Nonconforming Discontinuous Galerkin Methods for Nonlocal Variational Problems

Abstract

We present a nonconforming discontinuous Galerkin finite element scheme for nonlocal variational problems associated with some linear nonlocal diffusion and nonlocal peridynamic operators subject to volumetric constraints. The nonlocal operators under consideration have nonlocal interaction kernels that exhibit singularities at the origin so that the natural energy spaces do not allow conforming discontinuous finite element functions. The key idea in our method is based on the introduction of modified nonlocal interaction kernels near the origin. It works for multidimensional problems defined on bounded domains partitioned with general meshes and approximations using general basis functions. A convergence theory is established for two types of finite dimensional spaces, with the first type including finite element spaces that lead to unconditional convergence if they contain all continuous piecewise linear functions and the other type being the discontinuous piecewise constant function space that is shown to be conditionally convergent. The analysis is based on the recently developed framework of asymptotically compatible schemes for parametrized variational problems. It also relies crucially on a successful extension of a compactness result of Bourgain, Brezis, and Mironescu to function sequences associated with a sequence of kernels that is assumed to be convergent to a more general limiting kernel. Numerical experiments are carried out to offer additional observations on the performance of our proposed method.1. Introduction. This paper serves as a first attempt to develop discontinuous and nonconforming Galerkin finite element discretization of nonlocal diffusion (ND) and nonlocal peridynamic models defined on a bounded spatial domain in any given space dimension. Since first introduced by Silling in [45], peridynamics (PD), which is an integral-type nonlocal continuum theory, has demonstrated effectiveness in modeling material singularities [4,32,47,48]. As pointed out in [21], linear scalar PD operators also share similarities with ND operators, thus making the study of PD relevant to the study of ND models, as well as fractional diffusion models, in various applications [2,11,13,19,22,27,29,30,31,33,36,40]. Along with research efforts on the mathematical analysis of PD/ND models, a variety of numerical methods have been studied and implemented, including finite difference, finite element, quadrature, and particle based methods [1,8,14,24,25,32,35,42,44,46,50,51,52,53,54,56,57]. In terms of Galerkin type approximations, most of the existing analyses have focused on the conforming ones and very few studies have been carried out for nonconforming discontinuous Galerkin (DG) finite element discretizations.

A Multiscale Implementation Based on Adaptive Mesh Refinement for the Nonlocal Peridynamics Model in One Dimension

“…For more singular kernels that do not allow conforming DG approximations, nonconforming DG methods for nonlocal problems were developed in[34].…”

Peridynamics review

“…Interesting studies were presented by Du et al [135,136] in which an adaptive FEM was developed for non-local diffusion equations and PD models by establishing a posteriori error analysis. Tian and Du [137] introduced a non-conforming discontinuous Galerkin finite-element scheme for non-local, volumetrically constrained problems associated with some linear non-local diffusion and non-local PD operators.…”

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

“…Then, they further established in [34] an abstract mathematical framework to analyze a class of asymptotically compatible schemes for conforming Galerkin approximations of some parameterized linear nonlocal problems. Meanwhile, some numerical methods such as finite difference, finite element, Fourier spectral, and discontinuous Galerkin (DG) approaches have been designed and studied to satisfy the asymptotic compatibility (see, e.g., [14,16,18,33,35]). When the kernel function is chosen such that the solution of the nonlocal diffusion problem contains spatial discontinuities, the DG method could be an advantageous choice for its discretization in space.…”

“…Particularly, there exist various DG approximations for the classical elliptic problems (see, e.g., [1,5,24]). For the nonlocal diffusion and nonlocal mechanical models, different conforming and nonconforming Galerkin approximations using discontinuous elements have been considered in [4,19,25,27,35]. The DG scheme recently proposed in [14] for solving the ND equation is motivated by the local discontinuous Galerkin (LDG) method [11] and relies on the introduction of auxiliary variables.…”

“…Assume that the exact solution u of the problem (1) is smooth enough, then we have the following a priori error estimate:Proof We firstly consider the estimate of (u, v h ) . With the Cauchy-Schwarz inequality and Sobolev's inequality, we obtain(35)…”