Numerical analysis of a class of penalty discontinuous Galerkin methods for nonlocal diffusion problems

TL;DR

The work addresses one‑dimensional nonlocal diffusion problems governed by a horizon‑dependent operator $\mathcal{L}_\delta$ and proposes a penalty discontinuous Galerkin (DG) discretization with a discrete bilinear form $B_h=E+J+\mu P$ to handle potential discontinuities. It establishes boundedness, stability, and a priori error estimates for the scheme, and proves asymptotic compatibility with the local Poisson limit as $\delta,h\to0$; consistent schemes $\text{nIP}$ and $\text{nNIPG}$ are shown to converge in the energy norm and, for fixed horizon, in $L^2$ with optimal rates. The paper also demonstrates applicability to time‑dependent convection‑diffusion problems with nonlocal diffusion, providing semi‑discrete stability and suggesting IMEX time stepping. Numerical experiments in 1D confirm high‑order accuracy, robustness to singular kernels, and the practical effect of nonlocal diffusion in smoothing and controlling transitions, validating the theoretical guarantees and the asymptotic compatibility property.

Abstract

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 [Du-Ju-Lu-Tian-CAMC2020], 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.

Figures (3)

• Figure 1: Plots of the numerical solution in Example \ref{['exp2']} produced by the nIP scheme \ref{['dgscm']} and \ref{['eqn_nip']}, $\delta = 1/8, \alpha = 1/2$, $N = 60$. Solid line: exact solution. Red circles: numerical solutions.
• Figure 2: Plots of the numerical solution in Example \ref{['exp4']} produced by the scheme \ref{['dgscm_td']} and \ref{['eqn_nip']}, $\delta = 1/8, \alpha = 1/2$, $k = 2$, $h = 1/6$. Solid line: exact solution with $\sigma = 0$. Green dash line: $\sigma = 1/48$. Blue dash-dot line: $\sigma = 1/24$. Red long dash line: $\sigma = 1/12$. Light blue dash dot dot line: $\sigma = 1/6$.
• Figure 3: Plot of the numerical solution in Example \ref{['exp5']} produced by the scheme \ref{['dgscm_td']} and \ref{['eqn_nip']}, $\delta = \pi/6, \alpha = 1/2$, $k = 2$, $N=60$. Solid line: exact solution with $\sigma = 0$. Green dash line: $\sigma = \pi/120$. Blue dash dot line: $\sigma = \pi/60$. Red long dash line: $\sigma = \pi/30$. Light blue dash dot dot line: $\sigma = \pi/15$.

Theorems & Definitions (17)

• Remark 2.1
• Proposition 3.1
• Lemma 3.1
• Theorem 3.1
• Remark 3.1
• Theorem 3.2
• Remark 3.2
• Lemma 3.2
• proof : Proof of Theorem \ref{['thm_ac']}:
• Theorem 3.3