Fluxes as Dirichlet Boundary Conditions
Consider Maxwell’s equations in 3D
with the flux terms
and
and the divergence on ΓE(H) and ΓH(E) applied row-wise.
For Ampère’s law, the normal to the flux term on exterior boundaries reads
and for Faraday’s law
which means that normal fluxes on exterior boundaries can only prescribe tangential components for the fields.
Boundary Conditions
The boundary conditions for outer boundaries are computed from the normal fluxes n · ΓH(E) and n · ΓE(H).
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Perfect electric conductor N × E = 0, or zero tangential components for E, is obtained by setting n · ΓH(E) = 0.
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Perfect magnetic conductor N × H = 0, or zero tangential components for H, is obtained by prescribing n ⋅ ΓE(H) = 0.
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Electric field N × E = N × E0, or n · ΓH(E) = n × E0.
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Magnetic field N × H = N × H0, or −n · ΓE(H) = n × H0.
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For exterior boundaries, the surface currents BC means N × H = Js, or −n · ΓE(H) = Js.
Absorbing Boundary Condition
A simple absorbing boundary can be implemented by setting N × E = ZH.