- #1

- 46

- 1

[tex]

\nabla \cdot \vec{J} = 0

[/tex]

using the Ohm's law

[tex]

\vec{J} = \sigma \cdot (\vec{E} + \vec{U} \times \vec{B})

[/tex]

where [itex]\sigma[/itex] is a given 3x3 nonsymmetric conductivity matrix and [itex]U,B[/itex] are given vector fields. I desire the electric potential [itex]\Phi[/itex] where [itex]\vec{E} = -\nabla \Phi[/itex]. This leads to the inhomogeneous elliptic PDE:

[tex]

\nabla \cdot (\sigma \cdot \nabla \Phi) = f

[/tex]

where the right hand side [itex]f[/itex] is known and is [itex]f = \nabla \cdot (\sigma \cdot \vec{U} \times \vec{B})[/itex].

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:

[tex]

a \Phi + b \hat{n} \cdot \nabla \Phi = g

[/tex]

For my particular problem, I am using a spherical region

[tex]

\Omega = [r_1,r_2] \times [\theta_1,\theta_2] \times [0, 2 \pi]

[/tex]

which is like a spherical shell with the top and bottom cut off at some [itex]\theta_1,\theta_2[/itex]

Now I know that at the lower boundary,

[tex]

\vec{J}(r_1,\theta,\phi) = 0

[/tex]

which means

[tex]

\sigma \cdot \nabla \Phi(r_1,\theta,\phi) = (\sigma \cdot (\vec{U} \times \vec{B}))(r_1,\theta,\phi) = g(r_1,\theta,\phi)

[/tex]

where [itex]g[/itex] 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!