3.2.4.16. Potential Projection Equation

This equation is used to create a potential field which minimizes difference between the potential, \psi, and the potential source, \mathbf{S}, in the least-squares sense

(3.74)\left(\grad\psi - \mathbf{S}\right)^2=0.

This equation was derived to create a hydrostatic potential field, so the most common usage is where \mathbf{S}=\dens\mathbf{g}.

Equation (3.74) is integrated by parts, resulting in the residual

(3.75)\symRes^i = \underbrace{\int\limits_\Vol \grad\psi \bcdot \grad(\e_k\phi^i)\dV}_\mathrm{DEF} - \underbrace{\int\limits_\Vol \mathbf{S} \bcdot \grad(\e_k\phi^i)\dV}_\mathrm{SRC}