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 Ward’s 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”