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On Helmholtz decompositions and their generalizations—An overview
Author(s) -
Sprössig W.
Publication year - 2009
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1212
Subject(s) - mathematics , sobolev space , helmholtz equation , helmholtz free energy , connection (principal bundle) , mathematical analysis , hilbert space , pure mathematics , boundary value problem , elasticity (physics) , algebra over a field , geometry , physics , quantum mechanics , materials science , composite material
Helmholtz' theorem initiates a remarkable development in the theory of projection methods that are adapted to the numerical solution of equations in fluid dynamics and elasticity. There is a dense connection with Hodge‐de Rham decompositions of smooth 1‐forms. We give an overview of this type of decompositions and discuss their applications to vector, quaternionic and Clifford‐valued boundary value problems in the corresponding Hilbert–Sobolev spaces. Copyright © 2009 John Wiley & Sons, Ltd.