Efficient quadrature rules for finite element discretizations of nonlocal equations

Abstract

In this paper, we design efficient quadrature rules for finite element (FE) discretizations of nonlocal diffusion problems with compactly supported kernel functions. Two of the main challenges in nonlocal modeling and simulations are the prohibitive computational cost and the nontrivial implementation of discretization schemes, especially in three-dimensional settings. In this work, we circumvent both challenges by introducing a parametrized mollifying function that improves the regularity of the integrand, utilizing an adaptive integration technique, and exploiting parallelization. We first show that the "mollified" solution converges to the exact one as the mollifying parameter vanishes, then we illustrate the consistency and accuracy of the proposed method on several two-and three-dimensional test cases. Furthermore, we demonstrate the good scaling properties of the parallel implementation of the adaptive algorithm and we compare the proposed method with recently developed techniques for efficient FE assembly.

A numerical approach to the fractional Laplacian operator with applications to quasi-geostrophic flows

“…Thus, it is important to develop a scalable parallel algorithm to reduce the time needed to have a solution. Since every processor needs information from all the other processors, the implementation of a parallel code for non-local assembly algorithms is not a straightforward task [12].…”

“…to ease the numerical assemblying of the stiffness matrix. We remark that, since the kernel is a symmetric function in both x and y, A 11 ij = A 22 ij and A 12 ij = A 21 ij [12]. Thus, for the stiffness matrix we have…”

nlfem: A flexible 2d Fem Code for Nonlocal Convection-Diffusion and Mechanics

“…Alternatives to kernel truncations have been investigated in [3]. Apart from this, various structure exploiting approaches, finite-difference schemes and meshfree methods have been used to compute numerical solutions.…”

An optimization-based strategy for peridynamic-FEM coupling and for the prescription of nonlocal boundary conditions

“…Computational challenges are related to the integral form that may require sophisticated quadrature rules, yielding discretized systems whose matrices are dense or even full. Among the works dedicated to improving implementation of nonlocal discretizations and numerical solvers for discretized nonlocal problems we mention variational methods [4,9,16,21] and meshfree methods [45,58,63,65,67].…”