Description:
I want to solve the equation $$ \nabla \cdot \left( b \nabla c \right) = 0 $$ for $b(x,y)$ and $c(x,y)$ in the domain $x,y \geq 0$. This partial differential equation is supplemented by $$ b \nabla c = \vec{g} $$ where $\vec{g}(x,y)$ is a known vector that has non-zero curl. Furthermore, the values of $b$ and $c$ are known on the boundary $(x = 0, y = 0)$. As a consequence, the normal derivatives of $c(x,y)$ on the boundary are also known: $$ \nabla \times \vec{g} = R(x,y) \ne 0 \\ c = 0 \hspace{4mm} \text{at} \hspace{4mm} x=0 \\ c = 0 \hspace{4mm} \text{at} \hspace{4mm} y=0 \\ $$
What solution methods to this problem exist? Can you point me to any resources? If I try to solve this numerically, which software packages can you recommend? Thanks.
It is possible to derive the first-order pde's $$ \vec{g} \times \nabla c = 0 \\ \nabla \ln(b) \times \vec{g} = R \; . $$ Unfortunately, the characteristic lines of the pde's are tangential to the boundary on $\partial \Omega$. So the interior of the domain is not connected to the boundary. I was hoping there are other approaches.
You can start by noticing that we need $$ \begin{split} 0 &= \nabla\times\nabla c = \nabla\times\frac gb = \nabla\frac1b \times g + \frac1b \nabla\times g = -\frac{\nabla b}{b^2} \times g + \frac Rb . \end{split} $$ Multiplying by $b$ you get $$ \nabla(\log b) \times g = R. $$ This is a linear first order partial differential equation, whose solution is of the form $b=\exp(\tilde f+f_0)$, where $\tilde f$ is a particular solution to the inhomogeneous PDE $\nabla f \times g = R$ and $f_0$ is a solution to the homogeneous one. The PDE is solvable as long as $g$ is not orthogonal to $\partial\Omega$ (where $\Omega$ is your domain, the positive quadrant) and boundary value for $b$ is compatible with the characteristic lines. You can find more pieces of information here, here, and here.
Once you have found $b$, $g/b$ is irrotational, therefore you find $c$ such that $\nabla c=g/b$.