Intr

oduc

tion

toA

Pos

teri

oriE

rror

Est

imat

ion

base

don

Dua

lity

Mat

sG

Lar

son

Com

puta

tion

alM

athe

mat

ics,

Cha

lmer

s

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

1

Out

line

Ove

rvie

wof

tool

san

dap

plic

atio

nsof

com

puta

tiona

lm

athe

mat

ical

mod

elin

g.•

Rev

iew

ofFE

M•

Apo

ster

iori

Err

orE

stim

atio

nba

sed

ondu

ality

•E

xam

ples

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

2

Con

trib

utor

sPr

esen

tatio

nba

sed

onw

ork

by:

•R

icka

rdB

ergs

tröm

•D

onE

step

•G

eorg

ios

Fouf

as

•Jo

han

Hof

fman

•M

ike

Hol

st

•C

laes

John

son

•A

xelM

ålqv

ist

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

3

The

fini

teel

emen

tm

etho

dT

heFi

nite

Ele

men

tMet

hod

isa

gene

ralm

etho

dfo

rco

mpu

ter

solu

tion

ofdi

ffer

entia

lequ

atio

nsba

sed

on:

•W

eak

form

ulat

ion

ofth

edi

ffer

entia

lequ

atio

n.•

Piec

ewis

epo

lyno

mia

lapp

roxi

mat

ion

ofth

eso

lutio

n.

App

licat

ions

incl

ude:

solid

s,fl

uids

,ele

ctro

mag

netic

s,

heat

cond

uctio

n,et

c.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

4

Poi

sson

equa

tion

:st

rong

form

Find

usu

chth

at −∇·A

∇u=

fin

Ω,

n·A

∇u=

g Non

ΓN

,

u=

g Don

ΓD,

with

Ωa

dom

ain

inR

dw

ithbo

unda

ryΓ

=Γ

D∪

ΓN

.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

5

Poi

sson

equa

tion

:w

eak

form

Find

u∈

Vg D

such

that

a(u

,v)

=l(

v)

for

allv

∈V

0,

with

Vg

=v

∈H

1:v

=g

onΓ

Da

nd

a(u

,v)

=

∫ ΩA∇u

·∇vdx,

l(v)

=

∫ ΓN

g Nvds.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

6

Poi

sson

equa

tion

:F

EM

FE

M:

Find

uh∈

Vh

such

that

a(u

h,v

)=

l(v)

for

allv

∈V

h,

whe

reV

h⊂

Vis

afi

nite

elem

ent

spac

eof

piec

ewis

e

poly

nom

ials

de£n

edon

atr

iang

ulat

ionK

ofΩ

.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

7

Abs

trac

tse

ttin

gM

odel

prob

lem

:Fi

ndu∈

Vsu

chth

at

a(u

,v)

=l(

v)

for

allv

∈V,

whe

rea(·,

·)is

anel

liptic

bilin

ear

form

and

l(·)

isa

linea

rfu

nctio

nal.

FE

M:

Find

uh∈

Vh

such

that

a(u

h,v

)=

l(v)

for

allv

∈V

h,

whe

reV

h⊂

Vis

a£n

iteel

emen

tsp

ace

ofpi

ecew

ise

poly

nom

ials

de£n

edon

atr

iang

ulat

ionK.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

8

Wha

tdo

we

wan

tto

com

pute

?T

hego

alof

aco

mpu

tatio

nco

uld

be:

•a

glob

ally

accu

rate

solu

tion.

•va

lues

ofa

num

ber

ofqu

antit

ies

ofpa

rtic

ular

inte

rest

,for

inst

ance

the

lifta

nddr

agin

a¤o

wco

mpu

tatio

n.

Cor

resp

onds

toes

timat

ing

the

erro

rin

a:•

Glo

baln

orm

•Se

tof

func

tiona

lsm

(·)of

the

solu

tion.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

9

Goa

lori

ente

der

ror

esti

mat

esO

bjec

tive

:L

etm

(·)be

alin

ear

func

tiona

lon

V.

We

seek

toes

timat

eth

eer

ror

m(u

)−

m(u

h)

inth

efu

nctio

nali

nte

rms

ofth

eco

mpu

ted

solu

tion.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

10

Exa

mpl

esof

func

tion

als

•A

vera

geof

erro

rin

subd

omai

n

m(e

)=

∫ ω

eψdx

•A

vera

geof

erro

rin

deri

vativ

ein

subd

omai

n

m(e

)=

−∫ ω

e∂xψ

dx

•A

vera

geof

inte

grat

ed¤u

x

m(e

)=

−∫ γ

n·∇

eψdx

•U

selin

eari

zatio

nfo

rno

nlin

ear

func

tiona

ls.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

11

Dat

ato

dual

prob

lem

−1

−0.

8−

0.6

−0.

4−

0.2

00.

20.

40.

60.

81

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

(a)

Poin

tval

ue

−1

−0.

8−

0.6

−0.

4−

0.2

00.

20.

40.

60.

81

−1

−0.

8

−0.

6

−0.

4

−0.

20

0.2

0.4

0.6

0.81 (b)

Der

ivat

ive

poin

tval

ue

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

12

Dua

lpro

blem

Tore

pres

entt

heer

ror

inth

egi

ven

linea

rfu

nctio

nal

m(·)

we

intr

oduc

eth

edu

alpr

oble

m:

Find

φ∈

Vsu

chth

atm

(v)

=a(v

,φ)

allv

∈V

.

Rem

ark:

Not

eth

atth

edu

alpr

oble

mis

cont

inuo

us.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

13

Err

orre

pres

enta

tion

form

ula

Setti

ngv

=e

=u−

uh

inth

edu

alpr

oble

mw

ege

t

m(u

)−

m(u

h)

=m

(e)

=a(e

,φ)

=a(e

,φ−

πφ)

=l(

φ−

πφ)−

a(u

h,φ

−πφ)

=∑ K∈K

(rK

(uh),

φ−

πφ)

Her

ew

eus

edth

elin

eari

tyof

a(·,

·)an

dm

(·)an

dor

-

thog

onal

ityto

subt

ract

anin

terp

olan

tπφ

∈V

hof

φ.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

14

Res

idua

lbas

eder

ror

esti

mat

es

|m(u

)−

m(u

h)|

=

∣ ∣ ∣ ∣ ∣∑ K∈K

(rK

(uh),

φ−

πφ)∣ ∣ ∣ ∣ ∣E

st.

1

≤∑ K∈K

∣ ∣ ∣(r K(u

h),

φ−

πφ)∣ ∣ ∣E

st.

2

≤∑ K∈K

RK

(uh)·W

K(φ

)E

st.

3

Not

e:T

hefi

rst

ineq

ualit

ypr

even

tsca

ncel

atio

nbe

twee

nel

e-

men

ts.

The

seco

ndin

equa

lity

prev

ents

canc

elat

ion

betw

een

dif-

fere

ntpa

rts

ofth

ere

sidu

alon

anel

emen

tlev

el.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

15

Poi

sson

:re

sidu

alL

etv

=φ−

πφ

we

have

(f,v

)−

(∇u

h,∇

v)

=∑ K∈K

(f+

∆u

h,v

) K+

([n·∇

uh],

v) ∂

K/2

=∑ K∈K

(rK

(uh),

v).

With

the

wea

kre

sidu

al

(rK

(uh),

v)

=(f

+∆

uh,v

) K+

([n·∇

uh],

v) ∂

K/2

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

16

Poi

sson

:re

sidu

alan

dw

eigh

tT

hefo

llow

ing

estim

ate

hold

s

|(rK

(uh),

φ−

πφ)|≤

RK

(uh)·W

K(φ

)

with

RK

(uh)

=

[‖∆

uh

+f‖ K

h−1

/2K

‖[n·∇

uh]‖ ∂

K/2

]

WK

(φ)

=

[‖φ

−πφ‖ K

h1/

2K

‖φ−

πφ‖ ∂

K

]

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

17

Poi

sson

:re

sidu

alan

dw

eigh

tL

etv

=φ−

πφ

.T

hen

we

have

(rK

(uh),

v)

=(f

+∆

uh,v

) K+

([n·∇

uh],

v) ∂

K/2

≤‖f

+∆

uh‖ K

‖v‖ K

+h

1/2

K‖[n

·∇u

h]‖ ∂

Kh−1

/2K

‖v‖ ∂

K

=R

K(u

h)·W

K(v

).

Rem

ark:

Not

eth

atth

esc

alin

gw

ithh

Kin

the

boun

d-

ary

cont

ribu

tion

isne

cess

ary

for

the

two

cont

ribu

tions

toth

ere

sidu

alan

dw

eigh

tto

scal

ein

the

sam

ew

ay.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

18

The

resi

dual

and

wei

ght

Bas

edon

appr

oxim

atio

nth

eory

we

expe

ctth

at

|RK

(uh)|∼

Ch

α−2

K|u|

α,K

|WK

(φ)|∼

Ch

β K|φ|

β,K

with

1≤

α,β

≤p

+1,

whe

reis

the

orde

rof

poly

nom

ials

used

inth

eap

prox

imat

ion.

•T

here

are

pow

ers

ofh

inbo

thth

ere

sidu

alan

dw

eigh

t!

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

19

Stab

ility

fact

orfo

rfu

ncti

onal

Con

tinui

ngth

ees

timat

esw

ege

t:

|m(u

)−

m(u

h)|≤

∑ K∈K

RK

(uh)·W

K(φ

)E

st.

3

≤( ∑ K

∈Kh

2β

KR

2 K(u

h)) 1/

2( ∑ K

∈Kh−2

βK

W2 K(φ

)) 1/2

≤S m

( ∑ K∈K

h2β

K

K|R

K(u

h)|2

) 1/2

Est

.4

whe

re

S2 m=

∑ K∈K

h−2

βK

W2 K(φ

)

isth

est

abili

tyfa

ctor

ofth

eco

mpu

tatio

nof

m(·)

.In

trod

uctio

nto

APo

ster

iori

Err

orE

stim

atio

nba

sed

onD

ualit

y–

p.20

Com

puti

ngth

eer

ror

esti

mat

e•

RK

(uh)

dire

ctly

com

puta

ble.

•W

K(φ

)ca

nbe

appr

oxim

ated

byso

lvin

gth

edu

alpr

oble

mnu

mer

ical

ly.

Not

eth

atde

riva

tive

info

rmat

ion

isne

cess

ary

here

.

Impo

rtan

tre

sear

chto

pic:

Con

stru

ctef

£cie

ntad

ap-

tive

met

hods

usin

gon

lyro

ugh

know

ledg

eof

φ.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

21

Com

puti

ngth

eer

ror

esti

mat

eW

eco

nsid

erth

efo

llow

ing

tech

niqu

es:

•So

lve

the

dual

with

high

eror

der

elem

ents

oron

afi

ner

mes

h.•

Solv

eth

edu

alon

the

sam

eor

aco

arse

rm

esh

and

use

post

proc

essi

ng.

•U

sein

terp

olat

ion

erro

res

timat

esth

eory

toes

timat

eth

ew

eigh

t.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

22

How

can

we

use

the

erro

res

ti-

mat

es?

The

rear

etw

odi

ffer

ent

uses

for

erro

res

timat

es:

•To

estim

ate

the

erro

r(e

xpen

sive

)•

Toco

nstr

ucta

nad

aptiv

em

etho

d(n

otex

pens

ive)

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

23

Est

imat

esba

sed

onlo

cal

prob

-le

ms

The

reis

anal

tern

ativ

est

rate

gyfo

rco

mpu

ting

the

apo

ster

iori

erro

res

timat

ew

hich

isba

sed

on:

•St

artf

rom

the

sam

eer

ror

repr

esen

tatio

nfo

rmul

a.•

Use

the

para

llelo

gram

law

tow

rite

the

erro

rre

pres

enta

tion

form

ula

asa

diff

eren

ceof

two

ener

gylik

epr

oduc

ts.

•E

mpl

oyup

per

and

low

eren

ergy

norm

erro

res

timat

esba

sed

onlo

calp

robl

ems

tobo

und

the

diff

eren

ce.

•R

esul

tsin

uppe

ran

dlo

wer

boun

dson

the

func

tiona

l.•

Dev

elop

edby

:B

abus

ka,O

den,

Pere

ira,

and

othe

rs.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

24

Est

imat

esba

sed

onlo

cal

prob

-le

ms

Setti

ngv

=e

=u−

uh

inth

edu

alpr

oble

mw

ege

t

m(u

)−

m(u

h)

=m

(e)

=a(e

,φ)

=a(e

,φ−

φh)

=a(e

,ε)

Her

ew

eus

edth

elin

eari

tyof

a(·,

·)an

dm

(·)an

dor

thog

onal

ityto

subt

ract

the

fini

teel

emen

tap

prox

imat

ion

φh∈

Vh.

We

thus

conc

lude

:

m(u

)−

m(u

h)

=a(e

,ε)

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

25

Est

imat

esba

sed

onlo

cal

prob

-le

ms

Star

ting

from

the

erro

rre

pres

enta

tion

form

ula

(e,ψ

)=

a(e

,ε)

whe

ree

=u−

uh,

ε=

φ−

φh

and

usin

gth

eid

entit

y

4ab

=(a

+b)

(a+

b)−

(a−

b)(a

−b)

we

obta

inth

eal

tern

ativ

eer

ror

repr

esen

tatio

nfo

rmul

a

4(e,

ψ)

=a(e

+ε,

e+

ε)−

a(e

−ε,

e−

ε)

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

26

Est

imat

esba

sed

onlo

cal

prob

-le

ms

Rec

allt

hatt

here

are

uppe

ran

dlo

wer

estim

ator

s

ρ2 ±

=∑ K

ρ2 ±,

K,

η2 ±

=∑ K

η2 ±,

K

ρ±

and

η ±su

chth

at

η2 ±≤

a(e

±ε,

e±

ε)≤

ρ2 ±

For

the

Pois

son

equa

tion

thes

ees

timat

ors

corr

espo

ndto

stan

dard

ener

gyno

rmes

timat

ors

for

the

prob

lem

s

−∆v ±

=f±

ψin

Ω,

u=

0on

∂Ω

,

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

27

Est

imat

esba

sed

onlo

cal

prob

-le

ms

Com

bing

the

estm

ator

san

dth

eer

ror

repr

esen

tatio

nfo

rmul

a 4(e,

ψ)

=a(e

+ε,

e+

ε)−

a(e

−ε,

e−

ε)

We

obta

inth

eup

per

and

low

erbo

unds

η2 +−

ρ2 −≤

4(e,

ψ)≤

ρ2 +−

η2 −

Rem

arks

:•

Upp

eran

dlo

wer

boun

dsw

ithno

cons

tant

sin

the

estim

ate.

•M

ore

com

plic

ated

and

less

gene

ralt

echn

ique

.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

28

Ext

ensi

onto

nonl

inea

rpr

ob-

lem

sA

ssum

eth

atth

efo

rmsm

(·)an

da(·,

·)ar

eno

nlin

ear

then

we

may

intr

oduc

elin

eari

zed

form

sw

ithth

epr

oper

ties m

(u)−

m(u

h)

=m

(u,u

h)(

u−

uh)

(1)

a(u

,v)−

a(u

h,v

)=

a(u

,uh;u

−u

h,v

)(2

)

We

then

intr

oduc

eth

edu

alpr

oble

m:

Find

φ∈

Vsu

chth

atm

(v)

=a(v

,φ)

allv

∈V

.

Rem

ark:

Not

eth

atth

edu

alpr

oble

mis

cont

inuo

us.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

29

Err

orre

pres

enta

tion

form

ula

Setti

ngv

=e

=u−

uh

inth

edu

alpr

oble

mw

ege

t

m(u

)−

m(u

h)

=m

(e)

=a(e

,φ)

=a(u

,φ)−

a(u

h,φ

)

=l(

φ)−

a(u

h,φ

)

=l(

φ−

πφ)−

a(u

h,φ

−πφ)

=∑ K∈K

(rK

(uh),

φ−

πφ)

Her

ew

eus

edth

elin

eari

tyof

a(·,

·)an

dm

(·)an

dor

-

thog

onal

ityto

subt

ract

anin

terp

olan

tπφ

∈V

hof

φ.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

30

Ada

ptiv

em

etho

ds•

Com

pute

aso

lutio

non

aco

arse

grid

•E

valu

ate

elem

enti

ndic

ator

(com

pute

RK

(uh)·W

K(φ

))•

Ref

ine

mes

h,h

orp,

base

don

elem

ente

rror

indi

cato

r•

Com

pute

anim

prov

edso

lutio

n•

Rep

eatu

ntil

satis

fact

ory

accu

racy

isac

hiev

ed(t

ime,

mem

ory,

...)

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

31

Ref

inem

ent

crit

eria

:h

-met

hod

Ref

ine

•30

%of

the

elem

ents

with

larg

esti

ndic

ator

.•

refi

neal

lele

men

tsK

with

RK

(uh)·W

K(φ

)≥

κm

ax KR

K(u

h)·W

K(φ

)

with

0≤

κ≤

1.

Onl

yco

arse

info

rmat

ion

onel

emen

tind

icat

orne

cess

ary

toge

trou

ghly

the

sam

em

esh.

Rem

ark:

h−

pm

etho

dsm

uch

mor

ede

licat

e!

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

32

Exa

mpl

es•

Pois

son’

seq

uatio

n•

Hel

mho

ltzeq

uatio

n•

Nav

ier-

Stok

es•

Ela

stic

ity

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

33

Poi

sson

’seq

uati

onW

eco

nsid

er −∆u

=f

inΩ

,u

=0

on∂Ω

with

fch

osen

soth

at

u=

sin

πx

sin

πye−

(r1/R

1)2

with

r 1=

√ (x−

0.5)

2+

(y−

0.5)

2 ,R

1=

0.1

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

34

Poi

sson

’seq

uati

on

00.

20.

40.

60.

81

0

0.2

0.4

0.6

0.8

10

0.2

0.4

0.6

0.81

00.

10.

20.

30.

40.

50.

60.

70.

80.

91

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

Figu

re1:

The

exac

tsol

utio

nfo

rth

e2D

prob

lem

and

two

poin

ts

whe

reth

eer

ror

coul

dbe

mea

sure

d

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

35

Poi

sson

’seq

uati

on

102

103

104

10-4

10-3

10-2

10-1

|ei|

N

|e| |e 1|

|e 2|

|e 3|

(a)

The

erro

r.

102

103

104

100

101

102

N

|ei/e|

|e 1/e|

|e 2/e|

|e 3/e|

(b)

The

erro

rin

dica

tors

.

Figu

re2:

The

erro

ris

mea

sure

din

(0.5

,0.4

),lin

ears

,ad

aptiv

e

algo

rith

mba

sed

ones

timat

e2.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

36

The

Hel

mho

ltz

equa

tion

find

usu

chth

at

−∆u−

k2 u

=0,

toge

ther

with

suita

ble

boun

dary

cond

ition

s.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

37

Hel

mho

ltz:

Pla

new

ave

solu

tion

00.

10.

20.

30.

40.

50.

60.

70.

80.

91

00.10.20.30.40.50.60.70.80.91

−202 Fi

gure

3:k

=16

.56,

θ=

π/4

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

38

Hel

mho

ltz:

The

dual

prob

lem

Toco

ntro

lthe

erro

rin

apo

intx

0w

eco

nsid

er

−φ−

k2 φ

=δ x

0.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

39

Hel

mho

ltz:

Dua

lsol

utio

n

00.

10.

20.

30.

40.

50.

60.

70.

80.

91

0

0.20.

40.60.

81

−0.

2

−0.

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Figu

re4:

k=

16.5

6,z

=[0

.5,0

.5]

Her

eψ

=δ x

0to

cont

rolt

heer

ror

inth

epo

intx

0.In

trod

uctio

nto

APo

ster

iori

Err

orE

stim

atio

nba

sed

onD

ualit

y–

p.40

Nav

ier-

Stok

esIn

com

pres

sibl

eN

avie

r-St

okes

equa

tions

(NS)

u+

u·∇

u=

ν∆

u−

∇p(N

ewto

n’s

2:nd

law

)

∇·u

=0

(inc

ompr

essi

bilit

y)

Com

puta

tiona

lmet

hods

use

NS

toco

nstr

uctr

ules

for

how

¤uid

part

icle

sin

asi

mul

atio

nm

ove

with

resp

.to

each

othe

r

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

41

NS:

dual

prob

lem

The

sens

itivi

tyof

diff

eren

terr

orm

easu

res(e

,ψ)

isco

ntai

ned

in

the

dual

solu

tion

ϕth

roug

hth

eda

taψ

.

The

linea

rize

ddu

alN

avie

r-St

okes

equa

tions

−ϕ−

(u·∇

)ϕ+∇U

·ϕ+∇θ

−ν∆

ϕ=

ψ

∇·ϕ

=0

Info

rmat

ion

istr

ansp

orte

dba

ckw

ards

insp

ace

and

time

byth

e

exac

tsol

utio

nu

thro

ugh

the

term

−ϕ−

(u·∇

)ϕ

We

have

grow

thdu

eto

the

reac

tion

term

∇U·ϕ

(whi

chin

lam

inar

¤ow

isdo

min

ated

bydi

ffus

ion

effe

cts)

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

42

NS:

Exa

mpl

eE

stim

ate

the

erro

rin

spac

e-tim

eav

erag

esov

era

spat

ialc

ube

ω

with

side

d(ω

),co

rres

pond

sto

afo

rce

ψac

ting

onω

over

atim

e

inte

rval

I=

[T−

d(ω

),T

]:

ψ=

χω×I

|χω×I|⇒

(e,ψ

)=

1

d(ω

)4

∫ I

∫ ω

edx

dt

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

43

NS:

Blu

ffbo

dydu

also

luti

on

t=

2.0

t=

1.75

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

44

NS:

Blu

ffbo

dydu

also

luti

on

t=

1.5

t=

1.25

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

45

NS:

Step

dow

ndu

also

luti

on

t=

2.0

t=

1.5

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

46

NS:

Step

dow

ndu

also

luti

on

t=

1.0

t=

0.5

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

47

Ela

stic

ity

find

u:Ω→

R3

such

that

−∇·σ

=f

inΩ

,

σ=

λ∇

·uI

+2µ

ε(u)

inΩ

,

u=

g Don

ΓD,

n·σ

=g N

onΓ

N.

whe

reε(

u)

=(∇

u+∇u

T)/

2is

the

stra

inan

dλ

and

µ

are

the

Lam

epa

ram

eter

s.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

48

Ela

stic

ity:

dual

prob

lem

find

φ:Ω→

R3

such

that

−∇·σ

=ψ

inΩ

,

σ=

λ∇

·φI

+2µ

ε(φ)

inΩ

,

φ=

0on

ΓD,

n·σ

=0

onΓ

N.

Taki

ngψ

=δ x

0m

cont

rols

the

disp

lace

men

terr

or(u

−U

)(x

0)·m

.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

49

Ela

stic

ity:

solu

tion

todu

al

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

50

Ela

stic

ity:

solu

tion

todu

al

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

51

Ela

stic

ity:

solu

tion

topr

imal

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

52

Ela

stic

ity:

solu

tion

tolo

cald

ual

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

53

Ela

stic

ity:

solu

tion

tolo

cald

ual

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

54

Ela

stic

ity:

solu

tion

tolo

cald

ual

Go

to:

Gri

dor

Mul

tisc

ale

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

55

Lin

ear

Ela

stic

ity:

solu

tion

todu

al

Fig

ure:

von

Mis

esst

ress

cont

ours

ina

hois

t£tti

ng.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

56

Ext

rapo

lati

onE

rror

esti

mat

e:Fr

omth

eer

ror

estim

ate

usin

ga

disc

rete

dual

solu

tion

Φ

e est

=l(

Φ)−

a(u

h,Φ

)

we

geta

valu

ew

ithsi

gn!

Ext

rapo

lati

on:

we

have

mex

tra

=m

(uh)+

e est

Ifth

ees

timat

eof

the

erro

ris

shar

pw

ege

tan

impr

oved

valu

e.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

57

Ext

rapo

lati

onL

etV

1an

dV

2be

two

fini

teel

emen

tspa

ces.

Dis

cret

edu

alpr

oble

m:

find

φ2∈

V2

such

that

m(v

)=

a(v

,φ2)

allv

∈V

2.

Ass

ume

that

V1⊂

V2,

e.g.

V2

isan

h-

orp-

refi

nem

ent

ofV

1,th

en m(u

2)=

m(u

1)+

l(φ

2)−

a(u

1,φ

2)

whe

reu

1an

du

2ar

eth

eso

lutio

nsof

the

prim

al£n

ite

elem

ent

prob

lem

usin

gth

esp

aces

V1

and

V2,

resp

ec-

tivel

y.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

58

Ext

rapo

lati

on

Ifne

sted

spac

esar

eus

ed,t

hen

the

func

tiona

lcou

ldbe

com

pute

dby

eith

er•

Solv

ing

the

coar

sepr

imal

prob

lem

and

fine

dual

prob

lem

and

usin

gth

eer

ror

repr

esen

tatio

nfo

rmul

a. m(u

2)=

m(u

1)+

l(φ

2)−

a(u

1,φ

2)

or•

Solv

ing

the

fine

prim

alpr

oble

man

dev

alua

ting

the

func

tiona

l.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

59

Solv

ing

PD

Es

onN

etw

orks

Com

puta

tion

ofse

vera

lfun

ctio

nals

can

bedo

nein

para

llel:

•E

ach

com

pute

ris

assi

gned

the

com

puta

tion

ofon

e

func

tiona

l.

•E

ach

com

pute

rso

lves

adu

alpr

oble

man

dca

lcul

ates

onits

own

adap

ted

grid

and

fina

llyde

liver

son

lyth

eva

lue

ofth

e

func

tiona

l.

•A

ppro

xim

ate

solu

tions

whe

reea

chco

mpu

ter

calc

ulat

esan

appr

oxim

atio

nto

the

solu

tion

ina

subd

omai

nus

ing

itsow

n

grid

.

Pape

r:E

step

-Hol

st-L

arso

n:G

ener

aliz

edG

reen

’sF

unct

ions

and

the

Effe

ctiv

eD

omai

nof

In¤u

ence

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

60

Par

alle

l:E

xam

ples

−1 π2∆

u=

2+

4e−5

((x−.

5)2+

(y−2

.5)2

) ,(x

,y)∈

Ω,

u(x

,y)

=0,

(x,y

)∈

∂Ω

,

whe

reΩ

isth

e“s

quar

ean

nulu

s”Ω

=[0

,3]×

[0,3

]\

[1,2

]×[1

,2].

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

61

One

com

pute

r:in

itia

lan

dfi

nal

mes

hIn

itia

l M

esh

Fin

al M

esh

Figu

re5:

Plot

sof

the

initi

al(l

eft)

and

£nal

(rig

ht)

mes

hes

with

data

ψgi

ving

the

aver

age

erro

r.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

62

One

com

pute

r:so

luti

on

12

30

1

2

3

0123456

y

x

12

30

1

2

3

0

0.4

0.8

1.2

1.6

y

x

Figu

re6:

Plot

sof

the

£nal

solu

tion

(lef

t)an

ddu

also

lutio

n

(rig

ht)

with

data

ψgi

ving

the

aver

age

erro

r.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

63

Seve

ral

com

pute

rs:

deco

mpo

si-

tion

12

3456

7 8

Figu

re7:

Dom

ains

for

the

part

ition

ofun

ity.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

64

Seve

ral

com

pute

rs:

dual

solu

-ti

ons

0

12

30

1

2

3

1

y

x

12

30

1

2

3

0

0.2

0.611.4

y

x

Figu

re8:

Plot

sof

the

dual

solu

tions

corr

espo

ndin

gto

ψ6

(lef

t)

and

ψ7

(rig

ht)

for

the

part

ition

ofun

ityde

com

posi

tion.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

65

Seve

ralc

ompu

ters

:fi

nalm

eshe

s

Fin

al M

esh f

or

U3

^F

inal

Mes

h f

or

U4

^F

inal

Mes

h f

or

U6

^F

inal

Mes

h f

or

U7

^

Figu

re9:

Plot

sof

the

fina

lm

eshe

sfo

rth

elo

caliz

edso

lutio

ns

U3,U

4,U

6,a

ndU

7.

Intr

oduc

tion

toA

Post

erio

riE

rror

Est

imat

ion

base

don

Dua

lity

–p.

66