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Fig. 7.2 Combination technique for level 4 in two dimensions
7.2.4 The Sparse Grid Combination Technique
For the sparse grid combination technique, we proceed as follows: we discretize
and solve the problem on a certain sequence of grids
Ω
l
with uniform mesh sizes
h
t
¼
2
l
t
in the
t
th coordinate direction. These grids may possess different mesh
sizes for different coordinate directions. To this end, we consider all grids
Ω
l
with
l
1
þþl
d
¼ n þ d
1
ð
Þ q
,
q ¼
0,
...
,
d
1,
l
t
>
0
:
ð
7
:
25
Þ
In contrast to the definition (
7.21
), for reasons of efficiency, we now restrict the
level indices to
l
t
>
0. For the two-dimensional case, the grids needed in the
combination formula of level 4 are shown in Fig.
7.2
. The finite element approach
with piecewise
d
-linear test and trial functions
φ
l,j
(x) on grid
Ω
l
now would give
f
l
ðÞ¼
X
2
l
1
X
2
l
d
...
α
l
,
j
ϕ
l
,
j
ðÞ
j
1
j
d
and the variational procedure (
7.4
), (
7.5
), and (
7.6
) would result in the discrete
system
