A cookbook for finite element methods for nonlocal problems, including quadrature rules and approximate Euclidean balls

Abstract

The implementation of finite element methods (FEMs) for nonlocal models with a finite range of interaction poses challenges not faced in the partial differential equations (PDEs) setting. For example, one has to deal with weak forms involving double integrals which lead to discrete systems having higher assembly and solving costs due to possibly much lower sparsity compared to that of FEMs for PDEs. In addition, one may encounter non-smooth integrands. In many nonlocal models, nonlocal interactions are limited to bounded neighborhoods that are ubiquitously chosen to be Euclidean balls, resulting in the challenge of dealing with intersections of such balls with the finite elements. We focus on developing recipes for the efficient assembly of FEM stiffness matrices and on the choice of quadrature rules for the double integrals that contribute to the assembly efficiency and also posses sufficient accuracy. A major feature of our recipes is the use of approximate balls, e.g., several polygonal approximations of Euclidean balls, that, among other advantages, mitigate the challenge of dealing with ball-element intersections. We provide numerical illustrations of the relative accuracy and efficiency of the several approaches we develop.

61 pages, 4 tables, 17 figures

References in corpus (5)

• Numerical methods for nonlocal and fractional models
• Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver
• Design of generalized fractional order gradient descent method
• Lyapunov functions for nabla discrete fractional order systems
• The exit-time problem for a Markov jump process

Cited by in corpus (5)

• A machine-learning framework for peridynamic material models with physical constraints
• Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation
• A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems
• Convergence of solutions of a rescaled evolution nonlocal cross-diffusion problem to its local diffusion counterpart
• Error analysis of some nonlocal diffusion discretization schemes