I have a highly nonlinear set of coupled PDEs. I know I will eventually need to solve them numerically, but first I need to understand how equations of this type should be solved. Essentially, I have
$$ \nabla \cdot \mathbb{A} = \mathbf{b} $$
$$ \nabla \cdot \mathbf{b} = x $$
where $\mathbb{A}$ is a second-rank tensor, $\mathbf{b}$ is a vector, and $x$ is a scalar. There are three unknown functions, and three equations (due to a symmetry argument, one equation is redundant).
Solution attempts
- At first, I tried a brute-force approach using finite differences. Even when I simplified the equations as far as I could, this method did not work, so I feel like more mathematical thought is needed here.
- I have followed the notes here (see Section 4) for learning Charpit's method and possibly applying it. However, it does not seem to generalize to more than one unknown function.
My question
Is there a technique that is typically used to solve a PDE of the type I have described? Charpit's method looked promising, but I can only find examples for a PDE in one unknown function.