This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
8/10/2019 The Immersed Interface Method Using a Finite Element Formulation Zhilin
254 Z. Li /Ap pli ed Numerical Mathematics 27 (1998) 253-267
A
u(x)
x j
oL
x j+l
F ig . 1 . A d i ag ram sh o ws wh y a f i n i te e l em en t m e th o d can n o t b e seco n d o rd er accu ra t e i n t h e i n f in i t y n o rm i f t h e i n t e r f ace
is no t a g r id po in t .
[u]~ de j l im
u ( x ) -
l i m
u ( x ) =
O, (1.3)
x ~ ? x ~ ;
[~Ux]c~
de=f lim /3( x)
u'(x) --
l im /3(x)
u ' (x )
= 0. (1.4)
X ----+C~? X - ~O~
T he so l u t i on o f (1. l ) i s typ i ca l l y non - s m o o t h a t t he i n te r f ace s i f / 3 (x ) ha s a f i n it e j um p a t e ach i n t e r f ace .
T h e p r o b l e m c a n b e s o l v e d b y b o t h f i n it e d if f e r e n c e m e t h o d s a n d f in i te e l e m e n t m e t h o d s . T h e
i m m erse d in t e r f ace m e t h od ( I IM ) [6 , 7 , 9 -11 ] i s an e f f i c i en t f i n it e d i f f e r ence app roac h fo r in t e r f ace
p rob l em s w i t h d i scon t i nu i t ie s and s i ngu l a ri t ie s . T he so l u t ion ob t a i ned f ro m t he I IM i s typ i ca l l y s econ d
o rde r a ccu ra t e in t he i n f i n it y no rm reg a rd l e s s o f t he r e l a ti ve pos i t ion be t w e en t he g r i d po i n t s and
t he i n t e r f ace s . H ow eve r , fo r t w o o r h i ghe r d i m ens i ona l p rob l em s , t he r e su l t i ng l i nea r sy s t em ob t a i ned
f r o m t h e I I M m a y n o t b e s y m m e t r i c p o s it iv e d e f i n it e .
I f t he f in i te e l em en t m e t hod w i t h t he s t anda rd l i nea r ba s i s i s u sed fo r (1. l ) w i t h p re se nce o f i n te r f ace s ,
s econd o rde r a ccu ra t e so l u t i ons can s t i l l b e ob t a i ned i f t he i n t e r f ace s l i e on t he g r i d po i n t s . T h i s c an
be p roved s t r i c t l y i n one d i m ens i ona l space . F o r h i ghe r d i m ens i ona l p rob l em s , t he ana l y s i s i s u sua l l y
g i ven i n an i n t eg ra l no rm w h i ch i s w eak e r than t he i n f in i t y no rm , s ee [1 ,3 , 4, 13 ], e t c . I f any o f t he
i n t e r f ace s i s no t a g r i d po i n t , t hen t he so l u t i on ob t a i ned f rom t he f i n i t e e l em en t m e t hod i s on l y f i r s t
o rde r a ccu ra t e in t he i n f i n i ty no rm , s ee F i g . 1 . F o r t w o o r h i ghe r d i m en s i ona l p rob l em s , i t i s d i ff i cu lt
and cos t l y to c ons t ruc t a body - f i t ti ng g r i d so t ha t t he i n t e r f ace a l i gns w i t h t he t r iangu l a t ion , e spec i a l l y
f o r m o v i n g i n te r f a c e p r o b le m s .
I n t h is p a p e r , w e t r y t o d e v e l o p a n u m e r i c a l m e t h o d w h i c h m a i n t a i n s t h e a d v a n t a g e s o f t h e s i m p l e
g r i d s t ruc t u re o f t he f i n i t e d i f f e r ence m e t hod and t he n i ce t heo re t i c a l p rope r t i e s o f t he f i n i t e e l em en t
m e t hod . T he i dea i s t o t ake a s i m p l e C a r t e s i an g r i d , fo r exam pl e , a un i fo rm g r i d , and m od i fy t he
bas i s func t i ons so t ha t t he i n t e r f ace j um p r e l a t ions a r e s a t i s fi ed . W i t h t he s i m p l e g r i d , o r t r i angu l a t ion ,
t he f in i te e l em en t m e t ho d co r r e sp onds t o a f i n i te d i f f e r ence m e t ho d i n w h i ch t he r e su l t ing l i nea r sy s -
t e m o f e q u a t i o n s i s s y m m e t r i c p o s it iv e d e f in i te . B y c h o o s i n g m o d i f i e d b a si s f u n c t i o n s , s e c o n d o r d e r
a c c u r a c y i s a c h i e v e d in t h e i n f in i ty n o r m f o r o n e d i m e n s i o n a l p r o b l e m s . I n h o m o g e n e o u s j u m p c o n -
d i ti o n s t h e n c a n b e t a k e n c a r e o f e a s i ly b y a d d i n g s o m e c o r r e c t i o n t e r m s a c c o r d i n g to t h e i m m e r s e d
i n t er f a c e m e t h o d . W e a l so p r o p o s e s o m e n u m e r i c a l m e t h o d s f o r c o n s t r u c ti n g b a si s f u n c t i o n s f o r tw o
d i m e n s i o n a l p r o b l e m s i n v o l v in g i n t e r fa c e s . W h i l e s e c o n d o r d e r c o n v e r g e n c e is p r e s e r v e d i n t h e e n -
e r g y n o r m f o r t h o s e m e t h o d s , t h e c o n v e r g e n c e o f t h o s e m e t h o d s i n t h e i n f in i ty n o r m i s s til l u n d e r
inves t iga t ion .
8/10/2019 The Immersed Interface Method Using a Finite Element Formulation Zhilin
Z. Li /Ap pl ied Numerical Mathematics 27 (1998) 253-267 255
2 . M o d i f i c a t i o n o f t h e l i n e a r b a s i s
D ef i n e t h e s t an d a r d b i l i n ea r f o r m
a ( u , v ) = / ( / 3 ( x ) u ' ( x ) v ' ( x ) + q ( x ) u ( x ) v ( x ) ) d x , u ( x ) , v ( x ) E
H(~(0, 1), (2.5 )
0
w h er e H ~ ( 0 , 1 ) i s t h e S o b o l e v s p ace . T h e s o l u t i o n o f t h e d if f e r en t ia l eq u a t i o n u ( x ) ~ H ~ ( 0 , 1 ) i s a l s o
t h e s o l u t i o n o f t h e f o l l o w i n g v a r i a t io n a l p r o b l em :
a ( u ,v ) = ( f , v ) = / f ( x ) v ( x ) d x , V v E H ~ (O, 1) . (2.6)
.J
0
W i t h o u t l o s s o f g en e r a l i ty , w e a s s u m e t h a t t h e r e i s o n l y o n e i n t e r f ace a i n t h e i n t e r v a l ( 0 , 1 ) . I n t eg r a t i o n
b y p a r t s o v e r t h e s ep a r a t ed i n t e r v a ls ( 0 , a ) an d ( a , 1 ) y i e l d s
OL
= / { - ( / 3 u ' ) ' + q u - f } v + / 3 - u z v (2.7)
0
+ / { - ( / 3 u ' ) ' + q u f } v +~ + "+
- / 3 ~ . v . 2 . 8 )
O~
T h e s u p e r s c r i p t s - an d ÷ i n d i ca t e t h e l i m i t in g v a l u e a s x ap p r o ach es a f r o m t h e le f t an d ri g h t,
r e s p ec t i v e l y , an d u x = u ' . R e ca l l th a t v - = v + f o r an y v i n H d , i t f o l l o w s th a t th e d i f f e r en ti a l
eq u a t i o n h o l d s i n each i n t e r v a l an d t h a t
u + u 0 , + +
- - = = / 3 U x - / 3 - u : ; = 0 ,
w h e r e w e h a v e d r o p p e d t h e s u b s c r i p t a i n th e j u m p s s i n c e th e r e i s o n l y o n e i n t e r fa c e . T h e s e r e la t io n s
a r e t h e s am e a s i n ( 1 . 3 ) , ( 1 . 4 ) , w h i ch i n d i ca t e s t h a t t h e d i s co n t i n u i t y i n t h e co e f f i c i en t / 3 ( x ) d o es n o t
c a u s e a n y t r o u b l e f o r th e t h e o r e t ic a l a n a l y s i s o f t h e F E M a n d t h e w e a k s o l u ti o n w i l l s a ti s fy t h e j u m p
condi t ions (1 .3) , (1 .4) .
N o w l e t u s t u r n o u r a t t en t io n t o t h e n u m er i c s . F o r s i m p l i c it y , w e u s e a u n i f o r m g r i d x i = i h ,
i - - 0 , 1 ~ . . . , N , w i t h x 0 = 0 ,
X N
= 1 and h =
1 / N
i n o u r d i s cu s s i o n . T h e s t an d a r d l i n ea r b a s i s
f u n c t i o n s a t i s f i e s
1 , i f i = k , ( 2 . 9 )
¢ i ( x k ) = 0 , o th e rw i se .
T h e s o l u t i o n
Uh(X)
i s a s p ec i f i c l i n ea r co m b i n a t i o n o f t h e b a s i s f u n c t i o n f r o m t h e f i n i t e d i m en s i o n a l
s p a c e
Vh :
Vh = Vh: Vh = ~ i ¢ i ( x ) , ( 2 . 1 0 )
i=1
a(uh, Vh) = (f , Vh),
f o r V v h E
Vh.
(2 .11)
8/10/2019 The Immersed Interface Method Using a Finite Element Formulation Zhilin
Li /Ap pl ied Numerical Mathematics 27 (1998) 253-267
I f an i n t e r f ace i s n o t o n e o f g r i d p o i n t s x i , u s u a l l y t h e s o l u t i o n
U h
i s o n l y f i r s t o r d e r a ccu r a t e i n t h e
i n fi n it e n o r m , s ee F i g . 1 . T h e p r o b l em i s t h a t s o m e b as i s f u n c t i o n s w h i ch h av e n o n - ze r o s u p p o r t n ea r
t h e i n t e r f ace d o n o t s a t i s f y t h e n a t u r a l j u m p co n d i t i o n ( 1 . 4 ) a t t h e i n t e r f ace .
T h e s o l u t i o n i s t o m o d i f y t h e b a s i s f u n c t i o n s i n s u ch a w ay t h a t n a t u r a l j u m p co n d i t i o n s a r e s a t i s f i ed :
1 , i f i = k , ( 2 . 1 2 )
¢ i ( x k ) = O, o t h e rw i s e,
[¢i] = O, (2.1 3)
[fl¢'~] = O. (2 .1 4)
O b v i o u s l y , i f
x j < a < x j + j ,
t h e n o n l y C j a n d C j + l n e e d t o b e c h a n g e d t o s a t is f y t h e s e c o n d j u m p
c o n d i ti o n . U s i n g a n u n d e t e r m i n e d c o e f f i c ie n t m e t h o d , w e c a n c o n c l u d e t h a t
Z . L i /A pp l i e d Num er i ca l Mathem at i cs 27 (1998) 253-2 67
257
a n d
Z - Z +
_ / 3
P
= ~ -T , D = h /3+ (x j+ l - ~ ) ,
(2 .16)
O, O ~ X < X j ,
X - - X j
D ' x j ~ x < o : ,
(~j+l(X) = p ( x - X j + l ) -~ - 1 c~ <<. x < x j + l ,
D
x j + 2 - - x
h ' X j + l ~ X ~ X j + 2 ,
O,
X j+ 2 <<. X <<. 1.
(2 .17)
F i g . 2 show s seve ra l p l o t s o f t he m od i f i ed ba s i s func t i ons
C j ( X ) ,
0j+ l (X) , a n d s o m e n e i g h b o r i n g
bas i s func t i ons , t ha t a r e the s t anda rd ha t func t i ons . A t t he i n t e r f ace , w e c an see c l ea r l y t he k i nk i n t he
bas i s func t i on w h i ch r e f l e c t t he na t u ra l j um p cond i t i ons .
3 T h e t h e o r e t ic a l a n a l y s i s
In t h is s ec t i on , w e p ro ve t ha t t he so l u t ion ob t a i ned f ro m t he f i n it e e l em en t m e t h od w i t h t he m od i f i ed
bas i s func t i on i s s econd o rde r a ccu ra t e i n t he i n f i n i t e no rm .
F o r t he s ake o f c l ean and conc i se p roo f , w e de r i ve t he t heo re t i c a l ana l y s i s fo r t he s i m p l e m ode l :
- / 3 ( x ) u " = f ( x ) , f ( x ) E C [0 , 1], 0 <~ x ~< 1, (3 .18 )
u(0 ) = O, u( 1) = 0 , (3 .19)
w i t h
/3(x) = { / 3 - , i f O ~< x < c~,
/3+ , i f c ~ < x ~ < l ,
w h e r e / 3 - a n d / 3 + a r e t w o c o n s t a n t s . T h e s o l u t i o n u ( x ) E H i sa t i s f i e s t he na t u ra l j um p cond i t i ons
a t c~. I f t he va l ue o f t he so l u t i on a t c~ i s know n , s ay uc~ , t hen t he p r ob l em i s eq u i va l en t t o t he fo l l ow i ng
t w o s e p a r a t e d p r o b le m s :
- / 3 - u " = f ( x ) , 0 ~ < x < o ~ , an d u ( 1 ) = 0 .
~ (0 ) = 0 , ~ (~ ) = ~ , ~ (~ ) = ~ ,
T h e r e f o r e f r o m t h e r e g u l a r i t y t h e o r y w e k n o w t h a t
u ( x ) E
C2[0 , 1] i n each sub-domain and uz-~ =
l im x ~ a -
u " ( x )
a n d u +x = l i m x ~ +
u " ( x )
are f in i t e . We def ine
IL~ llo~ = - m a x I~xxl, I x~l, sup
lUxxl,
sup I~xxl , (3 .20)
0<x<a c~<x<l
w h i c h i s b o u n d e d .
3.1. The in terpolant o f the solut ion
A s i n t he s t anda rd F E M ana l y s i s , an i n t e rpo l a t i ng func t i on o f t he so l u t ion p l ays an i m po r t an t ro l e in
t he e r ro r ana l y s i s . I n th i s subsec t i on , w e w i l l de f i ne t he p i ecew i se li nea r func t i on i n the space ~ w h i ch
8/10/2019 The Immersed Interface Method Using a Finite Element Formulation Zhilin
L i / A p p l i e d N u m e r i c a l M a t h em a t i c s 2 7 (1 9 9 8 ) 2 5 3 - 2 6 7
a l s o i n t e r p o l a t e s u ( x ) a t t h e n o d e p o i n t s . A s s u m i n g t h a t xj <~ o~ < xj + l, w e d e f i n e a n i n t e r p o l a n t o f
u ( x )
a s f o l l o w s :
X i + l - - X X - - X i
h u (x i ) + - - -h -u (x i+ l ) , i # j , x~ <~ x <~ X~+l ,
u i ( x ) = u (x j ) + t ~ (x - u ( x j ) ) , x j ~< x < o~, (3 .2 1)
u(X j+ l) + ~ p ( x - x j+~ ), o~ <~ x <~
xj+, ,
w h e r e
9 - ~ x j + l ) -
~(~)
p - f l+ , t~ = . ( 3 .22 )
o ~ - x j - p o ~ - X j + l )
I t i s e a s y t o v e r i f y t h a t
U l ( X i ) = u ( x i ) , i = 0 , 1 , . . . , N - 1 , ( 3 . 2 3 )
[ui] = O, [ /3u}] = O, (3. 24 )
a n d h e n c e
ux(x ) E Vh.
B e f o r e g i v in g a n e r ro r b o u n d f o r ] ] u i ( x ) - u ( x ) ] ] , ¢ , w e n e e d t he f o l lo w i n g
l e m m a w h i c h g i v e s t h e e r ro r e s t i m a t e s f o r t h e f ir st d e r i v a t i v e s o f u i ( x ) a p p r o x i m a t i n g u ' ( x ) .
L e m m a 3 . 1 .
G iven u i ( x ) as de f ined in
(3 .21 ) ,
the fol lo win g inequali t ies hold:
OZ - - X j - - p ( O - - X j + I ) ~ m i n { ½ h , ½ h P } ,
I ,~ - ~ ; I ~ < c I l u l l ~ h ,
I p ~ - ux+l ~< C Ilu ll~ h,
w h e r e
2 max{ 1 , p}
C -
m i n { 1 , p }
( 3 . 2 5 )
( 3 . 2 6 )
( 3 . 2 7 )
( 3 . 2 8 )
S o C a c t s l i ke a c o n d i t io n n u m b e r f o r t h e in t e r f a c e p r o b l e m .
P r o o f . I t i s o b v i o u s th a t
{ 0~ - - X j - - p ( ( y - X j + l ) ~ p ( x j + l - o L ) ~ ½hP, i f ~ - x j < ½h,
OL - - X j - - p (O L - - X y + I ) ~ O - - X j
~ ½ h, if c~ -
x j > ½h,
w h i c h c o n c l u d e s t h e f i rs t i n e q u a l i t y . U s i n g t h e T a y l o r e x p a n s i o n a b o u t c~, w e h a v e
- x j - p c ~ - x y + l )
= ~ + + ~ + x j + ~ - ~ ) + ½ ~ x x ~ 1 ) x ~ + ~ - ~ ) 2
o ~ - x j - p o ~ -
x j + l )
_ _ U - d - U + x ( X j - - O l ) -'~ l U x x ( ~ 2 ) ( X j - -
O~)2 __ Ztx ,
o ~ - x a - p c ~ -
X j + l )
w here ~ 1 E
(o~ ,x j+l )
a n d ~2 E ( x j , o ~ ) . W i t h t h e j u m p c o n d i t i o n s u + = u - a n d
u + - - pu x ,
t h e
e x p r e s s i o n a b o v e i s s i m p l i f i e d t o
8/10/2019 The Immersed Interface Method Using a Finite Element Formulation Zhilin
Z. Li /Applied NumericalMathematics27 (1998) 253-267
261
R e m a r k 3 .1 . T h e c o n c l u s i o n c a n b e e a s i ly e x t e n d e d to t h e m o r e g e n e r a l c a s e (1 .1 ) , (1 .2 ) w i t h v a r ia b l e
/ 3 (x ) . T he key m od i f i c a t i on i s t he fo l l ow i ng :
Z. Li /Ap plied Numerical Mathematics 27 (1998) 253-267 263
0 3
0 2 5
0 2
0 1 5
0 1
0 0 5
0
0 0 5
n = 4 0 , [~ = 1 [ ~ + = 1 0 0 f~2/3
0 0 C o m p u t e d
E x a c t
C = = C C C O ~
I I I I I I t I I
0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9
× 1 4
E r r o r P l o t n : 4 0 , ~ 1 , { ~ * : 1 0 0 1 1 -- 2/ 3
(a) (b)
Fig. 3. Com parison of the com puted solution and exact solution when ~z = 40. (a) The solution plo t. (b) The erro r plot.
F i g . 3 ( b ) i s t h e e r r o r p l o t o f t h e e r r o r i n th e e n t i r e i n t e r v a l . W e s e e t h e e r r o r s a r e z e r o a t g r i d p o i n t s
and
O ( h 2 ) a t o t h e r p o i n t s .
5 C o n s t r u c t i n g b a s i s f u n c t i o n s i n t w o d i m e n s i o n s
T h e m o d e l e q u a t i o n i n t w o d i m e n s i o n s o n a r e c t a n g u l a r r e g i o n w i t h a c l o s e d i n t e r f a c e i s
V . ( / 3 V u ) = f ( x , y ) , ( x , y ) c f2 , ( 5 . 3 4 )
g i v e n B C o n 0 S ?, ( 5 .3 5 )
s e e th e d i a g r a m i n F i g . 4 . T h e n a t u r a l j u m p c o n d i t i o n s a c r o s s th e i n t e r f a c e F a r e
[u] = 0, [/3u~] = 0, (5 .36 )
w h e r e u ~ i s t h e n o r m a l d e r i v a t i v e .
A s d i s c u s s e d i n p r e v i o u s s e c t i o n s , w e w i l l u s e a u n i f o r m t r i a n g u l a t i o n , s e e F ig . 5 . I f a c e l l c o n t a i n s
n o i n t e r f a c e, w e c a n u s e t h e s t a n d a r d l i n e a r b a s is f u n c t i o n o v e r t h a t c e l l. I t i s m o r e d i f f ic u l t t o c o n s t r u c t
b a s i c f u n c t i o n i n t w o d i m e n s i o n s w h e n i n t e r f a c e c u t s t h r o u g h t h e u n i f o r m t r i a n g u l a t i o n . I t i s t r u e t h a t
w e c a n e a s i l y f i n d p i e c e w i s e l i n e a r , o r q u a d r a t i c , o r c u b i c f u n c t i o n w h i c h i n t e r p o l a t e s t h e s o l u t i o n
o f ( 5 . 3 4 ), ( 5 .3 5 ) t o s e c o n d o r h i g h e r o r d e r a c c u r a c y u s i n g t h e T a y l o r e x p a n s i o n . T h e d i f f i c u l t y i n
c o n s t r u c t i n g t h e b a s i s f u n c t i o n i s t h e r e q u i r e m e n t s o f c o n t i n u i t y in t h e e n t ir e r e g i o n a n d t h e j u m p
c o n d i t i o n s a c r o s s t h e i n t e r f a ce . T h e r e a r e s e v e r a l a p p r o a c h e s c u r r e n t l y u n d e r i n v e s t i g a t io n . B u b e a n d
K a u p e [ 2 ] a r e t y i n g t o u s e a q u a d r i l a t e r a l t r i a n g u l a t i o n . H o u a n d W u [ 5 ] a r e e x p e r i m e n t i n g w i t h b o t h
c o n f o r m i n g a n d n o n - c o n f o r m i n g b a s i s f u n c t i o n s . B e l o w w e p r o p o s e a n a p p r o a c h w h i c h i s i n t h e s a m e
s p ir it a s o u r d i s c u s s i o n f o r o n e d i m e n s i o n a l p r o b l e m . W e w i l l s ti c k w i th c o n f o r m i n g b a s is f u n c t io n s ,
t h a t i s , t h e b a s i s f u n c t i o n s b e l o n g t o H01 ( £ 2) .
8/10/2019 The Immersed Interface Method Using a Finite Element Formulation Zhilin
264 Z Li /Appli ed Numerical Mathematics 27 (1998) 253-267
~+
f~
F i g . 4. A d i a g r a m f o r a m o d e l p r o b l e m i n t w o d i m e n s i o n s .
F
A~
E
y~
G
h
xl
Fig . 5 . A typ ica l c e l l w i th the in t e r face cu t t ing th rough .
5.1. A coupled approach
N o w t a k e a t y p i c a l c a s e a s s h o w n i n F i g . 5 . A n a r b i t r a r y c l o s e d i n t e r f a c e c a n b e a p p r o x i m a t e d b y
p i e c e w i s e li n e s e g m e n t s . W e w a n t t o f in d a b a s i s f u n c ti o n ¢ ( x , y ) , f o r e x a m p l e , c e n t e r e d a t
(xi , y j)
w h i ch s a t i s f i e s
¢ ( x , y ) E C ( O ) ,
¢(x i , y j )
= l , (5 .37)
¢ ]
= o , = o . 5 . 3 8 )
I t i s q u i t e o b v i o u s th a t a li n ea r b a s i s f u n c t i o n w i l l n o t w o r k . I n s t ead , w e t r y to u s e a p i ecew i s e q u ad r a t i c
b as i s f u n c t i o n . I n F i g . 5 , t h e i n t e r f ace d o es n o t cu t t h r o u g h t h e t r i an g l e s 4 , 5 an d 6 . T h e r e f o r e t h e
b as i s f u n c t i o n can b e t ak en a s t h e s t an d a r d l i n ea r b a s i s f u n c t i o n o v e r t h o s e t r i an g l e s .
T h e t r i an g l e s l , 2 an d 3 co n t a i n a p o r t i o n o f t h e i n t e rf ace w h i ch d i v i d e s t h e r eg i o n i n t o s i x p i ece s ,
t h r ee tr i an g l e s an d t h r ee q u ad r i la t e r a l s. T h e p i e cew i s e q u ad r a t i c f u n c t i o n o v e r th e s i x p i ece s can b e
d e t e r m i n e d u s i n g t h e
undetermined coefficient method.
T h e t o t a l n u m b e r o f d e g r e e s o f f r e e d o m i s 3 6
w i t h o u t an y co n s t ra i n s . T h e co n t i n u i t y co n s t r a i n s i n v o l v i n g b o t h t h e s i d e s o f t h e tr i an g l e s an d t h e
i n te r f a ce a r e 3 3 . T h e r e m a i n i n g 3 d e g r e e s o f f r e e d o m a r e t h en u s e d t o s a t is f y th e f l u x j u m p c o n d i t i o n
[ /3¢n] --- 0 to cer t a in de gree . Fo r ins tanc e , we can f orce [ /3¢n] to be zero a t the m id-po in t s o f l inear
8/10/2019 The Immersed Interface Method Using a Finite Element Formulation Zhilin
Z L i / A p p l i e d N u m e r ic a l M a t h e m a t i c s 2 7 1 9 9 8 ) 2 5 3 - 2 6 7
265
s e g m e n t s
A B , B C ,
a n d
C D .
A t o t h e r p o i n t s o f t h e i n t e r face , I t C h ] w i l l a l s o b e v e r y s m a l l f r o m t h e
c o n t i n u i ty c o n d i t i o n o f t h e s o l u t i o n a n d t h e a s s u m p t i o n t h a t t h e n o r m a l d i r e c ti o n o f t h e i n t e rf a c e d o e s
n o t c h a n g e t o o m u c h i n t h e c e l l .
I n t h e en d , w e n eed t o s e t u p a l in ea r s y s t em o f eq u a t i o n f o r t h e co e f f i c i en t s o f t h e q u ad r a t i c b a s i s
f u n ct io n . T h e n u m b e r o f u n k n o w n s c a n b e g r e at ly r e d u c e d i f w e t a k e a d v a n t a g e o f th e b o u n d a r y
co n d i t i o n s . F o r ex am p l e , t h e f u n c t i o n ¢ ( x , y ) o n th e q u ad r i l a t e ra l
A B O E
can b e w r i t t en a s
¢ ( x , y ) = a o + a l ( x - x i - 1 ) + a 2 (y - - y j ) + a 3 ( x - - x i - 1 ) ( y - y j ) .
I t i s n o t s o e a s y t o i m p l e m e n t t h e a p p r o a c h d i s c u s s e d a b o v e b e c a u s e t h e b a s i s f u n c t i o n s o n s e v e r a l
ce l l s a r e co u p l ed t o g e t h e r . T h i s i s t h e p r i ce w h i ch w e n eed t o p ay f o r i n t e r f ace p r o b l em s t o o b t a i n
m o r e a c c u r a t e r e s u lt s . H o w e v e r w e c a n s i m p l i f y t h e p r o c e s s o f c o n s tr u c t in g t h e b a s is f u n c t i o n i f w e
c a n
p r e - d e t e r m i n e
t h e v a l u e s o f t h e b a s i s f u n c t i o n a t p o i n t s B , an d C a s in th e ex a m p l e o f F i g . 5 . A n
i n t e r p o l a t i o n ap p r o ach t o d e t e r m i n e t h o s e v a l u e s w i l l b e d i s cu s s ed l a t e r i n t h i s s ec t i o n .
5 .2 . A d eco u p l ed a p p ro a ch
S u p p o s e w e c a n p r e - d e t e r m i n e t h e v a l u e s o f th e b a s i s f u n c ti o n a t th o s e p o i n t s B , a n d C i n F i g . 5 ,
t h en w e can co n s t r u c t a b i l i n ea r f u n c t i o n i n each q u ad r i l a t e r a l w h i ch i n t e r p o l a t e s t h e f u n c t i o n v a l u e s
a t t h e v e r t i c e s an d t h o s e i n t e r s ec t i o n s s u ch a s t h e p o i n t s B , an d C i n F i g . 5 . T h e b i l i n ea r f u n c t i o n s
¢ ( x , y ) w i l l b e l i n e a r o n t h e b o u n d a r y o f th e r e g i o n a s s h o w n i n F ig . 5 . F o r e x a m p l e , t h e b i l in e a r
f u n c t i o n ¢ ( x , y ) o n t h e q u ad r i l a t e r a l A B O E is
¢ X , y ) - - X - h x i - 1 q - ~ Y - y j ÷ 1 ÷ q ( x - x i - 1 ) y - y f i ) ,
w h e r e q i s c h o s e n s u c h t h at ¢ ( x B , YB ) = C B , th e p r e - d e t e r m i n e d v a l u e o f t h e b a si s f u n c t i o n a t B .
O n c e w e h a v e d e t e r m i n e d t h e b i l i n e a r f u n c t i o n o n e a c h q u a d r i l a t e r a l , t h e n w e k n o w t h e v a l u e s
o f th e b a s i s f u n c t i o n a t m i d - p o i n t s o f a ll s i d e s o f e ach t r i an g l e s. T h u s a q u ad r a t i c f u n c t i o n i s e a s i l y
d e t e r m i n ed f r o m t h e s i x v a l u e s o f th e b a s i s f u n c t i o n o n each t r i an g l e .
I n t h e a p p r o a c h w e d e s c r i b e d a b o v e , w e a r e a l m o s t a b l e t o f i n d t h e b a s i s f u n c t i o n o n e a c h t r i a n g l e
s ep a r a t e l y , w h i ch m ak es i t e a s i e r t o a s s em b l e t h e s t i f f n e s s m a t r i x .
5 .3 . I n t e rp o l a t io n sch em e f o r t h e p re -d e t e rm i n ed va l u e s
T h e ap p r o ach d e s c r i b ed ab o v e r e l y o n t h e v a l u e s a t i n t e r s ec t io n s o f t h e i n t e r face an d t h e i n te r i o r
s i d e s o f t h e t ri an g l e s . I d ea l l y , t h e b a s i s f u n c t i o n can b e t ak en a s t h e s o l u t i o n o f th e f o l l o w i n g P o i s s o n
e q u a t i o n w i t h n a t u r a l j u m p c o n d i t i o n s :
v . ( 3 v ¢ ) = o ,
[¢] = o , [9 ¢ , ] = o .
T h e b o u n d a r y c o n d i t i o n i s
I X - - X i - - 1
X - - X i
¢ =
0
o n
E O ,
o n
E F ,
o n E G , G H a n d H F .
( 5 . 3 9 )
( 5 . 4 0 )
(5 .41)
8/10/2019 The Immersed Interface Method Using a Finite Element Formulation Zhilin
Li / Applied Numerical Mathematics 27 (1998) 253-267
O n c e w e k n o w t h e s o l u t io n o f th e P D E a b o v e , w e c a n g e t t h o s e p r e - d e t e rm i n e d v a l u e s . H o w e v e r , it
i s t o o co s t l y t o s o l v e t h e P D E ab o v e a t a l l t h e ce l l s w h i ch co n t a i n t h e i n t e r f ace . W h a t w e can d o ,
h o w e v e r , is t o u s e a n i n te r p o l a ti o n s c h e m e i n t e rm s o f t h e b o u n d a r y v a l u e s , t h e j u m p c o n d i t io n s , a n d
t h e p a r t i a l d i f f e r en t i a l eq u a t i o n i t s e l f , f o r ex am p l e ,
T h e co e f f i c i en t s c an b e d e t e r m i n ed u s i n g t h e w e i g h t ed l e a s t s q u a r e s i n t e r p o l a t i o n [ 1 0] an d t h e p r o ced u r e
i s b r i e f l y d e s c r i b ed b e l o w :
S e l ec t a p o i n t X o n t h e i n t e r face .
. . . . .+
U s e t h e l o ca l co o r d i n a t e s a t X i n t h e t an g en t i a l an d n o r m a l d i r ec t io n s o f t h e i n t e r f ace .
_____+*
U s e t h e T a y l o r e x p a n s i o n o v e r X t o e x p a n d t h e v a l u e s f r o m e a c h s i d e o f t h e i n te r fa c e .
E l i m i n a t e t h e q u an t i ti e s o f o n e s i d e , s u ch a s th e s o l u t i o n , th e d e r i v a t i v e s u p t o s eco n d o r d e r, i n
t e r m s o f an o t h e r u s i n g t h e j u m p c o n d i t i o n s a n d t h e d i f f e ren t i a l eq u a t i o n .
S e t u p an d s o l v e t h e li n ea r s y s t e m o f eq u a t i o n t o g e t t h e co e f f i c i en t s o f th e i n t e r p o l a ti o n .
T h e d e t a i l s c an b e f o u n d i n [ 8 , 1 0 ] w i t h s o m e m o d i f i c a t i o n .
T h eo r e t i c a l l y , i f t h e b a s i s f u n c t i o n s b e l o n g t o H 01 ( ~ ) s p ace an d s a t i s f y t h e n a t u r a l j u m p c o n d i t i o n s ,
t h e n t h e s t a n d a r d e r r o r a n a l y s i s u s i n g t h e e n e r g y n o r m w o u l d a p p l y . S o w e w o u l d h a v e t h e s t a n d a r d
co n v e r g en c e r e s u l t ev e n i f t h e r e i s an i n t e r f ace i n th e s o l u t i o n d o m a i n . I t is n o t e a s y t o s ee o r p r o v e
w h e t h e r t h e m e t h o d s d e s c r i b e d a b o v e a r e s t i l l s e c o n d o r d e r a c c u r a t e i n t h e i n f i n i t y n o r m . H o w e v e r ,
t h e a p p r o a c h e s d e s c r i b e d a b o v e c e r t a i n l y h a s b e t t e r a c c u r a c y c o m p a r e d t o t h e s t r a i g h t f o r w a r d f i n i t e
e l em en t m e t h o d w i t h n o m o d i f i c a t i o n s . T h e p r i ce i s t h e ex t r a co s t a t t h o s e ce l l s w h e r e t h e i n t e r f ace
cu t s t h r o u g h .
I n s u m m ar y , t h e m o d i f i ed f in i te e l em en t m e t h o d u s i n g t h e s i m p l e o r u n i f o r m t r i an g u l a t i o n i s v e r y
a c c u r a t e f o r o n e - d i m e n s i o n a l p r o b l e m s a n d v e r y p r o m i s i n g f o r t w o - d i m e n s i o n a l p r o b l e m s . T h e c o r -
r e s p o n d i n g f in i te d i ff e r e n c e m e t h o d w o u l d a l l o w u s to d e a l w i th i n h o m o g e n e o u s j u m p c o n d i t io n s .
H o w e v e r , t h e a n a l y s i s f o r t w o o r h i g h e r d i m e n s i o n a l i n t e r f a c e p r o b l e m s i s f a r f r o m c o m p l e t e . W e
h o p e t h e i d ea s p r e s en t ed i n t h i s p ap e r w i l l ev en t u a l l y l e ad t o t h e d ev e l o p m en t o f s o m e e f f i c i en t f i n i t e
e l e m e n t m e t h o d s w i t h s i m p l e t ri a n g u la t io n s f o r t w o a n d t h r ee d i m e n s i o n a l p r o b l e m s .
cknowledgements
T h e a u t h o r w o u l d li k e to th a n k T h o m a s H o u , T o n y C h a n , J i n c a o X u , X i a o h u i W u a n d o t h e r p e o p l e
f o r v e r y u s e f u l d i sc u s s i o n s a n d e n c o u r a g e m e n t s f o r t h is p r o j e c t s. T h e a u t h o r a l s o t h a n k s t h e r e f e r e e ' s
c o m m e n t s a n d s u g g e s t i o n s .
References
[1] I . Babu~ka, The finite element method for elliptic equations with discontinuous coefficients,
Computing 5
(1970) 207-213.
[2] K. Bube and T. Kaupe, Private communications.
[3] Z. Chen an d J. Zou , Finite e lem ent meth ods and their conv ergen ce for elliptic and parabo lic interface
problems, Research Repor t No. Math-96-25(99) , CU HK (1996).
8/10/2019 The Immersed Interface Method Using a Finite Element Formulation Zhilin