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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 meshless Galerkin method for non-local diffusion using localized kernel bases
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by R. B. Lehoucq, F. J. Narcowich, S. T. Rowe and J. D. Ward PDF
Math. Comp. 87 (2018), 2233-2258 Request permission

Abstract:

We introduce a meshless method for solving both continuous and discrete variational formulations of a volume constrained, non-local diffusion problem. We use the discrete solution to approximate the continuous solution. Our method is non-conforming and uses a localized Lagrange basis that is constructed out of radial basis functions. By verifying that certain inf-sup conditions hold, we demonstrate that both the continuous and discrete problems are well-posed, and also present numerical and theoretical results for the convergence behavior of the method. The stiffness matrix is assembled by a special quadrature routine unique to the localized basis. Combining the quadrature method with the localized basis produces a well-conditioned, symmetric matrix. This then is used to find the discretized solution.
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Additional Information
  • R. B. Lehoucq
  • Affiliation: Computational Mathematics, Sandia National Laboratories, Albuquerque, New Mexico 87185-1320
  • MR Author ID: 611433
  • Email: rblehou@sandia.gov
  • F. J. Narcowich
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
  • MR Author ID: 129435
  • Email: fnarc@math.tamu.edu
  • S. T. Rowe
  • Affiliation: Sandia National Laboratories, Albuquerque, New Mexico 87185
  • MR Author ID: 975133
  • Email: srowe@sandia.gov
  • J. D. Ward
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
  • MR Author ID: 180590
  • Email: jward@math.tamu.edu
  • Received by editor(s): January 11, 2016
  • Received by editor(s) in revised form: January 9, 2017, and May 12, 2017
  • Published electronically: February 6, 2018
  • Additional Notes: The first author’s research was supported by the Laboratory Directed Research and Development (LDRD) program at Sandia National Laboratories. Sandia is multi-program laboratory managed and operated by Sandia Corporation, wholly a subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000
    The second author’s research was supported by grant DMS-1514789 from the National Science Foundation
    The third author’s research was supported by grant DMS-1211566 from the National Science Foundation and Sandia National Laboratories.
    The fourth author’s research was supported by grant DMS-1514789 from the National Science Foundation
  • © Copyright 2018 American Mathematical Society
  • Journal: Math. Comp. 87 (2018), 2233-2258
  • MSC (2010): Primary 45P05, 47G10, 65K10, 41A30, 41A63
  • DOI: https://doi.org/10.1090/mcom/3294
  • MathSciNet review: 3802433