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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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A discontinuous Galerkin method for one-dimensional time-dependent nonlocal diffusion problems
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by Qiang Du, Lili Ju and Jianfang Lu HTML | PDF
Math. Comp. 88 (2019), 123-147 Request permission

Abstract:

Given that nonlocal diffusion (ND) problems allow their solutions to have spatial discontinuities, discontinuous Galerkin (DG) discretizations in space become natural choices when numerical approximations are considered. In this paper, we design and study a novel DG method for solving the one-dimensional time-dependent ND problem. The key idea of the method is the introduction of an auxiliary variable, analogous to the classic local discontinuous Galerkin (LDG) method but with some nonlocal extensions. Theoretical analysis shows that the proposed semi-discrete DG scheme is $L^2$-stable, convergent, and in particular asymptotically compatible. This latter feature implies that the classical DG approximation of the local diffusion problem can be recovered in the zero horizon limit. We also present various numerical tests to demonstrate the effectiveness and robustness of our method.
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Additional Information
  • Qiang Du
  • Affiliation: Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027
  • MR Author ID: 191080
  • Email: qd2125@columbia.edu
  • Lili Ju
  • Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
  • MR Author ID: 645968
  • Email: ju@math.sc.edu
  • Jianfang Lu
  • Affiliation: Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing, 100193, People’s Republic of China
  • MR Author ID: 1050703
  • Email: jflu@csrc.ac.cn
  • Received by editor(s): July 15, 2017
  • Received by editor(s) in revised form: September 28, 2017
  • Published electronically: April 12, 2018
  • Additional Notes: The first author’s research was partially supported by US National Science Foundation grant DMS-1719699, US AFOSR MURI Center for Material Failure Prediction Through Peridynamics, and US Army Research Office MURI grant W911NF-15-1-0562.
    The second author’s research is partially supported by US National Science Foundation grant DMS-1521965 and US Department of Energy grant DE-SC0016540.
    The third author’s research is partially supported by Postdoctoral Science Foundation of China grant 2017M610749.
  • © Copyright 2018 American Mathematical Society
  • Journal: Math. Comp. 88 (2019), 123-147
  • MSC (2010): Primary 65M60, 65R20, 45A05
  • DOI: https://doi.org/10.1090/mcom/3333
  • MathSciNet review: 3854053