Efficient quadrature rules for finite element discretizations of nonlocal equations

Eugenio Aulisa, Giacomo Capodaglio, Andrea Chierici, Marta D'Elia

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
JournalNumerical Methods for Partial Differential Equations
DOIs
StateAccepted/In press - 2021

Keywords

  • asymptotic behavior of solutions
  • finite element discretizations
  • nonlocal models
  • numerical quadrature

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