Let $f$ be a smooth function and $V$ a vector field on $\mathbb{R}^n$. I'm looking for an other vector field $W$ on $\mathbb{R}^n$ that satisfies $$\langle V, W \rangle + \nabla \cdot W = f$$ where $\langle,\rangle$ is the standard scalar product on $\mathbb{R}^n$ and $\nabla \cdot W=\sum_i \partial_i W_i$ is the divergence of $W$.
I would already be very happy if I could solve this equation. More generally, I want to solve this equation for many different pairs of $(f_i,W_i)$, ideally having an (computationally not too expensive) algorithm that takes $(f,W)$ as input and produces a solution $W$.