3 - Projection to divergence free vector field

• Consider a general vector field \(w\)
• Helmoltz decomposition: \(w = u + v\)
• • - \(u\): Divergence free vector field such that \(div(u)=0\)
  • - \(v\): Gradient field \(v=\nabla q\;\), \(\;\;q\) scalar field.

• \(q\) satisfies a Poisson equation
• • \(div(w) = \underbrace{div(u)}_{=0} + div(v) \Rightarrow div(w) = \underbrace{div(\nabla q)}_{\triangle q}\)

• Method- Given an input field \(w\)
• • 1. Compute \(q\) as solution of \(\triangle q=div(w)\)
  • 2. Compute \(u = w - \nabla q\)