On Helmholtz decompositions and their generalizations-An overview

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.