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==Helmholtz–Hodge decomposition==
The theoretical background of projection type method is the decomposition theorem of [[Olga Aleksandrovna Ladyzhenskaya|Ladyzhenskaya]] sometimes referred to as Helmholtz–Hodge Decomposition or simply as Hodge decomposition. It states that the vector field <math>\mathbf{u}</math> defined on a [[simply connected space|simply connected]] ___domain can be uniquely decomposed into a divergence-free ([[Solenoidal vector field|solenoidal]]) part <math>\mathbf{u}_{\text{sol}}</math> and an [[Conservative vector field#Irrotational vector fields|irrotational]] part <math>\mathbf{u}_{\text{irrot}}</math>.
<ref>{{cite book | title = A Mathematical Introduction to Fluid Mechanics | author1 = Chorin, A. J. | author2 = J. E. Marsden | edition = 3rd | publisher = [[Springer Science+Business Media|Springer-Verlag]] | year = 1993 | isbn = 0-387-97918-2}}</ref> {{Citation
| surname1 = Chorin
| given1 = A. J.
| title = A numerical method for solving incompressible viscous flow problems
| journal = Journal of Computational Physics
| volume = 22
| year = 1968
| pages = 12–26
| url =http://math.berkeley.edu/~chorin/chorin67.pdf
}}</ref>
Thus,
:<math>
\mathbf{u} = \mathbf{u}_{\text{sol}} + \mathbf{u}_{\text{irrot}} = \mathbf{u}_{\text{sol}} + \nabla \phi
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