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]

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1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
2. 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 Lab. (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, Vol. 299; ISSN 0045-7825
• Publisher: Elsevier
• Country of Publication: United States
• Language: English

Related Subjects

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