Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 51

We now show that the constraint v( 0 ) = 1 will be ignored by the solution of the

variational problem. The singular function

log log eR
r

w 0 (x) = with r = r(x) = x

has a finite Dirichlet integral (3.12). We smooth it to get

w 0 (x) for r(x) ≥ ε,

w ε (x) =

log log eR
ε

for 0 ≤ r(x)<ε.

Now | w ε | 1 ,B R ≤| w 0 | 1 ,B R , and w ε ( 0 ) tends to ∞ as ε → 0. Thus,

w ε (x)
w ε ( 0 )

u ε =

for ε = 1 , 1 / 2 , 1 / 3 ,... provides a minimizing sequence for J(v) which converges

almost everywhere to the zero function. This means that the requirement u( 0 ) =

1

was ignored.
The situation is different if we require that u 1 on a curve segment. This

would be the case if the tent were attached to a ring on the tent pole or if the tent
is put over a rope so that it assumes the shape of a roof. While the evaluation of an
H 1 function at a point does not make any sense, its evaluation on a line in the L 2
sense is possible. A condition on a function which is defined on a curve segment
will be respected almost everywhere.

=

Fig. 8. Tent attached to a loop of a rope to prevent an extreme force concentration
at the tip

For most larger tents, the boundary of the tent at the tip is a ring instead of
a single point. Or (see Fig. 8) the tent may be attached to a loop of rope. This
avoids very high forces, since the force applies to the ring or loop, rather than at
a single point. The trace theorem explains why the point is to be replaced by a
one-dimensional curve.