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

Abstract

Here, we introduce a meshless method for solving both continuous and discrete variational formulations of a volume constrained, nonlocal diffusion problem. We use the discrete solution to approximate the continuous solution. Our method is nonconforming 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.

Authors:
 [1];  [2];  [1];  [2]
  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
  2. Texas A & M Univ., College Station, TX (United States)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1247659
Report Number(s):
SAND-2016-0223J
Journal ID: ISSN 0025-5718; PII: S00255718201803294X
Grant/Contract Number:  
AC04-94AL85000
Resource Type:
Accepted Manuscript
Journal Name:
Mathematics of Computation
Additional Journal Information:
Journal Volume: 87; Journal Issue: 313; Journal ID: ISSN 0025-5718
Publisher:
American Mathematical Society
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; meshless method; localized Lagrange bases; radial basis functions; nonlocal diffusion; volume constraint

Citation Formats

Lehoucq, Richard B., Narcowich, Fran J., Rowe, Stephen T., and Ward, Joe D. A meshless Galerkin method for non-local diffusion using localized kernel bases. United States: N. p., 2018. Web. doi:10.1090/mcom/3294.
Lehoucq, Richard B., Narcowich, Fran J., Rowe, Stephen T., & Ward, Joe D. A meshless Galerkin method for non-local diffusion using localized kernel bases. United States. doi:10.1090/mcom/3294.
Lehoucq, Richard B., Narcowich, Fran J., Rowe, Stephen T., and Ward, Joe D. Tue . "A meshless Galerkin method for non-local diffusion using localized kernel bases". United States. doi:10.1090/mcom/3294. https://www.osti.gov/servlets/purl/1247659.
@article{osti_1247659,
title = {A meshless Galerkin method for non-local diffusion using localized kernel bases},
author = {Lehoucq, Richard B. and Narcowich, Fran J. and Rowe, Stephen T. and Ward, Joe D.},
abstractNote = {Here, we introduce a meshless method for solving both continuous and discrete variational formulations of a volume constrained, nonlocal diffusion problem. We use the discrete solution to approximate the continuous solution. Our method is nonconforming 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.},
doi = {10.1090/mcom/3294},
journal = {Mathematics of Computation},
number = 313,
volume = 87,
place = {United States},
year = {2018},
month = {2}
}

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