Boundary conditions for inhomogeneous non-sepearable 3D PDE

[vibe3]

Hello, I am looking to solve the 3D equation in spherical coordinates
$$\nabla \cdot \vec{J} = 0$$
using the Ohm's law
$$\vec{J} = \sigma \cdot (\vec{E} + \vec{U} \times \vec{B})$$
where $\sigma$ is a given 3x3 nonsymmetric conductivity matrix and $U,B$ are given vector fields. I desire the electric potential $\Phi$ where $\vec{E} = -\nabla \Phi$. This leads to the inhomogeneous elliptic PDE:
$$\nabla \cdot (\sigma \cdot \nabla \Phi) = f$$
where the right hand side $f$ is known and is $f = \nabla \cdot (\sigma \cdot \vec{U} \times \vec{B})$.

Now my question relates to how to express the boundary conditions. Many existing PDE software require inputs of Robin-type boundary conditions, which would be of the form:
$$a \Phi + b \hat{n} \cdot \nabla \Phi = g$$

For my particular problem, I am using a spherical region
$$\Omega = [r_1,r_2] \times [\theta_1,\theta_2] \times [0, 2 \pi]$$
which is like a spherical shell with the top and bottom cut off at some $\theta_1,\theta_2$

Now I know that at the lower boundary,
$$\vec{J}(r_1,\theta,\phi) = 0$$
which means
$$\sigma \cdot \nabla \Phi(r_1,\theta,\phi) = (\sigma \cdot (\vec{U} \times \vec{B}))(r_1,\theta,\phi) = g(r_1,\theta,\phi)$$
where $g$ is known.

What I can't see easily is now to convert this into the Robin-type equation above so it can be input into a PDE software. Does anyone have any ideas?

Many thanks in advance!

 

[pasmith]

Hello, I am looking to solve the 3D equation in spherical coordinates
$$\nabla \cdot \vec{J} = 0$$
using the Ohm's law
$$\vec{J} = \sigma \cdot (\vec{E} + \vec{U} \times \vec{B})$$
where $\sigma$ is a given 3x3 nonsymmetric conductivity matrix and $U,B$ are given vector fields. I desire the electric potential $\Phi$ where $\vec{E} = -\nabla \Phi$. This leads to the inhomogeneous elliptic PDE:
$$\nabla \cdot (\sigma \cdot \nabla \Phi) = f$$
where the right hand side $f$ is known and is $f = \nabla \cdot (\sigma \cdot \vec{U} \times \vec{B})$.

Now my question relates to how to express the boundary conditions. Many existing PDE software require inputs of Robin-type boundary conditions, which would be of the form:
$$a \Phi + b \hat{n} \cdot \nabla \Phi = g$$

For my particular problem, I am using a spherical region
$$\Omega = [r_1,r_2] \times [\theta_1,\theta_2] \times [0, 2 \pi]$$
which is like a spherical shell with the top and bottom cut off at some $\theta_1,\theta_2$

Now I know that at the lower boundary,
$$\vec{J}(r_1,\theta,\phi) = 0$$
which means
$$\sigma \cdot \nabla \Phi(r_1,\theta,\phi) = (\sigma \cdot (\vec{U} \times \vec{B}))(r_1,\theta,\phi) = g(r_1,\theta,\phi)$$
where $g$ is known

Your $g$ is a vector.

You need somehow to solve
$$\sigma \cdot \nabla \Phi = \vec g$$
for the radial component of $\nabla \Phi$, which is $\vec n \cdot \nabla\Phi$ for this boundary. That in general is possible only if $\sigma$ is invertible on the boundary ($\det \sigma \neq 0$), so that
$$\frac{\partial \Phi}{\partial r} = \hat r \cdot (\sigma^{-1} \cdot \vec g).$$

(Actually for $r = r_1$ we have $\vec n \cdot \nabla\Phi = - \dfrac{\partial \Phi}{\partial r}$, so on that boundary
$$-\frac{\partial \Phi}{\partial r} = \hat r \cdot (\sigma^{-1} \cdot \vec g)$$
and on $r = r_2$
$$\frac{\partial \Phi}{\partial r} = \hat r \cdot (\sigma^{-1} \cdot \vec g).$$

Otherwise you may need to find a different solution method.