diffusion. We conduct several numerical experiments, including accuracy tests and Burgers’ equation with nonlocal diffusion, and various horizons are taken to show the good performance of the proposed algorithm and validate the theoretical findings.
Mathematics Subject Classification: 65M60 / 65R20 / 45A05
Key words: Nonlocal diffusion / asymptotic compatibility / discontinuous Galerkin / interior penalty
© The authors. Published by EDP Sciences, SMAI 2024
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