Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

The iteration is carried out either alonq rows or colums. Generally
the GauO-Seidel iteration procedure is used with a successive over-
relaxation factor to speed up convergence..
3.3. P.nomalous slab as basic domain
In practice it is not necessary to solve the diffusion equation by
finite differences in t.he total conductor and the air half-space.
Instead it is sufficient to treat the equation only in that slab
which contains the anomalous domain.
Let the anomalous slab be confined to the depth range zl 5 z 5 z2.
Within this domain we have to solve the inho3nogeneous equation
(considering for the moment only the TE-case)
subject to two homogeneous boundary conditiolls at z = z and z2,
1
~Thich involve a for z < z and z > z2 respectively and account for
n 1
the vanishing anomalous field for z -t 2 m. Yihen (3.25) is solved
by finite differences, the discretization invol.ves also the field
values one grid point width above and below the anomalous slab.
The idea is to express these values in terms of a line integral
over Ea at z = z and z2 respectively.
I
Let V and V2 be the half-planes z - < zl and z -- > z2, respectively.
~ e G t r [r). 2, r, be Green's functions which satisfy
" "m
(-0 -
subject to the boundary condition
In V and V2, E is a solution of
1 a
Now Green's formula for two-dimensions states that