March 2 2018 Physical situations when solving a PDE for Div Free CurlFree fields Why do we care Step back why are RBFs so nice Any scattered data in any number of dimensions can be handled the same ID: 682733
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Slide1
Curl-Free and Div-Free Radial Basis Functions
March 2, 2018Slide2
Physical situations when solving a PDE for Div-Free, Curl-Free fieldsSlide3
Why do we care?Slide4
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 soundSlide5
An Example:
Question: Can we find a unique interpolant that fits the data and is divergence free?Slide6
An Example:Slide7
An Example:Slide8
Narcowich and Ward’s matrix valued RBFSlide9
General Curl-Free and Div-Free formulationsSlide10
Theoretical convergence propertiesSlide11
Using Div-Free RBFs to calculate MSlide12
Using
Div
-Free RBFs to calculate MSlide13
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”