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Piecewise Polynomial Spaces. The reason for introducing a mesh of a domain is that it allows for a construction of piecewise polynomial function spaces on this domain. Definition: linear function in x and y. Is a linear function in x and y. Example:. Piecewise Polynomial Spaces. - PowerPoint PPT Presentation
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Piecewise Polynomial Spaces
The reason for introducing a mesh of a domain is that it allows for a construction of piecewise polynomial function spaces on this domain
cbyaxyxp ),(
723),( yxyxp
Definition:
linear function in x and y
Is a linear function in x and y
Example:
Piecewise Polynomial Spaces
Definition:
be the space of linear functions on
Remark:
)(1 KP
We observe that any member in is uniquely determined by its nodal values
triangle with nodesK 321 ,, NNN
1N
2N 3N
)(1 KP
Example: Find a linear polynomial on the triangle K such that p(N1)=2 , p(N2) = 3, p(N3)=1
),( yxp
K
cbyaxyxp ),(
K
)3,2(1 N
)2,1(2 N
)1,3(3 N
K
1
3
2
113
121
132
c
b
a
4),( xyxp
Local basis functions
Example:
Find a linear function on the triangle K such that p(N1)=2 , p(N2) = 3, p(N3)=1
),( yxp
cbyaxyxp ),(
)3,2(1 N
)2,1(2 N
)1,3(3 N
K4),( xyxp
Example:
Find a linear function on the triangle K such that p(N1)=1 , p(N2) = 0, p(N3)=0p(N1)=0 , p(N2) = 1, p(N3)=0p(N1)=0 , p(N2) = 0, p(N3)=1
),( yxp
3/)52(),(1 yxyx3/)72(),(2 yxyx
3/)1(),(3 yxyx
Remark: 3211 ,,)( spanKP any member in can be expressed as a linear combination of these three functions
)(1 KP
Example:
4),( xyxp
321 32
K)( 1,1 yx
1
2
3 )( 3,3 yx
)( 2,2 yx
0
0
1
1
1
1
1
1
1
33
22
11
c
b
a
yx
yx
yx
)2/(
23
32
2332
1
1
1
K
xx
yy
yxyx
c
b
a
33
22
11
1
1
1
2
1
yx
yx
yx
K
0
1
0
1
1
1
2
2
2
33
22
11
c
b
a
yx
yx
yx
)2/(
31
13
3113
2
2
2
K
xx
yy
yxyx
c
b
a
1
0
0
1
1
1
3
3
3
33
22
11
c
b
a
yx
yx
yx
)2/(
12
21
1221
3
3
3
K
xx
yy
yxyx
c
b
a
ycxbaK111
)(1
ycxbaK222
)(2
ycxbaK333
)(3
The local basis functions for the triangle K are
Local basis functions
321 ,,
Local basis functions
Find three linear functions on the reference triangle such that
Reference triangle
Exercise3
1)1,0(~
,0)0,1(~
,0)0,0(~
0)1,0(~
,1)0,1(~
,0)0,0(~
0)1,0(~
,0)0,1(~
,1)0,0(~
333
222
111
321
~,
~,
~
Then find a linear function on the reference triangle K such that p(0,0)=2 , p(1,0) = 3, p(0,1)=1
),( yxp
Continuous Piecewise Polynomial Spaces
Definition:
the space of all continuous functions
KKPv
CvV
K
h trianglesallfor )(
),(
1
0
Definition:
h
)(0 C
be a triangulation of
the space of all continuous piecewise linear polynomials
An example of a continuous piecewise linear function
Global Basis Functions for
KKPv
CvV
K
h trianglesallfor )(
),(
1
0
the space of all continuous piecewise linear polynomials
hV
To construct a basis for this space we note that a function v in this space is uniquely determined by its nodal values
},,,,{ 321 nh spanV
where n is the number of nodes in the mesh
10
11)(1 k
kNk
jk
jkNkj 0
1)(
} , ,, { 1321 spanVh
Example (for global basis functions)
global basis functions
12
3 4
5
6
7
8
9
1011
12 13
),(5 yx
0)nodes o(
1)5.0,5.0(
5
5
ther
} , ,, { 1321 spanVh
12
3 4
5
6
7
8
9
1011
12 13
),(10 yx
0)nodes o(
1)75.0,75.0(
10
10
ther
global basis functions
} , ,, { 1321 spanVh
Global basis functions related to interior nodes
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
),(10 yx
0)nodes o( ,1)75.0,75.0( 1010 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
Exercise4: Find in explicit form 10
Remark:
Continuous Piecewise Linear Interpolation
Definition:
we define its continuous piecewise linear interpolant by
)(0 CfLet
n
kkkNff
1
)( hVf
approximates by taking on the same values in the nodes Ni.
f f
f
f
[p,e,t] = initmesh('squareg','hmax',0.7); % meshx = p(1,:); y = p(2,:); % node coordinatespif = x.^2+ y.^2; % nodal values of interpolantpdesurf(p,t,pif') % plot interpolant %pdeplot(p,e,t,'xydata',pif,'zdata',pif,'mesh','on');
to draw πf given fto draw πf given f
Piecewise Polynomial Spaces
KDefinition:
be the space of linear functions on
Remark: Reference triangle
)(1 KP
We observe that any function inP1(K) is uniquely determined by its nodal values
triangle with nodes
