## A radial basis function Galerkin method for inhomogeneous nonlocal diffusion

## Abstract

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.

- Authors:

- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- 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

- OSTI Identifier:
- 1237367

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

Journal ID: ISSN 0045-7825; 562414

- Grant/Contract Number:
- AC04-94AL85000

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Computer Methods in Applied Mechanics and Engineering

- Additional Journal Information:
- Journal Volume: 299; Journal ID: ISSN 0045-7825

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; nonlocal diffusion; radial basis functions; nonlocal vector calculus; Lagrange functions

### Citation Formats

```
Lehoucq, Richard B., and Rowe, Stephen T. A radial basis function Galerkin method for inhomogeneous nonlocal diffusion. United States: N. p., 2016.
Web. doi:10.1016/j.cma.2015.10.021.
```

```
Lehoucq, Richard B., & Rowe, Stephen T. A radial basis function Galerkin method for inhomogeneous nonlocal diffusion. United States. doi:10.1016/j.cma.2015.10.021.
```

```
Lehoucq, Richard B., and Rowe, Stephen T. Mon .
"A radial basis function Galerkin method for inhomogeneous nonlocal diffusion". United States. doi:10.1016/j.cma.2015.10.021. https://www.osti.gov/servlets/purl/1237367.
```

```
@article{osti_1237367,
```

title = {A radial basis function Galerkin method for inhomogeneous nonlocal diffusion},

author = {Lehoucq, Richard B. and Rowe, Stephen T.},

abstractNote = {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.},

doi = {10.1016/j.cma.2015.10.021},

journal = {Computer Methods in Applied Mechanics and Engineering},

number = ,

volume = 299,

place = {United States},

year = {2016},

month = {2}

}

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