In this paper, we propose an adaptive finite element algorithm for the numerical solution of a class of nonlocal models which correspond to nonlocal diffusion equations and linear scalar peridynamic models with certain nonintegrable kernel functions. The convergence of the adaptive finite element algorithm is rigorously derived with the help of several basic ingredients, such as the upper bound of the estimator, the estimator reduction, and the orthogonality property. We also consider how the results are affected by the horizon parameter δ which characterizes the range of nonlocality. Numerical experiments are performed to verify our theoretical findings.
Introduction.In this work, we consider numerical approximations of some nonlocal diffusion models which arise in many fields such as image analysis [10,25,28], nonlocal diffusion [8,19], and continuum mechanics [37]. These models offer new alternatives to traditional PDE based models. For instance, the peridynamic (PD) theory proposed in [37] is an integral-type nonlocal continuum theory which incorporates the nonlocal nature of material interactions. It also connects continuum mechanics and molecular dynamics within a single framework [39]. Meanwhile, there has been much development in the mathematical theory of nonlocal models; see, for instance, an extensive treatment on nonlocal diffusion problems in [4]. In [18,26], a nonlocal vector calculus was developed to provide a more general variational setting for nonlocal models. More theoretical studies of related volume-constraint problems can be found in [19,30,20]. Within the context of PD-based nonlocal models, there have been a variety of numerical methods implemented for their approximations including finite difference, finite element, quadrature, and particle-based methods [3,9,15,24,27,29,35,38,42]. Given the ability of nonlocal PD models to simulate cracks or fractures, an adaptive method is a natural way to reduce the computational cost. Indeed, adaptive refinement for nonlocal PD-type models has been studied in [9] with meshless methods. Utilizing the nice variational structures of the volume-constrained problems associated with the linear nonlocal diffusion or PD operators and the strong connection to the variational PDE problems associated with elliptic operators, it is also natural to study finite element and adaptive finite element approximations of
