Domain Decomposition Modified with Characteristic Finite Element Method for Numerical Simulation of Semiconductor Transient Problem of Heat Conduction

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1 Joural of Matmatis Rsar; Vol. 7, No. 3; 5 ISSN E-ISSN Publisd by Caadia Ctr of Si ad Eduatio Domai Domositio Modifid wit Caratristi Fiit Elmt Mtod for Numrial Simulatio of Smiodutor Trasit Problm of Hat Codutio Yirag Yua, Luo Cag, Cagfg Li, & Togju Su Istitut of Matmatis, Sadog Uivrsity, Jia, Cia Sool of Eoomis, Sadog Uivrsity, Jia, Cia Corrsod: Yirag Yua, Istitut of Matmatis, Sadog Uivrsity, Jia, SadaNalu 7, Cia. yryua@sdu.du. Rivd: May 5, 5 Atd: Ju 5, 5 Oli Publisd: July, 5 doi:.5539/jmr.v736 URL: tt://dx.doi.org/.5539/jmr.v736 T rsar is fiad by t Natioal Natural Si Foudatio of Cia (Grat Nos 44, 73, t Natioal Taklig Ky Program (Grat No. 569, ad t Dotorat Foudatio of t Miistry of Eduatio of Cia (Grat No Abstrat A aratristi fiit lmt algoritm basd o domai domositio is struturd i tis ar to aroximat umrially multi-dimsioal smiodutor trasit roblms of at odutio. Fiit lmt aroximatio is rstd for t ltri fild ottial quatio, ad a domai domositio disrtizatio wit aratristi fiit lmt is ut forward for t ltro otratio quatio, ol otratio quatio ad at odutor quatio. A otimal ordr rror stimat i L orm is drivd for t ould systm by usig som tiqus su as variatio, domai domositio, t mtod of aratristis, t riil of rgy, gativ orm stimats, idutio yotsis, rior stimats tory ad otr tiqus of artial diffrtial quatios. Fially, xrimtal data osistt wit tortial ovrg rat ar sow. Tis ty of umrial mtod is of ig omutatioal ffiiy ad a sussfully solv tis itratioal roblm. Kywords: Numrial simulatio of smiodutor dvi, domai domositio, aratristi fiit lmt, otimal ordr stimats i L orm, umrial xrimts. Itrodutio Baus of t grat dvlomt of smiodutor dvi idustry dsribd by a iitial-boudary valu systm of diffusio ty of oliar artial diffrtial quatios, umrial simulatio must b of ig ordr auray ad quit ffiit, ad traditioal umrial mtods ar ot osidrd i atual omutatio. T som w modr disrtizatio tiqus ar itrodud to solv multi-dimsioal roblms wit omliatd ad irrgular gomtri rgios (H & Wi,989; Si, ; Yua, 9, 3. Numrial simulatio of smiodutor dvi bgis at t us of t squ itratio rstd Gumml sussfully i 964 (Gumml, 964. Cosidrig atual omutatios, Douglas ad Yua ut forward a usful fiit diffr mtod for simlifid o-dimsioal ad two-dimsioal modls (ostat offiit, witout osidratio of tmratur ad giv t ris tortial rsults firstly (Douglas & Yua, 987; Yua, Dig & Yag, 98. T Yua alis t mtod of aratristis rstivly wit fiit lmt ad mixd fiit lmt to solv two-dimsioal roblms wit variabl offiits ad givs otimal ordr rror i H orm ad L orm (Yua, 99, 993, 99. I viw of aliatios, Yua disusss t fft of at odutio to smiodutor dvi ad givs t aratristi fiit diffr ad uwid fratioal sts fiit diffr of tr dimsioal roblm o uiform artitio ad t ovrg rat i disrt l orm (Yua, 996,, 5, 8. I umrial simulatio of modr smiodutor dvi, t omutatio sal is ug, ad t disrtizatio osists of ts of tousad or svral millios ods, so a owrful aralll tool is itrodud (H & Wi,989; Si, ; Yua, 9, 3. For t simlst araboli quatio, Dawso, Duot ad Du tak t lad i disussig Galrki domai domositio mtod ad tortial aalysis (Dawso & Duot, 6

2 Joural of Matmatis Rsar Vol. 7, No. 3; 5 99, 994; Dawso, Du & Duot, 99; Dawso & Du, 99. O t disussio ow to simulat trdimsioal smiodutor trasit roblm of at odutio ffiitly, t autors stat a ty of domai domositio wit modifid aratristi fiit lmt ad aalyz t ovrg ad otimal ordr stimats i l orm by usig t tiqus su as variatio, domai domositio, aratristis, rgy orm, gativ orm stimats, matmatial idutio yotsis ad rior stimat tory ad tiqu of artial diffrtial quatio. Numrial data of xrimtal xaml ar sow osistt wit tortial ovrg rat ad by wi it illustrats tat tis rodur is mor ffiit ad alid sussfully to solv itratioal famous roblms (H & Wi,989; Si, ; Yua, 9, 3; Bak, t al., 985; Jrom, 994; Sidma, 986; Lou, 995. It is most valuabl i modl aalysis, umrial mtod, riil rsar, tory ad aliatio of smiodutor dvi simulatio. Tis ar is orgaizd as follows. I t first stio, matmatial modl, ysial itrrtatio ad som rlatd rsar ar itrodud. Som otatios ad rimary work ar giv i t sod stio. I t tird stio, t autors rst t rodurs of domai domositio ad modifid aratristi fiit lmt. I t fourt stio, t autors attmt to giv ovrg aalysis. W xami umrially t auray ad aralllism of t sm by a xaml. Som olusios ad disussios ar statd i t last stio. I tis ar M ad ε xrss gral ositiv ostat ad gral ositiv small ostat, rstivly, ad ty av diffrt maigs at diffrt las.. Matmatial Modl ad Pysial Bakgroud T matmatial modl of smiodutor trasit roblm of at odutio is dsribd by a iitial boudary valu systm of four artial diffrtial quatios (Yua, 996,, 5, 8; Bak, t al., 985; Jrom, 994; Sidma, 986; Lou, 995, wr t ottial is dfid by a lliti quatio, t otratios of ltro ad ol ar dfid by ovtio-diffusio quatios ad t tmratur is dfid by a at odutio quatio. Eltroi fild ottial is gratd by t itsity i t ltro quatio, t ol otratio quatio ad t at odutio quatio, ad boms a systm wit iitial ad boudary oditios, wi is dfid o tr variabls satial domai Ω as follows, ψ = α( N(X, X = (x, x, x 3 T Ω, t J = (, T], ( t = (D (X µ (X ψ R (,, T, (X, t Ω J, ( t = (D (X µ (X ψ R (,, T, (X, t Ω J, (3 ρ(x T t T = (D (X µ (X ψ (D (X µ (X ψ } ψ, (X, t Ω J, (4 wr t ltroi fild ottial ψ, ltro otratio, ol otratio, ad tmratur T ar ukow futios. All t offiits of (-(4 ar gratr ta a ositiv umbr ad lss ta aotr ositiv umbr. T umbr α = q/ε dots t quotit of two ositiv ostats, ltroi load q ad diltri offiit ε. T rlatio of diffusio offiit D s (X ad mobility ratio µ s (X is formulatd by D s (X = U T µ s (X(s =,, wr U T mas t quatity of at (uit: volt. N(X, a giv futio, is t diffr of t door imurity otratio N D (X ad t ator imurity otratio N A (X. T valus of N(X vary quikly as X lis arby smiodutor kot P-N. R i (,, T(i =, rrsts t rombiatio rat iflud by t ltro otratio, t ol otratio ad t tmratur. ρ = ρ(x is t offiit of at odutio. Lt gradit orator ad Lala orator of a futio b dotd by = ( x, x, x 3 T ad =. x x x3 Iitial-valu oditios ar statd as follows, (X, = (X, (X, = (X, T(X, = T (X, X Ω. (5 Boudary-valu oditios ar dfid by ψ γ = Ω γ = Ω γ = T Ω γ =, Ω (X, t Ω J, (6 wr Ω dots t boudary of Ω, ad γ is t outward uit ormal vtor of Ω. 6

3 Joural of Matmatis Rsar Vol. 7, No. 3; 5 Numa boudary valu oditios dtrmi a family of ltri fild ottial futios diffrd by a ostat. So a additioal oditio is sulid to gt a uiqu formulatio ψdx =, t J, (7a A omatibility oditio is giv, I gral, t roblm (-(7 is suosd to b ositiv dfiit, Ω Ω ( N dx =, t J. (7b (C < D D s (X D, < µ µ s (X µ, s =, ; < ρ ρ(x ρ, (8 wr D, D, µ, µ, ρ ad ρ ar ositiv ostats. R i (,, T(i =, ar Lisitz otiuous i ε -igbor trd by (,, T. Assum tat xat solutios of (-(7 ar rgular, (R ψ L (J; W k (Ω,,, T L (J; W l (Ω, τ,, T τ t L (J; L (Ω. (9 3. Notatios ad Praratios For simlifiatio, lt t omutatioal domai Ω = (x, x, x 3 < x <, < x <, < x 3 < } b domosd ito two uboids Ω = (x, x, x 3 < x < /, < x <, < x 3 < }, Ω = (x, x, x 3 / < x <, < x <, < x 3 < }, wos itrior boudary is = (x, x, x 3 x = /, < x <, < x 3 < }, as i Fig.. W us aroximatio of drivativs of dlta futio at svral oits i tis work, ad for furtr x 3 (/,, x (/,, x Figur. Skt of domai domositio artitio Ω, Ω, rfr w dfi two sial futios Φ ad Φ 4 as follows (Dawso & Duot, 99, 994; Dawso, Du & Duot, 99; Dawso & Du, 99. x, x, Φ (x = x, x, (a, otrwis. (x /, x, 5x /4 7/6, x, Φ 4 (x = 5x /4 7/6, x, (b (x /, x,, otrwis. Not tat if (x is a olyomial of dgr at most o, t (x Φ (x dx = (, (a 63

4 Joural of Matmatis Rsar Vol. 7, No. 3; 5 ad if (x is a olyomial of dgr at most tr, t (x Φ 4 (x dx = (. Dfiitio : For ay umbr H (,, dfi a ormalizd futio (b Φ(x = Φ m ((x //H/H, m =, 4. ( Lt N, j b a fiit-dimsioal fiit lmt sa H (Ω j ( j =,, ad lt N (Ω b a fiit-dimsioal fiit lmt subsa of l dgr of L (Ω. Morovr, if a futio v oms from N, t v Ω j must blog to N, j. Not tat a giv futio v N (Ω as a wll-dfid jum [v] o, [v] (,x,x 3 = v(, x, x 3 v(, x, x 3. (3 Dfiitio : Dfi a biliar maig D s (u, v, D s (u, v = D s (X u vdx dx dx 3 λ s uvdx dx dx 3, s =,, T, (4 Ω Ω Ω Ω wr u, v H (Ω j, j =,, ad D s (X(s =, ar all ositiv dfiit futios, D T =, ad λ s is a ositiv ostat. Dfiitio 3: A itgral orator aroximatig ormal drivativ at itrior boudary is dfid as follows wr Φ formulatd by ( is usd. B(ψ(, x, x 3 = Φ (x ψ(x, x, x 3 dx, (5 Lt (, dot a rodut ir i L (Ω Ω, ad omit t subsrit otatio (ψ, ρ = (ψ, ρ Ω as Ω Ω = Ω. Cosidr a futio ψ of H (Ω ad H (Ω, ad giv a sial orm, Not w av (D(x, x, x 3 B(ψ, [ψ] = ψ s = D s (ψ, ψ H D s [ψ] L (, s =,, T. (6 D(, x, x 3 Φ (x ψ(x, x, x 3 dx [ψ](, x, x 3 dx dx 3, Φ (x ψ(x, x, x 3 dx = ψ(x, x, x 3 Φ(x Φ(x ψ x (x, x, x 3 dx (D(x, x, x 3 B(ψ, [ψ] = H = H [ψ](, x, x 3 Φ(x ψ x (x, x, x 3 dx, D(, x, x 3 [ψ] (, x, x 3 dx dx 3 D(, x, x 3 T sod trm is formulatd i aotr xrssio, D / (, x, x 3 H Φ(x ψ x (x, x, x 3 dx [ψ](, x, x 3 dx dx 3. D / ( H, x, x 3 Φ(x ψ x (x, x, x 3 dx [ψ](, x, x 3 dx dx 3 D / (, x, x 3 ( Φ (x dx / ( H [ψ](, x, x 3 dx dx 3 ( / ( D( 3H, x, x 3 [ψ] ( /, x, x 3 dx dx 3 H D( H, x, x 3 ψ x (x, x, x 3 dx dx dx 3 D( H, x, x 3 ψ x (x, x, x 3 dx / /. (7 64

5 Joural of Matmatis Rsar Vol. 7, No. 3; 5 Not tat t valu of D(x, x, x 3 at x = / is dfid by t w av = H H ( M H D(, x, x 3 = D(x, x, x 3 (x D x (ξ (x, x, x 3, H D(, x, x 3 ψ x (x, x, x 3 dx dx dx 3 [D(x, x, x 3 (x H D (ξ (x, x, x 3 ]ψ x x (x, x, x 3 dx dx dx 3 H wr M = max x ( H, H (x, x 3 (, (, Cotiu to fid a ositiv ostat M su tat D(x, x, x 3 ψ H x (x, x, x 3 dx dx dx 3, D x (ξ(x,x,x 3 D(x,x,x 3. D s (ψ, ψ (D s B(ψ, [ψ] M ψ s, s =,, T, (8a or Similarly, w av t followig stimats for t T, ψ s M D s (ψ, ψ (D s B(ψ, [ψ] }, s =,, T. B(ψ L ( M H 3 ψ, B(ψ L ( M H ψ,, u(, t B(u(, t γ M 3 H m, L ( (8b (9a (9b (9 wr M, M, M 3 ar ositiv ostats, m =, 4, ad u γ dots t ormal drivativ of u aross itrior boudary. 4. Domai Domositio Sm Modifid wit Caratristi Fiit Elmt T aralll rodurs ar illustratd i tis stio, wi osists of a fiit lmt sm for t ltri fild ottial ad a domai domositio sm modifid wit aratristis for t ltro otratio, ol otratio ad tmratur. Firstly, rwrit t ottial quatio (, ( ψ, v = α ( N, v, v H (Ω, t J, (ψ, =. (a (b Sodly, (-(4 ar rformulatd as follows t = (D (X µ u αµ ( N u µ R (,, T, (X, t Ω J, (a t = (D (X µ u αµ ( N u µ R (,, T, (X, t Ω J, (b ρ(x T t = ( T (D (X µ (Xu (D (X µ (Xu } u, (X, t Ω J, ( wr u = ψ. Lt W = W ψ dot a fiit lmt sa of k dgr, ad lt ψ b t artitio st, t w av if v v L v W (Ω M v k k ψ. ( 65

6 Joural of Matmatis Rsar Vol. 7, No. 3; 5 Notig tat t ltri fild ottial ags vry slow wit rst to t, w a adot a larg st i its alulatios. Wil i t omutatios of t ltro otratio, ol otratio ad tmratur small sts ar adotd. Som otatios ar giv as follows,, tim st of t otratio quatio, ψ, tim st of t ottial quatio, j = ψ /, t =, t m = m ψ, ψ = ψ(t, ad ψ m = ψ(t m. For a ottial futio ψ(x, t, dfi a liar xtraolatio Eψ of t valus at t losst two tim lvls to t, Eψ ψ, t = t ( γ/ jψ m γ/ jψ m, t m < t < t m, t = t m γ. wr t subsrits orrsod to ottial tim lvls, sursrits to otratio tim lvls. T xtraolatio a b alid for t vtor futio u = ψ, tat is to say Eu is dfid similarly to Eψ ad as som sam rortis. Eltri fild ottial quatio ( is aroximatd by t followig fiit lmt sm ( ( ψ,m, v = α,m,m N m, v, v W, (3a ( ψ,m, =. (3b T t gradit of ottial is omutd by U,m = ψ,m. T ltro otratio quatio (, ol otratio quatio (3 ad at odutio quatio (4 ar osidrd. Multily tm by ω N (Ω, aly itgratio by arts, t w av wak forms of domai domositio as follows ( t, ω Ω ( µ u, ω Ω ( D, ω Ω ( D γ, [ω] ( u µ, ω Ω = α ( µ ( N, ω Ω ( R (,, T, ω Ω, ω N (Ω, (4a ( t, ω Ω ( µ u, ω Ω ( D, ω Ω ( D γ, [ω] ( u µ, ω Ω = α ( µ ( N, ω Ω ( R (,, T, ω Ω, ω N (Ω, (4b ( T ρ t, ω Ω ( T, ω Ω ( T γ, [ω] = ([ (D µ ψ (D µ ψ ] ψ, ω Ω, ω N (Ω. (4 Notig tat t flow movs sstially alog t aratristis, w aly a modifid rodur of aratristis for t first-ordr yrboli art of ( ad (3 to ovrom umrial disrsio ad osillatio. Tis kid of mtod as a ig ordr auray ad a fi stability i umrial simulatio ad a larg tim st a b adotd (Ewig 983; Ewig, Russll & Wlr, 984; Douglas & Russll, 98; Russll, 985; Douglas & Yua, 986; Ewig, Yua & Li, 989. Lt τ = τ(x, t dot t uit vtor i aratristi dirtio ( µ u, µ u, µ u 3, [ ad lt τ = τ (X, t b t uit vtor i (µ u, µ u, µ u 3, -dirtio. Lt Φ s = µ s u ] /, s =,, ad omut t aratristi dirtioal drivativ by Φ T (4a ad (4b ar agd ito = τ t µ u, Φ = τ t µ u. ( Φ, ω Ω τ ( D, ω Ω ( D γ, [ω] ( u µ, ω Ω = α ( µ ( N, ω Ω ( R (,, T, ω Ω, ω N (Ω, (5a ( Φ, ω Ω τ ( D, ω Ω ( D γ, [ω] ( u µ, ω Ω = α ( µ ( N, ω Ω ( R (,, T, ω Ω, ω N (Ω. (5b Aroximat τ = τ (X, t by a bakward diffr quotit i τ -dirtio, (X (X µ u, τ ( µ u / 66

7 Joural of Matmatis Rsar Vol. 7, No. 3; 5 ad aroximat τ (X i τ, (X (X µ u. τ ( µ u / T t rodurs of domai domositio wit aratristi fiit lmt ar statd as follows to aroximat t otratios (5, ( ê, ω Ω ( D, ω Ω ( D B(, [ω ] = α ( µ ( ˆ ê N, ω Ω ( ê EU µ, ω Ω ( R (ê, ˆ, T, ω ( ˆ, ω Ω ( D, ω Ω ( D B(, [ω ] = α ( µ ( ˆ ê N, ω Ω ( ˆ EU µ, ω Ω ( R (ê, ˆ, T, ω Ω, ω N (Ω, Ω, ω N (Ω, wr ê = ( ˆX, ˆX = X µ EU, ˆ = ( ˆX, ˆX = X µ EU, U,m = ψ,m, ad U,m = ψ,m. Notig tat t valus of ˆX ad ˆX mayb li outsid t boudary Ω, wil w a us t rfltio of t boudary baus of Numa oditios (6 ad dfi t valus of ê ( ˆX ad ˆ ( ˆX (Douglas & Yua, 986; Ewig, Yua & Li, 989. Similarly, t at odutio quatio is disrtizd by t mtod of domai domositio wit fiit lmt, (6a (6b ( T ρ T, ω Ω ( T, ω Ω ( B(T, [ω ] = ([ (D µ EU (D µ EU ] EU, ω Ω, ω N (Ω. (6 T rogram works for a tim st as follows. Firstly, giv t iitial aroximatio,, T }, ψ,, U, } is omutd by t fiit lmt sm (3. Sodly,,, T },,, T },, i, i, T i } ar omutd i aralll by t domai domositio sm wit fiit lmt (6. It otius to gt ψ,, U, } by t valus j, j, T j } =,,,, T, } ad (3, ad as a rsult, t solutios j, j, T j },,,,,, T, } ar obtaid i aralll by (6. Fially, umrial solutios a b sow aftr a sris of abov omutatio ad ty xist ad ar sol by t ositiv dfiit oditio (C. 5. Covrg Aalysis Svral subsidiary lliti rojtios ar itrodud first i tis stio, ad lt ψ} : J N b dfid o J = (, T], ( (ψ ψ, v =, v W, (7a ( ψ ψ, =. (7b Lt θ = ψ ψ ad η = ψ ψ b rror futios ad ty ar stimatd as follows (Ciarlt, 978; Wlr, 973, θ ψ θ M ψ k k ψ, (8a θ θ t ψ t M ψ k ψ } t k ψ. (8b k Dfi lliti maig ẽ,, T} : J N N N as follows, D ((, t ẽ(, t, ω λ ((, t ẽ(, t, ω =, ω N, D ((, t (, t, ω λ ((, t (, t, ω =, ω N, D T (T(, t T(, t, ω λ T (T(, t T(, t, ω =, ω N, (9a (9b (9 wr t lliti orator is oriv i H (Ω for suffiitly larg ostat λ, λ ad λ T. Iitial aroximatio is giv by = ẽ(, = (, T = T(. (3 67

8 Joural of Matmatis Rsar Vol. 7, No. 3; 5 Itrodu rror futios dotd by ξ = ẽ, ζ = ẽ, ξ =, ζ =, π = T T ad σ = T T. By Galrki mtod (Ciarlt, 978; Wlr, 973, ζ ζ M l l, (3a ζ ζ M l l, (3b σ σ M T l l, (3 ζ ζ t t M l } t l ζ t ζ t M l } t l σ σ t t M T l T } t l l, (3a l, (3b l, (3 wr Ω = Ω Ω, ad dots t saial st of fiit lmt sa N (Ω. Torm Suos tat xat solutios of (-(7 ar suitably rgular, ψ L (J; W k (Ω,,, T L (J; W l (Ω,,, T L (J; L (Ω. Adot t aralll rodur modifid aratristi fiit lmt (3 ad τ τ t (6 o Ω ad Ω, ad suos tat t rlatio of artitio aramtrs is = O(, ψ = O(, M H, l = o(h, (33 wr M is a ositiv ostat ad k ad l ar t idxs of fiit lmt sa. W av ψ ψ L (J;W L (J;L (Ω L (J;L (Ω T T L (J;L (Ω M H m/ k ψ } l H l (34, wr g L (J;X = su g X ad t ostat M is ddt o ψ,,, T ad tir drivativs. T Proof. T ottial futio ψ is osidrd firstly, ad ψ ψ is stimatd. By ( (t = t m, (7 (t = t m ad (3, w av η m M ξ,m ξ,m } (l. (35 T ltro otratio is disussd sodly. Subtrat (6a from (4a (t = t, ad us (9a (t = t, ( ξ ξ, ω ( D ξ (, ω D B(ξ, [ω ] ( = ( µ EU, ω λ ζ, ω t α ( [ µ ( ˆ ê N ( N ] ([ê, ω EU u ] µ, ω (36 ( R (ê, ˆ, T R ( (,, T, ω µ (EU u, ω ( ˆξ ξ ( ζ ˆζ (, ω, ω (D γ B(, [ω ], ω N, wr ê = ( ˆX, ˆξ = ξ ( ˆX, ˆζ = ζ ( ˆX ad ˆX = X µ EU. Tak ω = ξ as a tst futio i (36, ( ξ ξ, ξ ( ( D ξ, ξ λ ξ, ξ = ( D B(ξ ξ, [ξ ] ( ( D γ γ ( D B(ζ, [ξ ] ( ( ˆξ ξ, ξ t ( ( λ ξ, ξ λ ζ, ξ ( D B(ξ, [ξ ], [ξ ] ( (D µ EU, ξ ( R (ê, ˆ, T γ B( ( ζ, [ξ ] ˆζ, ξ 68

9 Joural of Matmatis Rsar Vol. 7, No. 3; 5 ( [ R (,, T, ξ α µ ( ˆ ê N ( N ], ξ ([ ê EU u ] ( ( µ, ξ µ EU u (37, ξ. T trms o t lft-ad sid ar stimatd as follows, ( ξ ξ = ξ ξ } ξ ξ, (38a, ξ ( ( D ξ, ξ λ ξ, ξ ( D B(ξ, [ξ ] M ξ T trms o t rigt ad sid of (37 ar stimatd latr. Usig (9, w av for ositiv ostats M i (i =,, 3, ( D B(ξ ξ, [ξ ] M B(ξ ξ L ( [ξ ] L ( M H 3/ ξ ξ H / ξ M H ξ ξ ε ξ, (39a [ ] (D γ, [ξ ] M γ γ γ [ξ ] L ( L ( M H / ξ M ( H ε ξ, (39b [ ] (D γ B(, [ξ ] M 3 H m ε ξ, (39 ( D B(ζ, [ξ ] M 3H ζ ε ξ M3 H (l ε ξ. (39d Tak suffiitly small, T A idutio yotsis is giv,. (38b M H. (4 ξ ξ M H ξ ξ. (4 su η m,, su max ξ,, ξ, }, π,,. (4 m [(L / j] L Otr trms o t rigt ad sid ar stimatd. Not tat ê µ EU Φ t, ad Φ is boudd baus of (4, w av ( ê wr J = (t, t ]. t µ EU, ξ Ω M τ (X,t ( ˆX,t L (J ;L (Ω dtdx, τ ξ, (43 It follows from gativ orm stimat ad idutio yotsis (4, ( ζ ˆζ ( ζ, ξ ζ ( ζ =, ξ ˆζ, ξ M ( ζ t ξ } ξ ε ξ, (44a L (J ;L (Ω ( ˆξ ξ, ξ M ξ ε ξ, (44b ( ( } λ ξ, ξ ζ, ξ M (l ξ }, (44 ( R (ê, ˆ, T R (,, T, ξ M ( k ψ (l ξ ξ ξ } π, (44d 69

10 Joural of Matmatis Rsar Vol. 7, No. 3; 5 By (4, w a s tat, ad T ar boudd, t w av α ( [ µ ( ˆ ê N ( N ], ξ M ( k ψ (l ξ ξ ξ }, Colltig (38-(44, w driv ([ê EU u ] ( ( µ, ξ µ EU u, ξ M ( k ψ (l ξ ξ,[/ j] ξ,[/ } j]. ξ ξ } ξ M M ( ζ t L (J ;L (Ω τ k ψ (l ( ( H L (J ;L (Ω H (l H m ξ ξ ξ π ξ,[/ j] ξ,[/ } j] ε ξ ξ }. (44 (44f (45 It otius to stimat t ol otratio. Subtrat (6b from (4b (t = t, ad us (9b (t = t, ξ ξ, ω ( ( D ξ, ω D B(ξ, [ω ] ( = ( µ EU, ω λ ζ, ω t α ( [ µ ( ˆ ê N ( N ] ([, ω ˆ EU u ] µ, ω ( R (ê, ˆ, T R ( (,, T, ω µ (EU u, ω ˆξ ξ, ω ζ ˆζ ( (, ω D γ B(, [ω ], ω N, (46 wr ˆ = ( ˆX ad ˆX = X µ EU. Tst futio ω = ξ is substitutd i (46, ad it follows from a similar disussio ad aalysis, ξ ξ } ξ M M ( ζ t L (J ;L (Ω τ k ψ (l ( L (J ;L (Ω H (l H m ξ ξ ξ π ξ,[/ j] ξ,[/ } j] ε ξ ξ }. (47 Fially, rror quatio of tmratur is drivd from (4, (ρ π π, ω ( π, ω (B(π, [ω ] ( [ T T ] T ( = ρ, ω λ T σ ([, ω (D t µ EU D µ EU EU (D µ u D µ u u ], ω (ρ σ σ ( T, ω γ B(T, [ω ], ω N. (48 7

11 Joural of Matmatis Rsar Vol. 7, No. 3; 5 Tak ω = π as tst futio i (48, π π } M π M ( σ t L (J ;L (Ω T t L (J ;L (Ω k ψ (l ( H (l H m π ξ ξ ξ,[/ j] ξ,[/ j] }. (49 Colltig (45, (47 ad (49, multilyig bot sids of t rsultig quatio by, summig o ( L, ad otig tat ξ = ξ = π =, w av ξ L ξ L π L L ξ ξ π } = M ζ t ζ L (J;L (Ω t σ L (J;L (Ω t L (J;L (Ω ξ ξ,[/ j] ξ,[/ ] j] ( τ T t ( k ψ (l L (J;L (Ω L [ π ξ = L (J;L (Ω H (l H m. τ L (J;L (Ω (5 Usig Growall Lmma, ξ L ξ L π L L ξ ξ π } = M ( k ψ (l H (l H m}. (5 Trfor, it follows from (35 ad (5, ψ,m ψ m M k ψ l H l H m/}. (5 It rmais to tstify t idutio yotsis (4. It is rigt as is qual to baus of ξ = ξ = π =. Assum (4 olds for ay ositiv itgr btw ad a giv ositiv itgr L. By (5 (5 ad t rstritio (4, it is asy to s tat (4 olds for = L. Basd o t rsultig stimats (5 ad (5, ad (3, Torm is rovd. T mtod disussd i tis ar a b xtdd to tr-dimsioal as, su as Mst modl (s Fig. (H & Wi,989. T tiqu of domai domositio is vry imortat i umrial simulatio of atual aliatios, as sow i Fig., wr omutatioal rgio Ω is dividd ito fiv subdomais, Ω = 5 Ω i (H & Wi, 989; Si, ; Yua, 9, Numrial Examl I tis stio o umrial xaml is rstd to tstify t ffiiy of t aralll algoritm disussd abov. Cosidr t followig modl roblm, t u x ( D(x, t = f (, x, t, < x <, < t < T, (53a x x (x, = os(πx, x, (, t = (, t =, t T. x x (53 Som rlatd futios ad artitio aramtrs ar giv as follows, u = x t, D(x, t =.x t, = t os(πx, f = t os(πx π t x si(πx.4π 3t x ( si(πx πx os(πx, H = 4, = ad T =.5. i= (53b 7

12 Joural of Matmatis Rsar Vol. 7, No. 3; 5 sour grid lak z y x Figur. Domositio of omutatioal domai of tr-dimsioal Mst modl, Ω = 5 Ω i i= T omutatioal itrval is domosd ito two subitrvals, [, ] = [,.5] [.5, ], ad itrior boudary is =.5. Absolut rrors of umrial data ar sow i Tabl. Tabl. Absolut Errors x =.5 x =.5 x =.45 x =.55 x =.75 x =.95 = / E 3.364E E E 3.364E E 3 = /8 5.35E 3.89E E E 3.89E E 3 = / E 3.5E E E 3.5E E 3 Numrial data ar osistt wit tortial aalysis from Tabl, ad rror rsults of aroximatio to ormal drivativ at ir boudary, x (.5 = T si π = ar sow i Tabl. Tabl. Error rsults of aroximatio to orma drivativ at itrior boudary B = /4 6.78E 6 = / E 5 = / E 4 Tim osts (uit: sod i umrial omutatio of domai domositio mtod (DDM ar illustratd i Tabl 3 omarig wit aotr omutatio stratgy, o-domai domositio mtod (NDDM. Tabl 3. Tim osts omariso (uit: sod DDM NDDM = / = / = / = / From Tabl 3, it is asy to olud tat domai domositio boms mor ad mor ffiit as t artitio boms mor rfid ad t systm of algbrai quatios boms largr ad largr. 7. Colusio ad Disussio I tis ar a domai domositio of fiit lmt is disussd to simulat smiodutor trasit roblms of at odutor. Matmatial modl, ysial basi ad itratioal rsar ar itrodud i t first stio. I t sod stio som rimary lliti rojtios, otatios ad basi rortis ar statd. T t 7

13 Joural of Matmatis Rsar Vol. 7, No. 3; 5 rodurs of domai domositio modifid wit aratristi fiit lmt ar struturd i t tird stio. Tortial aalysis is sow ad a otimal ordr stimat i L orm is drivd i t fourt stio. I t last stio a umrial xrimt is illustratd to tstify t tortial rsult ad som advatags of tis mtod ar oludd as follows. Firstly, tis mtod a b alid i larg-sald umrial omutatio of tr-dimsioal roblm wit omliatd domai. Sodly, idustrial softwar is mad asily o tis mtod. Tirdly, t algoritm works o modr aralll omutrs ad givs umrial omutatio of ig ordr auray ad ig ffiiy i aralll for smiodutor trasit roblms. Fially, tis ar is a xasiv argumt of my arly rsar (Yua,, ad a mixd lmt mtod ombid wit domai domositio is argud firstly i t rfr (Yua, to solv t roblm of smiodutor dvi. I tis ar t otimal rror stimat i L orm is drivd i tortial aalysis but ot obtaid i t rvious work. Akowldgmts T autors xrss tir d ariatio to Prof. J. Douglas Jr, ad Prof. Jiag Lisag for tir lful suggstios i t srial of rsar o umrial simulatio of smiodutor dvi. Rfrs Bak, R. E., Cougra, W. M. Jr., Fitr, Wolfgag, Gross, E. H, Ros, D. J., & Smit, R. K. (985. Trasit simulatio of silio dvis ad iruits. IEEE Tras: Comutr-Aidd Dsig of Itgratd Ciruits ad Systm, 4(4, Ciarlt, P. G. (978. T fiit lmt mtod for lliti roblms. Amstrdam: Nort-Hollad. Dawso, C. N., Du, Q., & Duot, T. F. (99. A fiit diffr domai domositio algoritm for umrial solutio of t at quatio. Mat. Com., 57(95, Dawso, C. N., & Duot, T. F. (99. Exliit/Imliit osrvativ Galrki domai domositio rodurs for araboli roblms. Mat. Com., 58(97, -34. Dawso, C. N., & Du, Q. (99. A fiit lmt domai domositio mtod for araboli quatios. Ri Tial rort TR9-, Dt. of Matmatial Sis, Ri Uivrsity, Housto, Txas. Dawso, C. N., & Duot, T. F. (994. Exliit/Imliit, osrvativ domai domositio rodurs for araboli roblms basd o blok-trd fiit diffrs. SIAM J. Numr. Aal., 3(4, Douglas, Jr. J., & Russll, T. F. (98. Numrial mtod for ovtio-domiatd diffusio roblms basd o ombiig t mtod of aratristis wit fiit lmt or fiit diffr rodurs. SIAM J. Numr. Aal., 9(5, Douglas, Jr. J., & Yua, Y. R. (986. Numrial simulatio of immisibl flow i orous mdia basd o ombiig t mtod of aratristis wit mixd fiit lmt rodur. T IMA Vol. i Mat. ad its Al.,, 9-3. Douglas, Jr. J., & Yua, Y. R.(987. Fiit diffr mtods for trasit bavior of a smiodutor dvi. Mat. Ali. Com., 6(, Ewig, R. E. (983. T Matmatis of Rsrvoir Simulatio. SIAM, Piladlia. Ewig, R. E., Russll, T. F., & Wlr, M. F. (984. Covrg aalysis of a aroximatio of misibl dislamt i orous mdia by mixd fiit lmts ad a modifid mtod of aratristis. Com. Mt. Al. M. Eg., 47(-, Ewig, R. E., Yua, Y. R., & Li, G. (989. Tim stig alog aratristis of a mixd fiit lmt aroximatio for omrssibl flow of otamiatio by ular wast diosal i orous mdia. SIAM J. Numr. Aal., 6(6, Gumml, H. K. (964. A slf-osistt itrativ sm for o-dimsioal stady-stat trasistor alulatio. IEEE Tras: Eltro Dvi, (, H, Y., & Wi, T. L. (989. Comutr simulatio mtod for smiodutor dvi. Si Prss, Bijig. Jrom, J. W. (994. Matmatial tory ad aroximatio of smiodutor modls. Piladlia, SIAM. Lou, Y. (995. O basi smiodutor quatio wit at odutio. J. Partial. Diff. Eqs., 8(, Russll, T. F. (985. Tim Stig Alog Caratristis wit Iomlt Itratio for a Galrki Aroxi- 73

14 Joural of Matmatis Rsar Vol. 7, No. 3; 5 matio of Misibl Dislamt i Porous Mdia. SLAM J. Numr. Aal., (5, Sidma, T. I. (986.Tim ddt solutio of a oliar systm arisig i smiodutor tory II: bouddss ad riodiity. Noliar Aalysis: Tory, Mtods & Aliatios, (5, Si, M. (. Pysis of modr smiodutor dvi. Si Prss, Bijig. Wlr, M. F. (973. A rior L -rror stimats for Galrki aroximatios to araboli diffrtial quatios. SIAM J. Numr. Aal., (4, Yua, Y. R., Dig, L. Y.,& Yag, H. (98. A w mtod ad tortial aalysis of umrial aalog of smiodutor. Ci. Si. Bull., 7(7, Yua, Y. R. (99. Caratristis mtod wit mixd fiit lmt for trasit bavior of smiodutor dvi. Ci. Si. Bull., 36(7, Yua, Y. R.(99. T aroximatio of t ltroi ottial by a mixd mtod i t simulatio of smiodutor. J. Systms Si. Mat. Si., (, 7-. Yua, Y. R. (993. Fiit lmt mtod ad aalysis of umrial simulatio of smiodutor dvi. Ata Mat. Si., 3(3, 4-5. Yua, Y. R. (996. Fiit diffr mtod ad aalysis for tr-dimsioal smiodutor dvi of at odutio. Si. Cia Mat., 39(, 4-5. Yua, Y. R. (. Caratristi fiit diffr fratioal st mtods for tr-dimsioal smiodutor dvi of at odutio. Ci. Si. Bull., 45(, 3-3. Yua, Y. R. (5. Fiit diffr fratioal st mtod for trasit bavior of a smiodutor dvi. Ata Mat. Si.(Egl.Sr., 5(3, Yua, Y. R. (8. Modifiatio of uwid fiit diffr fratioal st mtods by t trasit stat of t smiodutor dvi. Numr. Mtods Partial Diffrtial Eq., 4(, Yua, Y. R. (9. Rt rogrss i umrial mtods for smiodutor dvis. Cis J. Comutatioal Pysis, 6(3: Yua, Y. R. (. T modifid mtod of aratristis wit mixd fiit lmt domai domositio rodurs for t trasit bavior dvi. Numr. Mtods Partial Diffrtial Eq., 8(, Coyrigts Coyrigt for tis artil is rtaid by t autor(s, wit first ubliatio rigts gratd to t joural. Tis is a o-ass artil distributd udr t trms ad oditios of t Crativ Commos Attributio lis (tt://rativommos.org/liss/by/3./. 74