Chapter 11 polar coordinate
Speaker: Lung-Sheng Chien
Book: Lloyd N. Trefethen, Spectral Methods in MATLAB
Discretization on unit disk
Consider eigen-value problem on unit disk
u   2u
with boundary condition u  r  1,   0
 x  r cos 
, then
y

r
sin


We adopt polar coordinate 
1
1
u  urr  ur  2 u   2u
r
r
, on r   0,1 and
 0, 2 
Usually we take periodic Fourier grid in  , and non-periodic Chebyshev grid in r
1
r 0,1
Chebyshev grid in x1,1
r
x 1
2
Chebyshev grid in r 0,1
Observation: nodes are clustered near origin r  0 , for time evolution problem,
we need smaller time-step to maintain numerical stability.
2
r 1,1
 x  r cos   r cos    

 y  r sin   r sin    
 x, y   0,0
1 to 2 mapping
 r , 
Asymptotic behavior of spectrum of Chebyshev diff. matrix
In chapter 10, we have showed that spectrum of Chebyshev differential matrix DN2
(second order) approximates
uxx  u, 1  x  1, B.C. u  1  0
1
2
with eigenmode k  
2
4
Eigenvalue of DN2 is negative (real number) and max  0.048N 4
Large eigenmode of D 2
N
does not approximate to k  
Since ppw is too small such that
resolution is not enough
Mode N is spurious and localized near
boundaries x  1
2
4
k2
k2
Grid distribution
1
r 0,1
2
r 1,1
Preliminary: Chebyshev node and diff. matrix [1]
 j 
Consider N  1 Chebyshev node on 1,1 , x j  cos 
 for j  0,1, 2, , N
 N 
Uniform division in arc
xN
x3 x2 x1 x0
1
xN / 2
1
x
Even case: N  6
1  x6 x5
x3  0
x4
x2
x1 x0  1
x
x1 x0  1
x
Odd case: N  5
1  x5
x4
x3
0
x2
Preliminary: Chebyshev node and diff. matrix [2]
 j 
Given N  1 Chebyshev nodes , x j  cos 
 and corresponding function value v j
 N 
N
We can construct a unique polynomial of degree N , called p  x    v j S j  x 
j 0
1 k  j is a basis.
S j  xk   
0 k  j
N
p  xi    v j S j
j 0
1
N
 xi    DijN v j
where differential matrix DN
j 0
 D  is expressed as
x j
2N 2 1
N
2N 2 1
N
D

, for j  1, 2,
D


,
D 
, NN
jj
2
6
2 1  x j 
6
N
00
c  1
DijN  i
, for i  j, i, j  0,1, 2,
c j xi  x j
i j
with identity D  
N
ii
,N
N

j 0, j i
DijN
Second derivative matrix is DN2  DN  DN
N
ij
, N 1
2 i  0, N
where ci  
1 otherwise
Preliminary: Chebyshev node and diff. matrix [3]
 
Let p  x  be the unique polynomial of degree  N with p  1  0 and p x j  v j
 
define w j  p x j
 
and z j  p x j
for 0  j  N
2
We abbreviate w  DN  v , then impose B.C. p  1  0 ,that is, v0  vN  0
In order to keep solvability, we neglect w0  wN ,that is, DN2  DN2 1: N 1,1: N 1
neglect
w0
w1
w2
wN 1
neglect
wN

DN2
v0
v1
v2
vN 1
vN
zero
zero
Similarly, we also modify differential matrix as DN  DN 1: N 1,1: N 1
Preliminary: DFT
v1, v2 ,
Given a set of data point
, vN   R
N
[1]
with N  2m is even, h 
 
2
N
Then DFT formula for v j
2
vˆk 
N
1
vj 
2
N
v
j 1
j
m

k  m 1
exp  ikx j 
for k  m, m  1,
vˆk exp  ikx j 
for j  1, 2,
, m  1, m
,N
Definition: band-limit interpolant of   function, is periodic sinc function SN  x 
m
sin  x / h 
h
1 sin  mx 
P  eikx 

2 k  m
 2 / h  tan  x / 2 2m tan  x / 2
SN  x 
If we write v j 
N
v 
k 1
k
j k
m
N
1
ikx
   v , then p  x  
P  e vˆk   vk S N  x  xk 
2 k  m
k 1
  v S   x
Also derivative is according to w j  p x j 
N
k 1
k
1
N
j
 xk 
Preliminary: DFT
[2]
sin  x / h 
, we have
Direct computation of derivative of S N  x  
 2 / h  tan  x / 2
0
j  0  mod N 

S N1  x j    1
j
 jh 

1
cot


  , j  0  mod N 

2
 2 

 

Example: D5  S5 x j  xk
1


0
S51   h  S51  2h  S51  3h  S51  4h  
 1

1
1
1
0
S5  h  S5  2h  S5  3h  
 S5  h 


  S51  2h  S51  h 
0
S51  h  S51  2h  
 1

1
1
1
S
3
h
S
2
h
S
h
0
S

h








5
5
5
 5

1
 S 1  4h  S 1  3h  S 1  2h 

S
h
0


5
5
5
5


is a Toeplitz matrix.
 1 2
j  0  mod N 
  2
6
3
h

 2
Second derivative is S N  x j   
j

1



, j  0  mod N 
 2sin 2  jh / 2 

Preliminary: DFT
[3]
 1 2
j  0  mod N 
  2
 6 3h
S N 2  x j   
j

1



, j  0  mod N 
 2sin 2  jh / 2 

  v S  x
For second derivative operation w j  p x j 
N
k 1
k
N
 xk    DN2  j , k  vk
N
 2
j
k 1
second diff. matrix is explicitly defined by using Toeplitz matrix (command in MATLAB)

DN2  S N 2  x j  xk 












 csc 2  2h / 2  / 2
csc2  h / 2  / 2
1 2
  2
6 3h
2
csc  h / 2  / 2
 csc 2  2h / 2  / 2




2:N
 1 2

1


  toeplitz   6  3h 2 , 2sin 2 1: N  1 h / 2 
 







Fornberg’s idea : extend radius to negative image [1]
r 1,1 and  0, 2 
 x, y  unit disk
1
Nr  5 (odd): to avoid singularity of coordinate transformation r  0
2
N  6 (even): to keep symmetry condition u  r ,    u  r ,     mod 2  
r
r   coordinate
x  y coordinate
1  r0
r1
r0
y
2
r2
r3
r4
1  rNr  r5
1
3

0 1 2 3 4 5 6
0

 N  2
6
4
5
x
Fornberg’s idea : extend radius to negative image [2]
In general Nr  2k  1 is odd, and N  2m is even, then r : non-unif and  
ri  1  ir : 1  i  Nr 1
Active variable
 j  j : 1  j  N
, total number is
 Nr 1 N
r   coordinate
r
x  y coordinate
1  r0
r1
r2
r0
2
N
y
2
1
rk
rk 1
m
rNr 1

1  rNr
0 1 2
0

m m1
 N 1
 N  2
 N
m1
 N 1
x
Redundancy in coordinate transformation [1]
2 to 1 mapping
 r ,  ,  r ,     mod
2     x, y 
x  y coordinate
r   coordinate
y
r
2
redundant
1  r0
r1
r0
3
r2
r3
r4
1  rNr  r5
1
4
6
x
6
x
5
y
2

0 1 2 3 4 5 6
0

 N  2
1
3
4
5
Redundancy in coordinate transformation [2]
2 to 1 mapping
 r ,  ,  r ,     mod
2     x, y 
x  y coordinate
r   coordinate
y
2
r
1  r0
r1
r0
3
r2
r3
r4
1  rNr  r5
1
x
6
x
5
4
y
redundant
2

0 1 2 3 4 5 6
0

 N  2
6
1
3
4
5
Redundancy in coordinate transformation [3]
Nr  2k  1  5 is odd, then Chebyshev differential matrix DNr
r0
r2
r3
r4
r1
DNr 
DNr 
is expressed as
r5
8.5
-10.4721
2.8944
-1.5279
1.1056
-0.5
r0
2.6180
-1.1708
-2
0.8944
-0.618
0.2764
r1
-0.7236
2
-0.1708
-1.6180
0.8944
-0.382
r2
0.3820
-0.8944
1.618
0.1708
-2
0.7236
r3
-0.2764
0.6180
-0.8944
2
1.1708
-2.618
r4
0.5
-1.1056
1.5279
-2.8944
10.4721
-8.5
r5
-1.1708
-2
0.8944
-0.618
2
-0.1708
-1.6180
0.8944
-0.8944
1.618
0.1708
-2
0.6180
-0.8944
2
1.1708
neglect
Redundancy in coordinate transformation [4]
Symmetry property of Chebyshev differential matrix :  DN i , j    DN  N i , N  j
DNr 
-1.1708
-2
0.8944
-0.618
2
-0.1708
-1.6180
0.8944
-0.8944
1.618
0.1708
-2
0.6180
-0.8944
2
1.1708
E1 
E2 
-1.1708
-2
is symmetric
2
-0.1708
-0.618
0.8944
is NOT symmetric
0.8944
-1.6180
Permute column by P  1, 2, 4,3
DNr P 
T
-1.1708
-2
-0.618
0.8944
2
-0.1708
0.8944
-1.6180
-0.8944
1.618
-2
0.1708
0.6180
-0.8944
1.1708
2
E
1
 u1 
 
u
E2  2  is faster ?
 u3 
 
 u4 

Redundancy in coordinate transformation [5]
2
Nr  2k  1  5 is odd, then Chebyshev differential matrix DNr
r0
r2
r3
r4
r1
DN2 r  DNr  DNr 
D 
2
Nr
is expressed as
r5
41.6
-68.3607
40.8276
-23.6393
17.5724
-8
r0
21.2859
-31.5331
12.6833
-3.6944
2.2111
-0.9528
r1
-1.8472
7.3167
-10.0669
5.7889
-1.9056
0.7141
r2
0.7141
-1.9056
5.7889
-10.0669
7.3167
-1.8472
r3
-0.9528
2.2111
-3.6944
12.6833
-31.5331
21.2859
r4
-8
17.5724
-23.6393
40.8276
-68.3607
41.6
r5
-31.5331
12.6833
-3.6944
2.2111
7.3167
-10.0669
5.7889
-1.9056
-1.9056
5.7889
-10.0669
7.3167
2.2111
-3.6944
12.6833
-31.5331
neglect
Redundancy in coordinate transformation [6]
Symmetry property of Chebyshev differential matrix :
D 
2
Nr
-31.5331
12.6833
-3.6944
2.2111
7.3167
-10.0669
5.7889
-1.9056
-1.9056
5.7889
-10.0669
7.3167
2.2111
-3.6944
12.6833
-31.5331
D1 
D 
2
N i, j
 
 DN2
-31.5331
N i , N  j
12.6833
is NOT sym.
7.3167
-10.0669
2.2111
-3.6944
Permute column by P  1, 2, 4,3
D2 
D P 
2
Nr
-31.5331
12.6833
2.2111
-3.6944
7.3167
-10.0669
-1.9056
5.7889
-1.9056
5.7889
7.3167
-10.0669
2.2111
-3.6944
-31.5331
12.6833
T
is NOT sym.
-1.9056
5.7889
Row-major indexing: remove redundancy
Define active variable uij
ui
u
i ,1
, ui ,2 ,
[1]
u  ri ,  j  for 1  i  Nr  1 and 1  j  N

, ui ,m  , ui ,m1, ui ,m2 ,

T
, ui , N
total number of active variables is k  N  2  6  12
NOT
 Nr 1  N  4  6  24
u1 
r
1  r0
r1
Index order
r0
r2
2
3
4
7 8
9
10 11 12
1
r3
r4
5
6
redundant

1  rNr  r5
0 1 2 3 4 5 6
0

 N  2
Index order
u2 
 u11 
 
 u12 
u 
 13 
1
 u14 
 
 u15 
u 
 16 
4
 u21 
 
 u22 
u 
 23 
7
2
3
5
6
8
9
 u24  10
 
 u25  11
u 
 26  12
Row-major indexing: remove redundancy
[2]
 i 
Nr  2k  1 is odd, and ri  cos 
 : 0  i  Nr
N
 r
suppose 0  j  k , then rk  j  rk 1 j  0 since
rk  j
k  j 

 cos
Nr

  k  j  1  
k  j    Nr 2 k 1

  cos   
 cos 
 
  rk 1 j
N
N
r
r




N  2m is even, and  
2 
 ,  j  j : 1  j  N
N m
Hence for 1  j  m,  j    m j and

m j
    mod 2    j
rNr 1  r1  0
rk  j  rk 1 j  0
rk  rk 1  0
1  rNr rNr 1
rk 1 j
rk 1
0
rk
rk  j
r1 r0  1
r
Row-major indexing: remove redundancy




From symmetry condition, we have u rk 1i , j  u rk 1i ,  j  
[3]
  mod 2 
u  rk 1i ,  j   u  rk i ,  m  j 
for 0  i  k and 0  j  m , symmetry condition implies
u  rk 1i ,  m  j   u  rk i ,  j 
Therefore, we have two important relationships
1
 uk 1, j   uk ,m  j  

 
 
u
u
k

2,
j
k

1,
m

j




 
 

 
 
u
u
 Nr 1, j   1,m  j   1
1   u1,m  j 


1   u2,m  j 



 
u
  k ,m  j 
2
 uk 1,m  j   uk , j  

 
 
u
u
k

2,
m

j
k

1,
j




 
 

 
 
u
u
 Nr 1,m  j   1, j   1
1   u1, j 


1   u2, j 



 
u
 k, j 
Kronecker product
[1]
r
Define active variable uij
u  ri ,  j 
for 1  i  2 and 1  j  3
1  r0
r1
r2
1  r3
1
2
3
4
5
6
 u11 
 
 u12 
u 
 13 
1
 u21 
 
u2   u22 
u 
 23 
4
u1 

0  0 1 2 3  
2
3
5
6
Separation of variable: assume matrix A acts on r-dir and matrix B acts on   dir
  r1 , j    a
   11
 Au  
  r2 , j    a21


a12   u1, j 


a22   u2, j 
  ri , 1    b11 b12


 Bu    ri ,2     b21 b22
  r ,    b
 i 3   31 b32
is independent of j
b13   ui ,1 
 
b23   ui ,2  is independent of i
b33   ui ,3 
 u1 
6
  R be row-major
 u2 
Let X  
index of active variable
Kronecker product
[2]
 a11B a12 B 
66
 R
 a21B a22 B 
Kronecker product is defined by A  B  
 a11B a12 B  u1   a11Bu1  a12 Bu2 
 A  B X  
   

a
B
a
B
22  u2 
 21
 a21Bu1  a22 Bu2 
Case 1: A  I
 Bu1 
I

B
X


  
 Bu2 
  ri , 1    b11 b12


 Bu    ri ,2     b21 b22
  r ,    b
 i 3   31 b32
b13   ui ,1 
 
b23   ui ,2 
b33   ui ,3 
Case 2: B  I
  u11 
 u21  
  
 
a
u

a
 11  12  12  u22  
 
u  
 a11u1  a12u2    u13 
 23 

 A I  X  

a
u

a
u
 u21  
 21 1 22 2    u11 
 a u   a u 
 21  12  22  22  
u 
 u 
 23  
  13 
  r1 ,  j    a
   11
 Au  
  r1 ,  j    a21


a12   u1, j 


a22   u2, j 
Kronecker product
[3]
Case 3: permutation, if permute 2  3
r
r
1  r0
r1
r2
1  r3
1
2
3
4
5
6
1  r0
r1
r2

1  r3
u 
X  1
 u21 
 u2 
 
 u22 
u 
 23 
P  1,3, 2
2
3
4
5
6

0  0 1 3 2  
0  0 1 2 3  
 u11 
 
 u12 
u 
 13 
1
u 
X  1
 u2 
 u11 
 
 u13   P  u1
u 
 12 
 u21 
 
 u23   P  u2
u 
 22 
 I  P  X
Pu 
  1 X
 P u2 
Kronecker product
[4]
Case 4: A  diag  a1, a2 
 a1Bu1 
 A  B X  

 a2 Bu2 
  r1 , 1  
 b11 b12



a1  Bu    r1 ,  2    a1  b21 b22
b
  r ,  
1
3
 31 b32


b13   u1,1 


b23   u1,2  i  1
b33   u1,3 
  r2 , 1  
 b11 b12



a2  Bu    r2 ,  2    a2  b21 b22
b
  r ,  
 31 b32
 2 3 
b13   u2,1 


b23   u2,2  i  2
b33   u2,3 
1
1
urr  ur  2 u   2u
r
r
1 1
diag  2 , 2 ,
 r1 r2
,
1
 D2, N
2 
rk 
Non-active variable  active variable
[1]
1
1
urr  ur  2 u   2u , on r 1,1 and  0, 2 
r
r
Nr  2k  1 is odd, and N  2m is even, then
Active variable is
 i 
ri  cos 
: 1 i  k
 Nr 
, that is ri  0, 0   j  2
 j  j : 1  j  N
Total number is kN , NOT  Nr 1  N
1
1
DN2 r u  DNr ur  2 DN u   2u
r
r
Note that differential matrix DNr is of dimension  Nr  1
,acts on
 u0, j 


u
1,
j






u
 N r 1, j 
u

 Nr , j 
y
2
Neglect due to B.C.
1
u  r  1  0 3
6
4
5
x
Non-active variable  active variable
[2]
 u1, j 


u
2,
j






u
 k, j 
However DN2 r
and DNr act on
 uk 1, j 


u
 k  2, j 




u
 N r 1, j 
NOT active variable, how
to deal with?
From previous discussion, we have following relationships which can solve this problem
 uk 1, j   uk ,m  j 
 u1,m  j 

 



u
u
u
k

2,
j
k

1,
m

j
2,
m

j


   k : 1:1 


 




 



u
u
u
 k ,m  j 
 Nr 1, j   1,m  j 
where
 k : 1:1
is a permutation matrix
and
 uk 1,m  j   uk , j 
 u1, j 

 



u
u
u
k

2,
m

j
k

1,
j
2,
j


   k : 1:1 


 




 



u
u
u
 k, j 
 Nr 1,m  j   1, j 
Non-active variable  active variable
uij
Recall
u  ri ,  j  for 1  i  Nr  1 and 1  j  N
Consider Chebyshev differential matrix Dr , Nr acts on u
D
r , Nr
[3]


and evaluate at

j

r0
r , Nr
i
u  ri , j   i  row of Dr , Nr  u  , j 
We write in matrix notation
D
 r , 

u  ri , j   di ,1 di ,2
r0
di , k
di ,k 1 di ,k  2
Question: How about if we arrange equations on
di , Nr 1
 r , 
i
j

 u1, j 


u
2,
j






u
 k, j 
r0
 uk 1, j 


u
 k  2, j 




u
 N r 1, j 
r0
 i when fixed j
Non-active variable  active variable
r0

  r1 , j  
d12

  d11
  r2 , j   
d 22

  d 21

 

 
r
,

d k ,2
  k j    d k ,1
DNr u 

  rk 1 , j    d k 1,1 d k 1,2

 d
  rk  2 , j    k  2,1 d k  2,2

 

 
 r ,   d Nr 1,1 d Nr 1,2
 Nr 1 j 
 E1 E2 
abbreviate DN r  

 E3 E4 



[4]
u  r , j 
r0
d1,k
d1,k 1
d1,k  2
d 2,k
d 2,k 1
d 2,k  2
d k ,k
d k ,k 1
d k ,k  2
d k 1,k
d k 1,k 1
d k 1,k  2
d Nr 1,k
d Nr 1,k 1
d1, Nr 1 

d 2, Nr 1  r  0


d k , N r 1 

d k 1, Nr 1 

r0


d Nr 1, Nr 1 
We only keep operations on active variable u  r  0, 
That is, only consider equation
Later on, we use the same symbol DNr   E1
 D u   r , 
Nr
E2 
i
j
for ri  0
 u1, j 


u
2,
j






u
 k, j 
 uk 1, j 


u
 k  2, j 




u
 N r 1, j 
 uk , m  j 


u
k

1,
m

j






u
 1,m  j 
Non-active variable  active variable
[5]
I
 k


1

Define permutation matrix P   1: k  ,  N r  1: 1: k  1   


k


1


 u1, j 
 u1, j 




u
u
2,
j


 2, j 








P u  r , j 
u
u
 k, j 
 k, j 
Let u  r ,  j  
 u1,m  j 
 uk 1, j 




u
u
 2,m  j 
 k  2, j 
Active variable with the same




indexing in r-direction




 uk ,m  j 
u


 N r 1, j 
 uk , m  j 


u
 k 1,m  j 




u
 1,m  j 


Non-active variable  active variable
[6]
Moreover, we modify differential matrix according to permutation P by

 
DNr u  r , j   DNr PT
  Pu  r, 
j
r0
r0
 d11 d12

 d 21 d22
T
DNr P  

 d k ,1 d k ,2

where
d1,k
d1, Nr 1
d1,k 2
d2,k
d2, Nr 1
d2,k 2
d k ,k
dk , Nr 1
d k ,k  2
d1,k 1 

d2,k 1 
  E1

dk ,k 1 

E2

such that for 1  j  m

 u1, j 
 u1,m  j 




u
u
2, j 
2, m  j 
Dr , Nr u  r ,  j   E1 
 E2 








u
u
 k, j 
 k ,m  j 

Evaluated at
and

 u1,m  j 
 u1, j 




u
u
2, m  j 
2, j 
Dr , Nr u  r ,  m  j   E1 
 E2 








u
u
 k ,m  j 
 k, j 

  r ,  ,  r ,  , ,  r ,  
1
j
2
j
k
j
Non-active variable  active variable
D
r , Nr
D
r , Nr
u
u
 u1,1 u1,2

u2,1 u2,2

,  r ,  m    E1


 uk ,1 uk ,2
   r ,  ,  r ,  ,
1
2
m 1
u1,m 
 u1,m 1 u1,m  2


u2, m 
u2, m 1 u2, m 2

 E2




uk ,m 
 uk ,m 1 uk ,m  2
 u1,m 1 u1,m  2

u2,m 1 u2,m  2

,  r ,  2 m    E1


 uk ,m 1 uk ,m  2
   r ,  ,  r ,  ,
m2
u1,2 m 
 u1,1 u1,2


u2,2 m 
u2,1 u2,2

 E2




uk ,2 m 
 uk ,1 uk ,2
 ui ,1 
 ui ,m 1 




u
u
define U   i ,2  , V   i , m  2  and active variable X   X
1
i

 i 





u
u
 i ,m 
 i ,2 m 
To sum up
D
r , Nr
u
   r ,  , ,  r ,    E  X
1
2m
[7]
1
1
u1,2 m 

u2,2 m 


uk ,2 m 
u1, m 

u2, m 


uk ,m 
 U1T 
 V1T 
 T
 T
U
V
X 2  , X1   2  , X 2   2 
 
 
 T 
 T 
U
 k 
 Vk 
| X 2   E2  X 2 | X1 
Non-active variable  active variable
[8]
Note that under row-major indexing, memory storage of X   X 1

is
mem  X   U1T | V1T | U 2T | V2T |
but
mem  X 2 | X1   V1T | U1T | V2T | U 2T |
| U Tj | V jT |

| U kT | VkT
| V jT | U Tj |

T
| VkT | U kT
D
r , Nr
u
   r ,  ,
1



E



2
Im 
 Im
V1T 

V2T 


T 
Vk 

T


If we adopt Kronecker products, then mem  X 2 | X 1    I k  
 Im


 Im
,  r ,  2 m     E1  


 U1T
 T
U
X2    2

 T
U k
I m  
   mem  X 
 
I m  
  mem  X 
 
 a11 B a12 B

a12 B a22 B
Definition: Kronecker product is defined by A  B  


 am1 B am 2 B
a1n B 

a2 n B 


amn B 
Non-active variable  active variable
[9]
The same reason holds for second derivative operator
2
r , Nr
D
 D1

 D3
then
D2 

D4 

neglect equation on
r0
Dr2, Nr u
Dr2, Nr   D1 D2 


  r ,  , ,  r,     D  
1
2m

1

1
1

T
D
u
r
,



i

row
of
matrix
D
P


r
,
N
i
j
r
,
N
 r

r
r
ri




 Im
1 

R
E

we have
 1 
r r


where



E



2
Im 
 Im
1/ r1

1/ r2

R






D



2
Im 
 Im
Im


Dr2, Nr PT  D1
D2

I m  
  mem  X 
 
collect equations on each ri :1  i  k
I m  

 


  diag  1 , 1 ,


 r1 r2

1/ rk 
1
, 
rk 
Non-active variable  active variable
[10]
We write second derivative operator on   direction as

D2,N  S N 2  xk  x j 

for 1  i  k, 1  j  2m

 1 2
j  0  mod N 
  2
 6 3h
where S N 2  x j   
j

1



, j  0  mod N 
 2sin 2  jh / 2 

 ri  0



D2, N u  ri , j   j  row of  D2, N



so



D2, N u  ri ,    D2, N
 ui ,1 


u
i
,2






u
 i ,2 m 
 ui ,1 


u
i
,2

  D2
 , N




u
 i ,2 m 



  j  row of



 2
 D , N

Ui 
 
 Vi 



1 2
Ui 
1 2

D
u
r
,


D , N


  implies  r 2  , N  i
2
r


i
 Vi 
Ui 
 
 Vi 
Non-active variable  active variable
 1 2  U1    1 2
 2 D , N     2 D , N
 V1    r1
 r1
  r1 ,    


  1 2 U 2   
D , N   
1 2
   r2 ,   
2
D
u

r

 V2    
 2  , N  
2

r


 

 
 
  rk ,    
1 2 U k   
 2 D , N    
r

 Vk   
 k
1 2
D , N
r22
1 2
D , N
rk2
If we adopt Kronecker products, then
  r1 ,  


1 2
   r2 ,  
 R 2  D2, N mem  X 
 2 D , N u  

r



  rk ,  

[11]

   U1  
   V1  


  U 2  
 
   V2  




  U k  
    
   Vk  
Non-active variable  active variable
[12]
summary



D



2
Im 
 Im
D
2
1 
1

  Dr , Nr u    r ,1  ,
r r
r

3
  r1 ,  


r
,



1 2
 2
  R 2  D 2 mem  X 
 , N
 2 D , N u  

r



r
,



 k

u
   r ,  ,

 Im
,  r ,  2 m     D1  


1
2
r , Nr
1

Note that
  r ,   r , 
1
2

 Im
,  r ,  2 m    R  E1  


I m  
  mem  X 
 



E



2
Im 
 Im
I m  
  mem  X 
 

  r1 ,     r1 ,1 

 
r
,



 r2 ,1 
2




 r , 2 m    
 

 
r
,

  k     rk ,1 
 r1 ,2 
 r2 ,2 
 rk ,2 
so that all three system of equations are of the same order.
 r1 ,2 m  

 r2 ,2 m  


r
,

 k 2m  
Non-active variable  active variable
[13]
1
1
urr  ur  2 u   2u , on r 1,1 and  0, 2 
r
r
Discretization on mem  X 
L  mem  X    2mem  X 
then
I

L   D1   m






D



2
Im 
 Im
 R 2  D2, N
I
  D1  RE1    m

where

I m  
 Im


R
E

 1 


 




D

RE

  2

2
Im 
 Im
1/ r1

1/ r2
R






E



2
Im 
 Im
Im 
2
2
  R  D , N




  diag  1 , 1 ,


 r1 r2

1/ rk 
,

1

rk 
I m  

 
Example: program 28
u   2u
with boundary condition u  r  1,   0
Nr  25 is odd and N  20
is even, let eigen-pair be
 Im
sparse structure of L   D1  RE1   

k ,Vk 



D

RE





2
2
Im 
 Im
Im 
2
2
  R  D , N



Nr 1
N  12  20  240
Dimension :
2
Example: program 28 (mash plot of eigenvector)
1 Eigenvalue is sorted, monotone increasing and normalized to first eigenvalue
1  2 
 n 
Vk
2 Eigenvector is normalized by supremum norm, Vk  V
k 
1  5.7832
2  3  14.682
4  5  26.3746
6  30.4713
7  8  40.7065
9  10  42.2185
Example: program 28 (nodal set)
Exercise 1
u   2u
on annulus
1  r  2
Nr  25 is odd and N  20
with boundary condition u  r  1, 2  0
is even, let eigen-pair be
r  1
Chebyshev node on x1,1

k ,Vk 
x 1
2
Chebyshev node on


L  Dr2, Nr  RDr , Nr  I N  R 2  D2, N

1  r  2
Exercise 1 (mash plot of eigenvector)
1 Eigenvalue is sorted, monotone increasing and normalized to first eigenvalue
1  2 
 n 
Vk
V

2 Eigenvector is normalized by supremum norm, k
Vk 