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Efficient quadrature rules for finite element discretizations of nonlocal equations

Journal Article · · Numerical Methods for Partial Differential Equations
DOI:https://doi.org/10.1002/num.22833· OSTI ID:1873333
 [1];  [2];  [3];  [4]
  1. Texas Tech University, Lubbock, TX (United States)
  2. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  3. University of Bologna (Italy)
  4. Sandia National Lab. (SNL-CA), Livermore, CA (United States)
+ Show Author Affiliations

Here, 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.

Research Organization:
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States); Sandia National Laboratories (SNL-CA), Livermore, CA (United States)
Sponsoring Organization:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE Laboratory Directed Research and Development (LDRD) Program; National science Foundation (NSF); USDOE National Nuclear Security Administration (NNSA)
Grant/Contract Number:
89233218CNA000001; NA0003525
OSTI ID:
1873333
Alternate ID(s):
OSTI ID: 1818821
Report Number(s):
LA-UR-21-20505; SAND2021-0672
Journal Information:
Numerical Methods for Partial Differential Equations, Journal Name: Numerical Methods for Partial Differential Equations Journal Issue: 6 Vol. 38; ISSN 0749-159X
Publisher:
WileyCopyright Statement
Country of Publication:
United States
Language:
English

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Related Subjects

97 MATHEMATICS AND COMPUTING
asymptotic behavior of solution
finite element discretizations
nonlocal models
numerical quadrature