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

Journal Article · Tue Feb 06 00:00:00 EST 2018 · Mathematics of Computation

DOI:https://doi.org/10.1090/mcom/3294· OSTI ID:1247659

Lehoucq, Richard B. ^[1]; Narcowich, Fran J. ^[2]; Rowe, Stephen T. ^[1]; Ward, Joe D. ^[2]

---

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

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.

• Research Organization: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
• Sponsoring Organization: USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: AC04-94AL85000
• OSTI ID: 1247659
• Report Number(s): SAND--2016-0223J; PII: S00255718201803294X
• Journal Information: Mathematics of Computation, Journal Name: Mathematics of Computation Journal Issue: 313 Vol. 87; ISSN 0025-5718
• Publisher: American Mathematical Society
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
localized Lagrange bases
meshless method
nonlocal diffusion
radial basis functions
volume constraint