14 APPROXIMATE METHODS
1.4 THE INVERSE PROBLEM
Sometimes it is convenient to take into account the conditions on Γ
ι
(u = ü for the Laplace equation) in the weighted residual statement.
This produces an approximation for these conditions and the error
integrals can now be written as follows,
I ewdQ = I s
2
wdr- I ^i^cir (1.31)
or
(V
2
w)wdfl= (q-q)wdr- (u-ü)-^dr (1.32)
h
Jr
2
Jr,
dn
The inclusion of the last integral with the weighting function dw/dn
can be justified in different ways but it is simpler to accept (1.32) on
dimensional grounds and to prove that the minus sign is correct by
integrating by parts. Integrating (1.32) by parts once, we obtain
J
du dw
1Λ
f ,„ f
tr
^ C dw
fr
,
—-—-άΩ= qwdr+ qwdr - ΰ—-άΓ +
Q
dx
k
dx
k
J
r
J
r
J
r
dn
^df (1.33)
on
This expression is seldom used as the boundary conditions on Γ
χ
are usually easy to satisfy when working with domain methods such
as finite elements. In this case the three integrals on Γ
χ
disappear.
Integrating equation (1.33) again by parts we obtain the following
statement:
^dr + (V
2
w)wdß= - qwdr- qwdr + u^-dr
h Jr
2
Jr, Jr
2
dn
+ I »^dr (1.34)
Formula (1.34) is the inverse of
the
original expression (1.32) and it
offers the possibility of working only on the boundary if a solution is
found for w such that V
2
w = 0. Note that now the weighting function
w
needs to be continuous up to the second derivatives but
u
only needs
to have continuity of the function itself (but not of the derivatives).