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7/21/2019 Fastest Fourier Transform in the West Documentation 2.0
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7/21/2019 Fastest Fourier Transform in the West Documentation 2.0
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P1 P1 P1 P1 P1
T0
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2
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u
div( u) + u + u= f ,u= uD ,
u
n
=g
\.
= I = 0 = 0 f1 uD0 g0 \ =
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1
0.5
0
0.5
1
1
0.5
0
0.5
10
0.05
0.1
0.15
0.2
Nr of degrees of freedom: 139
Discrete P1 Solution
101
102
103
101
100
1
0.5
0
0.5
1
1
0.5
0
0.5
10.05
0
0.05
0.1
0.15
0.2
Nr of degrees of freedom: 159
Discrete CR Solution
101
102
103
101
100
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7/21/2019 Fastest Fourier Transform in the West Documentation 2.0
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7/21/2019 Fastest Fourier Transform in the West Documentation 2.0
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L2
P1
P1
u= 1 u= 0
O(h) N1/2
00.2
0.40.6
0.8 1
0
0.5
10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Nr of degrees of freedom: 697
Discrete P1 Solution
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Nr of Nodes 763
Grid on Level 7
101
102
103
104
105
102
101
100
bulk, P1, estimator
maximum, P1, estimator
uniform, P1, estimator
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P1
u= 1 u= 0
H5/3
()
H1 O(h2/3) N1/3
10.5
00.5
1 10.5
00.5
10
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Discrete P1 Solution
Nr of degrees of freedom: 564
1 0.5 0 0.5 11
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Nr of Nodes 637
Grid on Level 5
100
101
102
103
104
105
102
101
100
101
bulk, P1, estimator
maximum, P1, estimator
uniform, P1, estimator
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P1
u= 2(x(1 x) + y(1 y)) u= 0
u = x(1x)y(1y)
O(h) O(h2) L2
00.2
0.40.6
0.8 1
0
0.5
10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Nr of degrees of freedom: 919
Discrete P1 Solution
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Nr of Nodes 999
Grid on Level 7
101
102
103
104
105
106
105
104
103
102
101
100
bulk, P1, estimator
bulk, P1, H1
bulk, P1, L2
uniform, P1, estimator
uniform, P1, H1
uniform, P1, L2
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P1
u= 0 u= 0 D ={(x= 0 y[0, 1]) (y= 0 x[0, 1])}
u = g(x)
N =\D
u= r2/3 sin(2/3)
O(h2/3) O(h4/3) L2
10.5
00.5
1 10.5
00.5
10
0.2
0.4
0.6
0.8
1
1.2
1.4
Discrete P1 Solution
Nr of degrees of freedom: 806
1 0.5 0 0.5 11
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Nr of Nodes 843
Grid on Level 7
101
102
103
104
105
105
104
103
10
2
101
100
bulk, P1, estimator
bulk, P1, H1
bulk, P1, L2
uniform, P1, estimator
uniform, P1, H1
uniform, P1, L2
7/21/2019 Fastest Fourier Transform in the West Documentation 2.0
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P1
u= f(x, y) u= 0
u= xy(1 x)(1 y) (k((x 5/4)2 + (y+ 1/4)2 1)) O(h) L2 O(h2) k
00.2
0.40.6
0.81
0
0.5
10.06
0.04
0.02
0
0.02
0.04
0.06
0.08
Nr of degrees of freedom: 767
Discrete P1 Solution
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Nr of Nodes 815
Grid on Level 7
101
102
103
104
105
105
104
103
102
101
100
bulk, P1, estimator
bulk, P1, H1
bulk, P1, L2
uniform, P1, estimator
uniform, P1, H1
uniform, P1, L2
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= (x, y) x x y y N
f
N
g N
uD N
(22N) (:, :, i) i (xi, yi) = z(:, :, i)
(2
N) (:, i)
i (xi, yi) = z(:, i)
N
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u
f
uD g
div( u) + u + u= f ,u= uD ,
u
n=g \ ,
u
f
u
f f=div u
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fdummy
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Ax= b
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x
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div(
u) +
u + u= f ,
u= uD D,
u = g N,
uD H1(;R) f L2(;R) g L2(N;R) L(;R22) L(,R2) L(,R) uV
(u ( v) + u v+ u v)dx =
f vdx +
N
g v d sx.
vV
Vh V Vh V vh Vh P1
P2 P3
CR
Vh V
2m m Lu = G L =AB A B p:=Bu Ap= G Bu= p (, u) V
div + (1 ) + u= f ,= u , u= uD D,
(1 ) = g N
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div v+ (1 ) v+ u v dx =
f v dx,
(1 ) + div u dx = D uD( ) dsx, vV
RT0P0 hRT0(T)H(div; ) a x+b aR2 b R uhP0(T)L2()
RT0P0
div C (u) = f ,u= uD D,
(C (u)) = g N,
uD H1(;R2) f L2(;R2) g L2(N;R2) (v) = 12( v + ( v)T) C
C
C := 2 + 0
2 + 00 0 C1 :=
+24(+)
4(+) 0
4(+)
+24(+) 0
0 0 1 .
u
uh
uV
(u) : C(v) dx=
f v dx +
N
g v dsx vV.
P1 P1 uH1()2 P1
P1CR u P1 CR
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(, u) V
div = f = C (u) , u= uD D = g N.
: C1 dx +
u div dx=D
uD ( ) dsx ,
v div dx=
f v dx vV.
h
f
uD g
u
f
uD g
uh u
E
u= 1 uD = 0 g= 0
u(r, ) = r2/3 sin(2/3) f
u= 1 uD = 0 g= 0
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u(x, y) = x(1 x) y(1 y)
f
u(x, y) = sin(x3) cos(y) + x8 y9 + x6y10 =
5 sin(x+y)6 cos(x+y)
= 7
f uD g
u(r, ) = r1/4 sin(1/4) f0
uD = u|D
u(x, y) = xy(1 x)(1 y) arctan
k(
(x 5/4)2 + (y+ 1/4)2 1)
k
u
k
u = f f u
f, uD g uh
u
(x, y) ,,
f
uD g
f= 0 g(x, y) = (0, 1000) (x, y)N x= 48 g = 0 N E= 2900
= 0.3
u(x, y) = 105 cos((x+1)(y+1)2)sin((x+1)) cos(y+1)
E= 105 = 0.3
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(r, ) < u ur, u ur(r, ) =
r
2(( + 1)cos(( + 1)) + (C2 ( + 1))C1cos(( 1))),
u(r, ) = r
2(( + 1) sin(( + 1)) + (C2+ 1)C1sin(( 1))).
C1 =cos((+ 1))/ cos(( 1)) C2 = 2(+ 2)/(+) = 0.54448... sin2 +sin2 = 0 = 3/4 E= 105 = 0.3 f0
f, uD g
,,
uh
u
(x, y) f uD g
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= conv{(0, 0), (1, 0), (0, 1)}
= [0, 1]2
N= conv{(1, 0), (1, 1)}
= [1, 1]2 \ {(0, 1] (0, 1]}
D ={conv{(0, 0), (1, 0)} conv{(0, 0), (0, 1)}}
=conv{(0, 0), (48, 44), (48, 60), (0, 44)} D conv
{(0, 0), (0, 44)
}
\D
[1, 1]2 {0} [0, 1]
(0, 1)
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P1 PNC1
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(j, k) j k
(j, k) j k
3 2
32
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3 2 P1
3 2 CR
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(j, k) j k
j k
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k i j j k k k k
j k (j, k) (k, j)
j j
j
j
j
j j
j j
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j
j
j j
j 3 2
j
j 3 2 j
j
j
j j
j j
j j 2
j 3 2 P1 j
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3 2
j 3 2 PNC1 j 3 2
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k
k
P1
A
A
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P1
P1
PNC1 H1(T) H1()H1(T)
P2
P3
RT0(T)H( ; )
P1 P1
P1 PNC1
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b
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m m
m
m
uh
m
uh
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m m
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= = = .
P1
T0 T0 Rn
T0
T T0 E(T)
T T0
E(T)
KT0 E(T) = TK=E(K) T0 E(T) : T T0 H1 L2
E
T osc
M
M =E
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M E E M
E> maxKE
K,
[0, 1] = 0.5
T M T > max
KTK.
M EM
2E KE
2K,
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T K TK
2T KT
2K,
M
T = Tref N N >0
T T0 = conv{(0, 0), (N, 0), (0, N)}
T T \{T0} xT diam(T)c 1N|x|1 + >2
||(u ITu)||L2()cNmin{1,} . u
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1/3
M E M EM, ET=E(T)M.
T E E(T) E(T) : T T E(T) E(T) T Tm T T T k = 0, 1, 2 k 1
T+1
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new1
new3 new2
new1
new1
new2
new1
new2
new1 new2 new3
new1
new3 new2new4
new1
new2 new3 new4
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T1 = conv(2, 3,new1) T1 = conv(3, 1,new1) 2, 3 3, 1
T T0
T T T E(T) T
K T0 T|K :={T T | T K} : K Tref Tref = {(0, 0), (0, 1), (1, 0)} (E(K)) = {(0, 0), (1, 0)} det D > 0 TK :={(T) :T T, T K} K
L2
V H1
V :=P1(T)
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P1(TRm) :={vC(T;Rm) : v T} ,P1(TRm) :={vL(;Rm) :T T, v|T P1(T;Rm)} .
vH1
0 ()
L2
v
V
vL2()C1vL2() h1T vL2()C2h1T vL2().
C1 C2 T0 L2
TT
h1T (v v)2L2(T)+EE
h1/2E (v v)2L2(E)C3v2L2().
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u
u
1
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= f dx + N g dx
f : Rk Rm,n
L2
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{0, 1}
I(f) =
10
(x)f(x)dx
I(f) := ni=1
if(xi),
i xi
p= 2n 1 (x) = 1 [1, 1]
xi i n N xi, i= 1, . . . , n pn
Jn:=
1 22 2 .
. . .. . n
n n
. i, i
i+1 := (xpi, pi)/(pi, pi) i0
2i+1 :=
1 i= 0(pi, pi)/(pi1, pi1) i1 .
i+1 = 0 i0 i+1 = i4i2 1 i1 .
xi
wk = (v(k)1 )
2, k = 1, . . . , n v(k) = (v(k)1 , . . . , v
(k)n )
xk Jk |v(k)|=ba
(x)dx
b
a (x) dx = 1
11 dx = 2
[1, 1] [0, 1]
10
1dx = 1
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(0, 0), (0, 1) (1, 1)
x1 = y1
x2 = y2(1
y1) = y2(1
x1).
0 x1 1 0 x1 1 x2 0yi 1, i = 1, 2
10
f(r)drni=1
aif(ri), 10
(1 s)f(s)dsnj=1
bjf(sj).
(x) = (1x)
i = 14i2 1 i+1 =
i(i + 1)
2(i + 1) 1 .
[0, 1]
n2
(sj, ri(1 sj)) aibj i= 1, . . . , n j = 1, . . . , n .
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H1
L2