Triangulation h No of elements = 16No of nodes = 13No interior nodes = 5No of boundary nodes = 8
With the triangulation we associate the function space consisting of continuous, piecewise linear functions on vanishing on i.e
Triangulation h
No interior nodes = 5No of global basis functions = 5
hτh0S
Γ vτKK):vΩC(vS hh on 0 ,each for in linear 0
543210 ,,,, SpanS h
1
2
3
4
5
6
78
9
1011
12
13
1312111050 ,,,, SpanS h
Element Labeling
14
15
16
Node Labeling (global labeling)
12
3 4
5
6
7
8
9
1011
12 13
1312111050 ,,,, SpanS h
global basis functions
12
3 4
5
6
7
8
9
1011
12 13
1312111050 ,,,, SpanS h
),(5 yx
0)nodes o(
1)5.0,5.0(
5
5
ther
12
3 4
5
6
7
8
9
1011
12 13
1312111050 ,,,, SpanS h
),(10 yx
0)nodes o(
1)75.0,75.0(
10
10
ther
global basis functions
Global basis functions
global basis functions),(5 yx
0)nodes o(
1)5.0,5.0(
5
5
ther
12
3 4
6
7
8
9345 x
0
0
0
0
0
0
0
0
0
0
0 0
145 x
345 y1011
5
145 y1312
1
23
4
56
78
910
1112
13
14
15
16
),(5 yx
4,15,166,7,8,13,11,2,3,4,5,ii
9
12
11
10
5
K0
K34
K14
K14
K34
),(
in
inx
iny
inx
iny
yx
global basis functions),(10 yx
0)nodes o(
1)75.0,75.0(
10
10
ther
12
3 4
6
7
8
910
0
0
0
0
0
0
0 0
1011
5
1312
0
0 10
10
10
10
10
)l(
)l(
)l(
)l(
)l(
c13
c12
c11
c10
c5
*****13
*****12
*****11
*****10
*****5
131211105
13
12
11
10
5
1
23
4
56
78
910
1112
13
14
15
16
12
3 4
5
6
7
8
9
1011
12 13
dxdya yyxx ,10,5,10,5105 ),(
109
,10,5,10,5,10,5,10,5
K
yyxx
K
yyxx dxdydxdy
16
1,10,5,10,5
i K
yyxx
i
dxdy
Assemble linear system
)l(
)l(
)l(
)l(
)l(
c13
c12
c11
c10
c5
*****13
*****12
*****11
*****10
*****5
131211105
13
12
11
10
5
dxdya yyxx ,10,5,10,5105 ),(
109
,10,5,10,5,10,5,10,5
K
yyxx
K
yyxx dxdydxdy
Assemble linear system
12
3 4
6
7
8
9345 x
0
0
0
0
0
0
0
0
0
0
0 0
145 x
345 y1011
5
145 y
1312
),(5 yx
12
3 4
6
7
8
9222 yx
0
00
0
00
0 0
1011
5
1312 00 10
10 10
10
),(10 yx
222 yx
999
8)2()4()2()0(KKK
dxdydxdy
101010
8)2()0()2()4(KKK
dxdydxdy
1
1
Approximation of u
10
20
30
40
50.069
60
70
80
90
100.049
110.049
120.049
130.049
u
Node Label (local labeling)
1 2
3
Each triangle has 3 nodes. Label them locally inside the triangle
Node and Element Label
Local label .vs. global label
Matrix t(3,#elements)
12345678910111213141516
141231234555510111213
29678101112131310111213101112
3131011129678101112139678
t
X-coordinate and y-coordinate
Matrix p(2,#elements)
12345678910111213
x10010.50.500.510.750.250.250.75
y11000.510.500.50.750.750.250.25p
Boundary node
vector e(#boundary node)
e1e2e3e4e5e6e7e8
start12346789
end67892341e
131211105nodeinterior
Approximation of u
10
20
30
40
50.069
60
70
80
90
100.049
110.049
120.049
130.049
u
Global basis functions
Triangulation
Recommended
