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8.5
The above figure shows a line with a refinement, as could be found along
a vertical grid line in Fig. 12. Extend this to a triangulation consisting of right
isosceles triangles which connect to a coarse grid.
8.6
Suppose we want to solve the elliptic differential equation
div[
a(x)
grad
u
]
=
f
in
1
0
. Show
that we get the same solution if
a(x)
is replaced by a function which is constant on
each triangle. How can we find the right constants?
with suitable boundary conditions using linear triangular elements from
M
2
we can obviously decompose every triangle into four congruent subtrian-
gles. With the help of a sketch, verify that the analogous assertion for a tetrahedron
in
8.7
In
R
3
does not hold.
R
8.8
Let
λ
1
,λ
2
,λ
3
be the barycentric coordinates of a triangle
T
with vertices
z
1
,z
2
,z
3
. Show that
3
µ(T )
p(z
i
)
=
(
3
λ
i
−
λ
j
−
λ
k
)p dx
for
p
∈
P
1
,
T
if
i, j, k
is a permutation of 1
,
2
,
3.
8.9
The implementation of the Neumann boundary-value problem was elucidated
for the case that the finite element space contains the kernel of the differential
operator. What happens if that conditions is not satisfied?
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