Understanding Curl-Free and Div-Free Radial Basis Functions in Physical Situations

This content explores the applications of Curl-Free and Div-Free Radial Basis Functions in solving partial differential equations for fields, the theoretical soundness of using RBFs, and examples illustrating divergence-free interpolation. It also delves into matrix-valued RBF formulations, convergence properties, and utilizing Div-Free RBFs for calculations. Numerous references from 1980 to 2011 are provided on related research in various fields like Magnetohydrodynamics and interpolation applications.

• Radial Basis Functions
• Divergence-Free
• Curl-Free
• Interpolation
• Matrix-Valued

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1. Curl-Free and Div-Free Radial Basis Functions March 2, 2018

2. Physical situations when solving a PDE for Div-Free, Curl-Free fields

3. Why do we care?

4. Step back: why are RBFs so nice? Any scattered data in any number of dimensions can be handled the same Interpolation matrices can be proved to be invertible There exist convergence results that imply use of RBF is theoretically sound

5. An Example: Question: Can we find a unique interpolant that fits the data and is divergence free?

6. An Example:

7. An Example:

8. Narcowich and Wards matrix valued RBF

9. General Curl-Free and Div-Free formulations

10. Theoretical convergence properties

11. Using Div-Free RBFs to calculate M

12. Using Div-Free RBFs to calculate M

13. References 1980 Brackbill, Barnes Effect Nonzero Div Magnetohydrodynamics 1994 Narcowich, Ward Matrix Valued Positive Definite 2002 Lowitzsch Div Free Interpolation Applications 2006 Fuselier Characterization Native Space Matrix Valued RBF 2006 Fuselier Error Estimates Matrix Values RBF 2007 Narcowich, Ward Div Free RBF Surfaces 2008 Fuselier Sobolev Approximation Rate Div Free Curl Free RBF 2009 Fuselier Error Stability Div Free Sphere RBF 2009 Fuselier Error Stability Vector Field Sphere RBF 2011 McNally Div Free Magnetic Field RBF