Fastest Fourier Transform in the West Documentation 2.0

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


Discrete CR Solution

101

102

103

101

100


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8/51


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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



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



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



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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



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, 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



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, H1

uniform, P1, L2


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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



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, 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


31/51

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

Fastest Fourier Transform in the West Documentation 2.0

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