A meshless Galerkin method for nonlocal 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 infsup conditions hold, we demonstrate that both the continuous and discrete problems are wellposed, 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 wellconditioned, symmetric matrix. This then is used to find the discretized solution.
 Authors:

 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Texas A & M Univ., College Station, TX (United States)
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1247659
 Report Number(s):
 SAND20160223J
Journal ID: ISSN 00255718; PII: S00255718201803294X
 Grant/Contract Number:
 AC0494AL85000
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Mathematics of Computation
 Additional Journal Information:
 Journal Volume: 87; Journal Issue: 313; Journal ID: ISSN 00255718
 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 nonlocal 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 nonlocal 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 nonlocal 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 nonlocal 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 infsup conditions hold, we demonstrate that both the continuous and discrete problems are wellposed, 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 wellconditioned, 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}
}