@TechReport{	  it:2012-020,
  author	= {Elisabeth Larsson and Erik Lehto and Alfa Heryudono and
		  Bengt Fornberg},
  title		= {Stable Computation of Differentiation Matrices and
		  Scattered Node Stencils Based on {G}aussian Radial Basis
		  Functions},
  institution	= {Department of Information Technology, Uppsala University},
  department	= {Division of Scientific Computing},
  year		= {2012},
  number	= {2012-020},
  month		= aug,
  abstract	= {Radial basis function (RBF) approximation has the
		  potential to provide spectrally accurate function
		  approximations for data given at scattered node locations.
		  For smooth solutions, the best accuracy for a given number
		  of node points is typically achieved when the basis
		  functions are scaled to be nearly flat. This also results
		  in nearly linearly dependent basis functions and severe
		  ill-conditioning of the interpolation matrices. Fornberg,
		  Larsson, and Flyer recently developed the RBF-QR method
		  which provides a numerically stable approach to
		  interpolation with flat and nearly flat Gaussian RBFs. In
		  this work, we consider how to extend this method to the
		  task of computing differentiation matrices and stencil
		  weights in order to solve partial differential equations.
		  The expressions for first and second order derivative
		  operators as well as hyperviscosity operators are
		  established, numerical issues such as how to deal with
		  non-unisolvency are resolved, and the accuracy and
		  computational efficiency of the method is tested
		  numerically. The results indicate that using the RBF-QR
		  approach for solving PDE problems can be very competitive
		  compared with using the ill-conditioned direct solution
		  approach or using variable precision arithmetic to overcome
		  the conditioning issue.}
}