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A numerical study of divergence-free kernel approximations

Institution:	1. Department of Physics, Lanzhou University of Technology, Lanzhou 730050, China;2. College of Mechano-Electronic Engineering, University of Technology, Lanzhou 730050, China;1. Department of Basic Sciences, Faculty of Architecture, Allied Sciences and Humanities, University of Engineering and Technology, Peshawar, Pakistan;2. Scientific Computing, Department of Information Technology, Uppsala University, Sweden;1. Institute of Mathematical Sciences, Ewha Womans University, Seoul 120-750, Republic of Korea;2. Department of Mathematics, Korea University, Seoul 136-713, Republic of Korea;1. Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, V5A1S6, Canada;2. BCAM—Basque Center for Applied Mathematics, Bilbao, Basque Country, 48009, Spain;3. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

Abstract:	Approximation properties of divergence-free vector fields by global and local solenoidal bases are studied. A comparison between interpolants generated with radial kernels and multivariate polynomials is presented. Numerical results show higher rates of convergence for derivatives of the vector field being approximated in directions enforced by the divergence operator when a rectangular grid is used. We also compute the growth of Lebesgue constants for uniform and clustered nodes and study the flat limit of divergence-free interpolants based on radial kernels. Numerical results are presented for two- and three-dimensional vector fields.

Keywords:	Radial basis functions Divergence-free Finite-differences Spectral methods
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