POTENTIAL PROBLEMS 33
potentials around Γ identical to those of our initial problem, within Ω,
that is u = u' on Γ. Hence we can add equations (2.27) and (2.28)
which gives, for an internal point in Ω,
".- u*(q + qf)dr + bu*dQ = 0 (2.29)
or
Ui=
u*adr-
bu*dQ (2.30)
This is the form of the indirect boundary element method
statement, σ represents the initially unknown density distribution of
u* over Γ necessary to generate w, via equation (2.30). Notice that in
the indirect formulation the singular source
u*
is always considered to
be located at the boundary point rather than at any internal boundary
point or even an external point as in the direct formulation. The
physical interpretation of σ is the difference between the fluxes
generated by the two solutions (internal and external).
We can also deduce another indirect formulation from equations
(2.27) and (2.28). This is the indirect approach in terms of dipoles.
Instead of continuity of potentials we can assume continuity of fluxes,
i.e. q + q = 0. Subtracting equations (2.27) and (2.28) we now obtain
U,;= ~ (U
■
u')q*df
—
bu*dQ (2.31)
Ω
or
U: =
ßq*df-
bu*dQ (2.32)
μ are called dipoles and give rise to an alternative indirect
formulation.
The steps for any self-adjoint case are similar to the ones described
for the above equations and they have been described in detail in
Brebbia and Butterfield.
1
Similar arguments can also be applied to
problems such as those present in elasticity (Chapter 4) and the
argument can even be extended to non-self-adjoint problems but for
these cases the fundamental solution for the adjoint operator may be
difficult to find.
Note that in order to apply natural boundary conditions on Γ, in