Hi,
What is the correct method for including spatially varying conductivity (i.e. multiple materials), \kappa(\boldsymbol{x}), in the mixed form of the Poisson equation?
In particular, in my derivation below I am unsure about the \int (\nabla\cdot\kappa(\boldsymbol{x})\tau)u\,d\Omega term in Equation (6). Is it correct to include \kappa within the divergence operator? This is currently solving but I want to make sure I am not missing something.
Strong form
\begin{aligned}
\nabla\cdot (\kappa(\boldsymbol{x})\nabla u) &= -f \;\;\;\;\;\;\;\;\;\;\;\;\;(1) \\
u &= u_D\; \textrm{on}\; \Gamma_D\;\;\;(2) \\
\kappa(\boldsymbol{x})\nabla u\cdot n &= g \; \textrm{on}\; \Gamma_N \;\;\;\;\;\;(3)
\end{aligned}
Converting to mixed form:
\begin{aligned}
q - \kappa(\boldsymbol{x})\nabla u &= 0\;\;\;\;\;\;(4)\\
\nabla\cdot q &= -f\;\;\;(5)
\end{aligned}
Weak form (after integration by parts)
Find q, u \in W = Q \times U , with test functions: \tau, v \in W such that:
\begin{aligned}
a &= \int q\cdot \tau\,d\Omega + \int (\nabla\cdot q)v \,d\Omega + \int (\nabla\cdot\kappa(\boldsymbol{x})\tau)u\,d\Omega \;\;\;(6) \\
L &= \int u_D \kappa(\boldsymbol{x})\tau \cdot n\,ds - \int fv\,d\Omega \;\;\;(7)
\end{aligned}
A similar question was posted a few years ago, but an explicit solution was not provided.
Thanks!