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Mathematics of Computation

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A meshless Galerkin method for non-local diffusion using localized kernel bases


Authors: R. B. Lehoucq, F. J. Narcowich, S. T. Rowe and J. D. Ward
Journal: Math. Comp. 87 (2018), 2233-2258
MSC (2010): Primary 45P05, 47G10, 65K10, 41A30, 41A63
DOI: https://doi.org/10.1090/mcom/3294
Published electronically: February 6, 2018
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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
Email: rblehou@sandia.gov

F. J. Narcowich
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: fnarc@math.tamu.edu

S. T. Rowe
Affiliation: Sandia National Laboratories, Albuquerque, New Mexico 87185
Email: srowe@sandia.gov

J. D. Ward
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: jward@math.tamu.edu

DOI: https://doi.org/10.1090/mcom/3294
Keywords: Meshless method, localized Lagrange bases, radial basis functions, non-local diffusion, volume constraint
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
Article copyright: © Copyright 2018 American Mathematical Society

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