A radial basis function Galerkin method for inhomogeneous nonlocal diffusion

Lehoucq, Richard B.; Rowe, Stephen T.

doi:10.1016/j.cma.2015.10.021

A radial basis function Galerkin method for inhomogeneous nonlocal diffusion

Journal Article · Mon Feb 01 00:00:00 EST 2016 · Computer Methods in Applied Mechanics and Engineering

DOI:https://doi.org/10.1016/j.cma.2015.10.021· OSTI ID:1237367

Lehoucq, Richard B. ^[1]; Rowe, Stephen T. ^[2]

Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Texas A & M Univ., College Station, TX (United States)

We introduce a discretization for a nonlocal diffusion problem using a localized basis of radial basis functions. The stiffness matrix entries are assembled by a special quadrature routine unique to the localized basis. Combining the quadrature method with the localized basis produces a well-conditioned, sparse, symmetric positive definite stiffness matrix. We demonstrate that both the continuum and discrete problems are well-posed and present numerical results for the convergence behavior of the radial basis function method. As a result, we explore approximating the solution to anisotropic differential equations by solving anisotropic nonlocal integral equations using the radial basis function method.

Research Organization:: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE

Grant/Contract Number:: AC04-94AL85000

OSTI ID:: 1237367

Report Number(s):: SAND--2015-0511J; 562414

Journal Information:: Computer Methods in Applied Mechanics and Engineering, Journal Name: Computer Methods in Applied Mechanics and Engineering Vol. 299; ISSN 0045-7825

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (10)

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Constraints in thermodynamic extremal principles for non-local dissipative processes Hackl, Klaus; Fischer, Franz Dieter; Svoboda, Jiri Continuum Mechanics and Thermodynamics, Vol. 32, Issue 5 https://doi.org/10.1007/s00161-019-00846-3	journal	November 2019

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Related Subjects

97 MATHEMATICS AND COMPUTING
Lagrange functions
nonlocal diffusion
nonlocal vector calculus
radial basis functions

A radial basis function Galerkin method for inhomogeneous nonlocal diffusion

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