Find vector field whose divergence is a scalar field

Question

Say we have a Poisson problem:

$\nabla^2 \varphi = S$

As a boundary value problem, it requires the definition of boundary conditions on all surfaces of the domain. If we assume there is a vector field $\vec{v}$ that satisfies:

$\nabla \cdot \vec{v} = S$

Then solving the Poisson problem can be rewritten:

$\nabla \cdot (\nabla \varphi) = \nabla \cdot \vec{v}$

$\nabla \varphi = \vec{v}$

The latter equation can be integrated up to a constant with an omnidirectional integrator without requiring any boundary conditions.

My questions are:

1. Is $\vec{v}$ unique to a constant?
2. If infinite solutions for $\vec{v}$ are possible, do all of them yield the same $\varphi$?
3. How to solve for $\vec{v}(x,y,z)$ for a general scalar field $S(x,y,z)$?

Answer 1

After doing some more research I found this answer on Quora:

https://www.quora.com/How-can-I-find-the-vector-field-having-its-divergence/answer/Tipper-Rumpf?ch=10&oid=106729750&share=4afbbd75&srid=hNudm7&target_type=answer

I guess what I was asking is if the divergence operator can be inverted as Tipper Rumpf replied. Looks like for a continuous field there are multiple solutions because it's an underdefined problem. In the discrete form, we can write $\nabla \cdot \vec{v}=S$ as:

$\begin{bmatrix} D_x & D_y & D_z \end{bmatrix}\begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix}=\big[S\big]$

where $D$ is the discrete divergence operator.For a grid with $N$ elements, we have $N$ equations and $3N$ unknowns, so there are infinite solutions.

Anyways, hope this helps others in the future. Thanks!

Answer 2

Suppose $\nabla ^2 \phi=S$ and $\nabla \cdot \vec{v} = S$.

By definition $\nabla^2 \phi = \nabla \cdot (\nabla \phi)$.

We also have for any vector field $\vec{A}$, $\nabla \cdot (\nabla \times \vec{A})=0 $.

So if $\vec{v}=\nabla \phi + \nabla \times \vec{A}$, then $\nabla \cdot \vec{v}=S$.

So $\vec{v}$ and $\nabla \phi$ can differ by more than a costant.

There are different techniques for solving for such a $\vec{v}$ depending on your coordinate system, symmetries inherent to the problem, etc.

Suppose you have an infinite line of charge with charge density $\lambda$. Imagine a cylinder centered on the line of charge having radius r and height $h$.

Gauss' Law tells us that $\int\int\int \nabla \cdot \vec{E}d\tau = \int\int \vec{E}\cdot \hat{n} dA$, and $\nabla \cdot \vec{E}=\rho/\epsilon_0$. This means the total charge in volume is the flux through its boundary surface.

So $\lambda h/\epsilon_0=E\cdot 2\pi r h\implies \vec{E}=\frac{\lambda }{2\pi \epsilon_0 r}\hat{r}$

This solves the problem $\nabla \cdot \vec{E} = \delta^2(r)\lambda/\epsilon_0. $

Green's Method could also give the same result.