A Discontinuous Galerkin Method with Penalty for One-Dimensional Nonlocal Diffusion Problems

Abstract

There have been many theoretical studies and numerical investigations of nonlocal diffusion (ND) problems in recent years. In this paper, we propose and analyze a new discontinuous Galerkin method for solving one-dimensional steady-state and time-dependent ND problems, based on a formulation that directly penalizes the jumps across the element interfaces in the nonlocal sense. We show that the proposed discontinuous Galerkin scheme is stable and convergent. Moreover, the local limit of such DG scheme recovers classical DG scheme for the corresponding local diffusion problem, which is a distinct feature of the new formulation and assures the asymptotic compatibility of the discretization. Numerical tests are also presented to demonstrate the effectiveness and the robustness of the proposed method.

Numerical methods for nonlocal and fractional models

“…Asymptotically compatible (AC) schemes, being either conforming Galerkintype approximations of weak forms (Chen and Gunzburger 2011, Xu, Gunzburger, Burkardt and Du 2016b, nonconforming discontinuous Galerkin approximations (Du, Ju, Lu and Tian 2019c, Du, Ju and Lu 2019b, Du and Yin 2019, or collocation or quadrature based approximations of strong forms , Seleson et al 2016, Du et al 2019d, Zhang, Gunzburger and Ju 2016a, Zhang, Gunzburger and Ju 2016b offer the potential to solve for approximations of a model of interest with different choices of parameters to gain efficiency and to avoid the pitfall of reaching inconsistent limits.…”

“…We note that the nonconforming approximations discussed in in the local limit do not yield a standard nonconforming finite element approximation nor a DG approximation to the local problem. One may construct other alternative formulations that can give rise to the conventional nonconforming and DG discretization of local PDEs, see, e.g., (Du et al 2019c) for a study based on the DG with penalty formulation.…”

Interior estimates for the eigenfunctions of the fractional Laplacian on a bounded domain

“…For examples, methods based on the Fourier/spectral representation include finite difference (FD) methods [17,23,24,25,30,31,39], spectral element method [35], and sinc-based method [6]. Methods based on the singular integral representation include FD methods [14,28,36], finite element methods [1,2,3,4,8,15,37], discontinuous Galerkin methods [12,13], and spectral method [26]; and FD methods [11,32,38] based on the Grünwald-Letnikov representation. Loosely speaking, most of the existing FD methods have been constructed on uniform grids, have the advantage of efficient matrix-vector multiplication via the fast Fourier transform (FFT), but do not work for domains with complex geometries and have difficulty to incorporate with mesh adaptation.…”

Uniqueness of the inverse reaction coefficient problems for nonlocal diffusion models

“…As for the direct problems for nonlocal diffusion models, i.e., the volume-constrained problem, which have been studied extensively in the past few years [19,20,21,22,23,24]. However, about the corresponding inverse problems, the results are very limited (see [25,26,27]).…”