Localized collocation methods based on radial basis functions (RBFs) for elliptic problems appear to be nonrobust in the presence of Neumann boundary conditions. In this paper, we overcome this issue by formulating the RBF-generated finite difference method in a discrete least squares setting instead. This allows us to prove high-order convergence under node refinement and to numerically verify that the least squares formulation is more accurate and robust than the collocation formulation. The implementation effort for the modified algorithm is comparable to that for the collocation method.


  1. radial basis function
  2. least squares
  3. partial differential equation
  4. elliptic problem
  5. Neumann condition
  6. RBF-FD

MSC codes

  1. 65N06
  2. 65N12
  3. 65N35

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Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A1441 - A1471
ISSN (online): 1095-7197


Submitted: 6 March 2020
Accepted: 8 February 2021
Published online: 26 April 2021


  1. radial basis function
  2. least squares
  3. partial differential equation
  4. elliptic problem
  5. Neumann condition
  6. RBF-FD

MSC codes

  1. 65N06
  2. 65N12
  3. 65N35



Funding Information

National Science Foundation https://doi.org/10.13039/100000001 : DMS-2012011
Svenska Forskningsrådet Formas https://doi.org/10.13039/501100001862 : 2016-04849

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