The FFTRR-based fast decomposition methods for solving complex biharmonic problems and incompressible flows

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1 IMA Joual of Numeical Aalysis 26) 36, doi:.93/imaum/dv33 Advace Access publicatio o July 2, 25 The FFTRR-based fast decompositio methods fo solvig complex bihamoic poblems ad icompessible flows Aditi Ghosh ad Pabi Daipa Depatmet of Mathematics, Texas A&M Uivesity, College Statio, TX 77843, USA Coespodig autho: pabi.daipa@math.tamu.edu aditi.ghosh@math.tamu.edu [Received o 2 Mach 24; evised o 25 Mach 25] I this wok, we peset seveal computatioal esults o the complex bihamoic poblems. Fist, we deive fast Fouie tasfom ecusive elatio FFTRR)-based fast algoithms fo solvig Diichlet- ad Neuma-type complex Poisso poblems i the complex plae. These ae based o the use of FFT, aalysis-based RRs i Fouie space, ad high-ode quadatue methods. Ou secod esult is the applicatio of these fast Poisso algoithms to solvig fou types of ihomogeeous bihamoic poblems i the complex plae usig decompositio methods. Lastly, we apply these high-ode accuate fast algoithms fo the complex ihomogeeous bihamoic poblems to solvig Stokes flow poblems at low ad modeate Reyolds umbe. All these algoithms ae iheetly paallelizable, though oly sequetial implemetatios have bee pefomed. These algoithms have theoetical complexity of the ode Olog N) pe gid poit, whee N 2 is the total umbe of gid poits i the discetizatio of the domai. These algoithms have may othe desiable featues, some of which ae discussed i the pape. Numeical esults have bee peseted which show pefomace of these algoithms. Keywods: complex Bihamoic equatio; complex Poisso equatio; fast algoithms; stokes equatios; icompessible flows; FFT; ecusive elatios; umeical implemetatio i Matlab.. Itoductio Bihamoic poblems aise i may fields of classical physics ad egieeig, icludig the theoy of elasticity, slow viscous flows, sedimetatio ad so o. They also aise as a itemediay step i solvig some liea ad oliea elliptic ad paabolic poblems such as poblems ivolvig the Navie Stokes equatios ad so o; see Guazzelli & Mois 22), Love 927) ad Muskhelishvili 977). Theefoe, thee has bee extesive eseach o the developmet of efficiet umeical algoithms to solve such poblems usig vaious methods such as fiite diffeece/fiite elemet methods; see Bjostad 983), Baess & Peiske 986), Cheg et al. 2), Lai 25), Mok 987) ad Peiske 988), ad itegal equatio methods; see Geebaum et al. 992), Geegad & Kopiski 998), Geegad et al. 996), Kopiski 999) ad Mayo 992), to ame a few of the methods. To the best of ou kowledge, howeve, thee ae o solves to-date of the bouday value poblems fo the ihomogeeous bihamoic equatio z z ) 2 ω = f z, z), i the complex z-plae which is the physical x y plae though the assigmet z = x + iy. We efe this equatio as the complex bihamoic equatio below. Whe the souce tem f is eal ad pescibed bouday values ae eal, we ecove the covetioal ihomogeeous bihamoic poblem i the physical plae. Thus, the complex bihamoic equatio is a moe geeal case, ad ca be used to solve c The authos 25. Published by Oxfod Uivesity Pess o behalf of the Istitute of Mathematics ad its Applicatios. All ights eseved.

2 THE FFTRR-BASED FAST DECOMPOSITION METHODS 825 may poblems i the plae icludig poblems i plae elasticity ad Stokes flow. Hee the poblems ae i the physical plae, ad ca as well be solved with may othe fast solves fo eal bihamoic equatio. Howeve, may poblems i plae elasticity see Muskhelishvili, 977; Wag et al., 99) ae fomulated as complex bihamoic poblems whose solutios ca be computed usig the fast algoithms developed i this pape. Ou effot i developig these algoithms is fowad-lookig, ad we hope that they fid applicatios i a vaiety of aeas icludig Stokes s flow. It is cojectued hee that this equatio ca play a impotat ole i applied mathematics, i paticula i fluid dyamics ad elasticity theoy, aalogous to the ole of Cauchy Riema equatios i two-dimesioal potetial flow. I this pape, fast algoithms fo the complex ihomogeeous bihamoic poblems ae built fom usig the decompositio method i which each of these poblems is decomposed ito two complex Poisso poblems, ad also oe complex homogeeous bihamoic poblem i some cases as we will see late. Ou algoithms fo the Poisso poblems ae deived fom aalysis of sigula itegal epesetatios of thei solutios based o classical Gee s fuctio appoach. Such epesetatios fo the complex Poisso poblems wee made available oly vey ecetly, as ecetly as the yea 28, by Begeh 27b). The use of quadatue methods to compute the complex sigula itegals i these epesetatios is computatioally vey expesive fo lage-scale poblems ad has poo accuacy. The asymptotic complexity of staightfowad implemetatio of the quadatue method is usually ON 2 ) pe gid poit, whee N 2 is the total umbe of gid poits i the discetizatio of the domai which is take to be the uit disk. I ode to educe this complexity ad to obtai high-ode accuate solutios, we fist develop fast Fouie tasfom ecusive elatio FFTRR)-based fast complex Poisso solve. This makes use of the FFT ad aalysis-based RRs i the Fouie space esultig i a complexity of the ode Olog N) pe gid poit. This complexity is a sigificat impovemet ove ON 2 ) pe gid poit. This FFTRR-based appoach to solvig complex poblems ivolvig Cauchy Riema ad Beltami equatios Daipa, 992, 993) ad eal Poisso poblems Boges & Daipa, 2) has bee itoduced ealie by oe of the authos P.D.). We summaize some featues of these algoithms. ) The algoithms have vey low computatioal complexity: Olog N) pe gid poit. 2) The costat hidde behid the ode estimate of computatioal complexity of these algoithms is also vey small as show i this pape. 3) The RRs allow oe to defie highe-ode itegatio methods i the adial diectio without the iclusio of additioal gid poits. 4) The ode of accuacy of the solutio pimaily depeds o the umeical itegatio scheme of oe-dimesioal itegals ivolved i the RRs of these fast algoithms. It is secod thid)- ode whe tapezoidal Simpso s) itegatio fomulae ae used. The ode of accuacy ca be iceased usig Eule Maclaui fomulas fo itegatio as show i this pape. The complexity estimate of these algoithms is idepedet of the ode of accuacy. 5) These algoithms ae paallelizable by costuctio see Boges & Daipa, 2, 2). 6) A pio selectio of the umbe of gid poits based o desied accuacy. 7) These algoithms ae easily implemeted. These algoithms fo the complex Poisso ad the complex ihomogeeous bihamoic poblems have bee implemeted usig MATLAB. The Matlab code has bee tested usig some caoical

3 826 A. GHOSH AND P. DARIPA examples, ad most of the featues of the algoithms metioed above have bee validated. Usig ou fast bihamoic solve, steady-state solutios of Navie Stokes equatios have bee computed withi a disk fo a vaiety of bouday coditios at low ad modeate Reyolds umbes. The accuacy of these solutios have bee validated agaist kow solutios of these poblems i the liteatue. Numeical esults ae also peseted, validatig the eo estimate ad its use i a pio selectio of mesh size based o a pescibed eo. This has bee doe fo the Poisso poblems ad the extesio to bihamoic case is staightfowad. The pape is the laid out as follows. I Sectio 2, we defie the Diichlet ad Neuma poblems fo Poisso equatios i the complex plae ad peset itegal epesetatios of thei solutios. Some theoems alog with thei poofs ad RRs esultig fom the aalysis of these itegal epesetatios ae give i this sectio. Simila developmet of the itegal epesetatios, RRs ad theoems with egad to the fou bihamoic poblems ae give i Sectio 3. These theoems ad RRs ae used to develop the fast algoithms fo Poisso ad bihamoic poblems which ae peseted alog with thei computatioal complexities i Sectio 4. The MATLAB code ecapsulatig these algoithms has bee used to pefom some umeical calculatios. Numeical esults o the validatio ad accuacy check of these algoithms fom computatio of some caoical Poisso ad bihamoic poblems ae give i Sectio 5. I Sectio 6, a cude but useful eo estimate fo the fast Poisso algoithms is give. Numeical esults ae peseted validatig the eo estimate ad its use i a pio selectio of mesh size based o a pescibed eo. I Sectio 7, these bihamoic algoithms have bee applied to solvig steady Navie Stokes equatios iside the uit disk at low ad modeate Reyolds umbes. Numeical esults ae peseted thee. Fially, we coclude i Sectio Complex Poisso poblems I this sectio, we descibe the mathematical fomulatio simila to the pevious woks of Daipa ad collaboatos Daipa, 992, 993; Boges & Daipa, 2) fo the FFT ad RR-based fast ad highode accuate algoithms to solve complex Poisso poblems i the uit disc i the complex plae. This is ecessay i ode to build fast ad high-ode accuate algoithms fo the complex ihomogeeous bihamoic poblems which is discussed i Sectio 3. We should metio hee that the existece of solutios to poblems P) ad P2) defied below is give ude the assumptio that the ihomogeeous tem f L D; C) see Begeh, 27a). Howeve, we develop the algoithms fo costuctig solutios of these poblems umeically ude the assumptio that f is expadable i a Fouie seies. This is a stoge estictio tha the oigial theoem, ad hece justified. 2. Diichlet poblem fo the complex Poisso equatio i the uit disk The Diichlet poblem fo the complex Poisso equatio i the uit disc D ={z C, z } i the complex plae is give by z z ω = f z, z) i D, ω = γ o D. P) Note hee that f i geeal is a fuctio of both the vaiables z ad z. Below, fo the sake of coveiece, we wite f z) eve though it is a fuctio of both the vaiables. This poblem is uiquely solvable fo f L D; C), γ C D; C) see Begeh, 27a). If u is the solutio of the homogeeous poblem, z z u = o D, u = γ v o D, 2.)

4 THE FFTRR-BASED FAST DECOMPOSITION METHODS 827 whee v satisfies the equatio z z v = f, 2.2) the the solutio of the poblem P) is give by ω = u + v. 2.3) A picipal solutio of 2.2) is obtaied usig stadad Gee s fuctio method see Geebeg, 97; Boges & Daipa, 2; Begeh, 27b) ad ca be witte as vz) = 4 Gz, ζ)f ζ ) dξ dη, 2.4) whee D Gz, ζ)= log ζ z, 2.5) 2π is the fee space Gee s fuctio fo the Laplace equatio. Specifically, we expad ω.) the solutio of poblem P), i tems of Fouie seies, ad deive the Fouie coefficiets of the associated sigula itegals i tems of RRs utilizig oe-dimesioal itegals i the adial diectio. To demostate this, we evaluate the sigula itegal vz = e iα ); <α 2π see Boges & Daipa, 2). Theoem 2. If vz) is the picipal solutio of the P) poblem ad f e iα ) = = f ) e iα with z = e iα, the the Fouie coefficiets v ) ae give by 4 ρ log f ρ) dρ + 4 ρ log ρf ρ) dρ if =, ρ ρ ) v ) = 2 f ρ ρ) dρ 2 ρ f ρ) dρ if >, 2.6) ) ρ ρ 2 ρ f ρ ρ) dρ + 2 f ρ) dρ if <. Poof. We itoduce the followig otatios below: Hece, Ω vz) = 2 π Bσ, ) ={z C : z σ < }, = B, ), Ω+ɛ D + 2 π lim ɛ ɛ = B, + ɛ) B, ɛ), Ω log ζ z f ζ ) dξ dη = 2 π lim ɛ Ω +ɛ ɛ Bσ, ) ={z C : z σ }. Ω ɛ log ζ z f ζ ) dξ dη + 2 π lim ɛ ɛ +ɛ = Ω+ɛ ɛ Bz, ɛ). log ζ z f ζ ) dξ dη Ω +ɛ log ζ z f ζ ) dξ dη. 2.7) Let I ɛ = 2 π log ζ z f ζ ) dξ dη. Ω ɛ +ɛ

5 828 A. GHOSH AND P. DARIPA The I ɛ 2 π sup ζ Ω +ɛ ɛ f ζ ) log ζ z )π + ɛ) 2 π ɛ) 2 ) 8 sup ζ Ω +ɛ ɛ f ζ ) ɛ log ɛ. Theefoe, the secod itegal i 2.7), amely lim ɛ I ɛ =. This simplifies the epesetatio of vz = e iα ) i 2.7) whose Fouie coefficiets v ) ae ow give by v ) = 2π ) π lim e iα log ζ z f ζ ) dξ dη + log ζ z f ζ ) dξ dη dα. 2.8) 2 ɛ Ω ɛ Ω+ɛ The it follows that v ) = f ζ )Q, ζ)dξ dη + f ζ )Q, ζ)dξ dη, 2.9) Ω Ω Q, ζ)= 2π e iα log ζ z dα. 2.) π 2 Fo z = e iα, ζ = ρ e iθ, >ρ, we have ρ ) e iθ if =, π Q, ζ)= 2 log if =. π Similaly, fo <ρwe have Fo >, we have f ζ )Q, ζ)dξ dη = Simila calculatio fo <, D v ) = 2 v ) = 4 ) e iθ if =, π ρ Q, ζ)= 2 log ρ if =. π = 2 Ω f ζ )Q, ζ)dξ dη + f ζ )Q, ζ)dξ dη Ω ρ ρ f ρ) dρ 2 =, espectively, yields ) ρ ρ f ρ) dρ + 2 ρ log f ρ) dρ + 4 ρ ) ρ ρ f ρ) dρ. 2.) ρ f ρ) dρ, 2.2) ρ log ρf ρ) dρ. 2.3)

6 THE FFTRR-BASED FAST DECOMPOSITION METHODS 829 The solutio u of the homogeeous poblem 2.) is obtaied usig the Poisso itegal fomula ad with z = e iα, it is give by u, α) = 2π φθ)k, α θ)dθ. 2π Hee Kρ, θ) is the Poisso keel ad φθ)= γθ) vθ) o D. Ifφ ae the Fouie coefficiets of φθ), the the Fouie coefficiets u, α) of uz) ae give by see Boges & Daipa, 2) u ) = φ. 2.4) Usig 2.6) ad 2.4), the solutio ω of the Diichlet poblem P) is obtaied fom its Fouie coefficiets ω = u + v see 2.3)). As show i the ext sectio, developmet fo the Neuma poblem is simila ad the computatio of solutio of the Neuma poblem ivolves the same itegals as i 2.6). 2.2 Neuma poblem fo the complex Poisso equatio i the uit disk We coside hee the complex Poisso equatio with Neuma bouday coditio i the uit disk D i the complex plae. The itegal epesetatio of the solutio of this poblem is give i Begeh 27a). Ou fast algoithm is deived fom aalysis of the itegals ivolved i this epesetatio. The followig theoem take fom Begeh 27a) gives this epesetatio. Theoem 2.2 The Neuma poblem fo the complex Poisso equatio defied by z z w = f i D, ν w = g o D, P2) f L D; C), g D; C), /2πi) D wz)dz/z) = k C, is uiquely solvable iff gζ ) dζ 4i D ζ = f ζ ) dξ dη. 2.5) D The solutio is give by whee wz) = k + 4πi D G 3 z, ζ)gζ ) dζ ζ G 3 z, ζ)f ζ ) dξ dη, 2.6) π D G 3 z, ζ)= log z ζ )ζ z) 2 2.7) is the Gee s fuctio fo the Neuma poblem ad k is a abitay costat. whee We ewite u N z) = πi D wz) = k + u N z) + IN z) + vz), 2.8) log z ζ gζ ) dζ ζ, IN z) = 2 π D log z ζ f ζ ) dξ dη, 2.9) ad vz) is give by 2.4). Usig the appoach give i Daipa 992), we aalyse the itegals ivolved i the fomula 2.6) to desig the fast algoithm. Towads this ed, we expad the solutio w ) i tems of Fouie seies, ad expess its adius depedet Fouie coefficiets i tems of Fouie coefficiets

7 83 A. GHOSH AND P. DARIPA of the bouday data g ad of the ihomogeeous tem f, the latte appeaig as itegads of oedimesioal itegals. This is embodied i the followig theoem poof of which ca be easily obtaied. Theoem 2.3 If w, α) is the solutio of the Neuma poblem i Theoem 2.2, z = e iα, f e iα ) = = f ) e iα, ad ge iα ) = = g e iα, the the Fouie coefficiets w ) ca be witte as w ) = kδ + u N, ) + IN, ) + v ), 2.2) whee if =, δ = u N, ) = g if =, if =, if =, I, N ) = 2 f ρ) ρ) ρ dρ if =, if =, ad expessios fo v ) ae give i Theoem ) 2.22) Both the poblems Diichlet ad Neuma) thus equie evaluatios of the itegals 2.6). To impove upo the computatioal complexity fo solvig these poblems, the itegals i 2.6) ae evaluated usig RRs which ae deived ext. 2.3 Recusive elatios We develop some RRs below. We discetize the uit disc ito M N gid poits with M equidistat poits i the adial diectio ad N equidistat poits i the agula diectio. We defie the followig itegals: ρ ρ ) p, ) = 2 f ρ ρ) dρ, p 2, ) = 2 ρ f ρ) dρ if >, ) ρ ρ s, ) = 2 ρ f ρ ρ) dρ, s 2, ) = 2 f ρ) dρ if <, 2.23) ρ ) t, ) = 4 f ρ) dρ, t 2, ) = 4 ρ log ρf ρ) dρ. Coollay 2.4 It follows that p, ) = p 2, ) = s, ) = s 2, ) = t, ) = t 2, ) =. Coollay 2.5 Let = < 2 < < M = ad j > i, i, j M. Defie ad A i,j = 2 j i RA A i,j = 4 j i ρ ρ f ρ) dρ, ρ j ) f ρ) dρ, B i,j = 2 j i B i,j = 4 j ρ R B ρ f ρ) dρ 2.24) i ρ log ρf ρ) dρ, 2.25)

8 THE FFTRR-BASED FAST DECOMPOSITION METHODS 83 whee i R A = { i if >, j if <, R B = { j if >, i if <. The fo i, j, l M, l < i < j ad usig Coollay 2.4 we have the followig RRs: l p, i ) = p, l ) B l,i, p i 2, i ) = p 2, j ) A i,j if >, ad s, i ) = t, i ) = i l l s, l ) + A l,i, s 2, i ) = i j j ) t, l ) + A l,i, t 2, i ) = t 2, j ) + B i,j l log l )t, l ) + t 2, l ) if =, v l ) = p, l ) + p 2, l ) if >, s, l ) + s 2, l ) if <. i s 2, j ) + B l,j if <, 2.26) 2.27) Poof. Simple algebaic maipulatio of the elatios i ) poves the above esult i Coollay 2.5. Coollay 2.6 Fo j > i, i, j M, ad usig the RRs we have l ) ) M i l log l A i,i + B i,i+ if =, i=2 l v l ) = l i i=2 l l l i=2 i B i,i A i,i + M l i=l i M i i=l l i=l A i,i+ if >, B i,i+ if <. Poof. Algebaic maipulatio of the elatios i 2.26) yields Coollay ) 3. Bihamoic poblems I this sectio, we descibe the mathematical foudatio fo the fast algoithms to solve complex ihomogeeous bihamoic equatios with fou types of bouday coditios see Begeh, 27a). Each of these fou poblems is uiquely solvable ude coditios that ae metioed i the sectios below. This has bee established by Begeh 28). Solutios of two see poblems D2) ad D4) i Sectios 3.2 ad 3.4) of these fou poblems ae based o decomposig each of these two poblems ito two complex Poisso poblems, wheeas each of the emaiig two poblems see poblems D) ad D3) i Sectios 3. ad 3.3) equie solutios of two complex Poisso poblems ad oe complex homogeeous bihamoic poblem. We should metio hee that the existece of solutios to poblems D),

9 832 A. GHOSH AND P. DARIPA D3) ad D4) defied below is give ude the assumptio that the ihomogeeous tem f L D; C) see Begeh, 27a). Howeve, we develop the algoithms fo costuctig solutios of these poblems umeically ude the assumptio that f is expadable i a Fouie seies. This is a stoge estictio tha the oigial theoem, ad hece justified. 3. Diichlet poblem of type D) The fist poblem fo the bihamoic equatio, we coside is the followig Diichlet D) poblem to be called D) bihamoic poblem hecefoth). z z ) 2 ω = f i D, ω = h o D, D) ω z = h o D, which is uiquely solvable see Begeh 28)) fo f L D; C), h C 2 D, C) ad h C D, C). The solutio to this poblem is based o the decompositio of the poblem ito two complex Poisso poblems 3.),3.) 2, ad a complex homogeeous bihamoic poblem H) which is a homogeeous vesio of the bihamoic poblem D). We wite ω = ω + ω 2, whee ω ad ω 2 satisfy } } z z )u = f i D, z z )ω = u i D, 3.) u = o D, ω = h o D. z z ) 2 ω 2 = id, ω 2 = o D, H) ω 2 z = h ω z = h o D. Thus, the method of solvig the above bihamoic poblem D) ivolves the followig steps: i) Solve the complex Poisso poblem 3.) usig the method of Sectio 2 to obtai uz) which is a iput fo the complex Poisso poblem 3.) 2 ; ii) The solve the complex Poisso poblem 3.) 2 to obtai ω z) i D ad ω z o D which is equied i the bouday data of the H) poblem defied above; iii) The fially solve the H) bihamoic poblem to obtai ω 2 as descibed below. Fially, combiig these two solutios ω ad ω 2, we obtai solutio ωz) of the D) bihamoic poblems. The solutio to the H) poblem is give by the followig bouday itegal see Begeh, 27a). ω 2 z) = z 2 ) g z, ζ)h 2πi ζ ) d ζ, g z, ζ)= z ζ + zζ. D Usig simila appoach as befoe, this bouday itegal is also aalysed usig Fouie seies of the itegads that lead to desig of fast algoithms. The esult of this aalysis is embodied i the followig theoem. We omit the poof hee because it ca be easily obtaied. Theoem 3. If ω 2, α) is the solutio of the H) poblem fo z = e iα i the uit disc ad if h eiα ) = = b e iα, the the th Fouie coefficiet ω 2, ) of ω 2, ) ca be witte as { b + 2 ) if =, ω 2, ) = b 2 3.2) ) if =.

10 THE FFTRR-BASED FAST DECOMPOSITION METHODS 833 We discuss the desig of the fast algoithm fo the costuctio of solutio of the complex bihamoic poblem 3. i Sectio Diichlet poblem of type D2) We coside hee the secod Diichlet D2) bihamoic poblem to be called D2) bihamoic poblem hecefoth). This poblem is sometimes called Riquie ad also Navie poblem. z z ) 2 ω = f i D, ω = h o D, z z ω = h 2 o D. This poblem has bouday coditios diffeet fom the D) hamoic poblem give above, ad is uiquely solvable see Begeh, 28) fof L 2 D; C), h C D; C) ad h 2 C D; C). We ote hee that this D2) bihamoic poblem ca be decomposed ito the followig two complex Poisso poblems. } } z z )u = f i D, z z )ω = u i D, 3.3) u = h 2 o D, ω = h o D. These ae Diichlet P) type complex Poisso poblems discussed i Sectio 2.. The fast Poisso algoithm discussed i Sectio 4. based o the theoy of Sectio 2. is used twice i successio, fist fo solvig 3.3) fo u ad the fo solvig 3.3) 2 fo ω. 3.3 Diichlet Neuma poblem of type D3) The thid poblem fo the bihamoic equatio we coside is the followig Diichlet Neuma D3) poblem to be called D3) bihamoic poblem hecefoth). z z ) 2 ω = f i D, ω = h o D, D3) ν ω = h o D. This poblem is uiquely solvable fo f L D; C), h C 2 D; C), ad h C D; C). The method fo solvig D3) bihamoic poblem is the same as fo the poblem D) see Sectio 3.), except that poblem H) thee is eplaced by followig H2) poblem. z z ) 2 ω 2 = id, ω 2 = o D, H2) ν ω 2 = h ν ω = h o D. D2) The solutio to the H2) poblem is give by the followig bouday itegal see Begeh, 27a). ω 2 z) = z 2 ) g z, ζ)h ζ )dζ 4πi ζ. D

11 834 A. GHOSH AND P. DARIPA Theoem 3.2 If ω 2, α) is the solutio of the H2) poblem i D, ad h eiα ) = = b e iα, the the Fouie coefficiets ω 2, ) of ω 2, ) ca be witte as 2 ) b if =, 2 ω 2, ) = 2 ) b if =. 2 Poof. The poof is simila to the poof of Theoem Diichlet Neuma poblem of type D4) The fouth poblem fo the bihamoic equatio we coside is the followig Diichlet Neuma D4) poblem to be called D4) bihamoic poblem hecefoth). z z ) 2 w = f i D, w = h o D, ν w zz = h 2 o D. This poblem is also uiquely solvable fo f L D, C), h, h 2 D; C) iff h 2 ζ ) dζ 4i D ζ = f ζ ) dξ dη. D This poblem ca similaly be decomposed ito two Poiso poblems as follows: } } z z )u = f, id, z z )w = u, id, ν u = h 2, o D, ω = h, o D. As befoe, the fast Poisso algoithm discussed i Sectio 4. is used twice i successio, fist fo solvig the complex Neuma Poisso poblem 3.5) fo u ad the fo solvig the complex Diichlet Poisso poblem 3.5) 2 fo ω. 4. Fast algoithms We fist build FFT ad RRs-based fast algoithms fo the complex Poisso poblems i Sectio 4., followed by the desciptio of the fast algoithms fo the bihamoic poblems i Sectio 4.3. Itis woth ecallig discetizatio of the uit disk D usig M N gid poits, M i the adial diectio ad N i the azimuthal diectio. 3.4) D4) 3.5) 4. Fast algoithm fo the complex Poisso poblems We fist coside the P) poblem. Iitializatio: Choose M ad N = 2, whee is a itege. Defie K = N/2. Iputs: M, N, γe 2πik/N ), f l e 2πik/N ), l [, M ], k = N ad [ K +, K]. Step. Compute the Fouie coefficiets γ, ad f l ), [ K +, K], l [, M ].

12 THE FFTRR-BASED FAST DECOMPOSITION METHODS 835, A i,i+, B i,i+ 2.25), ad 2.25) 2. Step 3. Usig 2.26) i Coollay 2.5, compute the RRs as show i the flowchat. Step 2. Compute A i,i+, B i,i+ fo i [, M ], [ K +, K] usig2.24),2.24) 2, Set p, = ) =, fo =,..., K fo l = 2,..., M p, l ) = l l p, l ) B l,l ed ed Set p 2, M = ) = fo =,..., K fo l = M ),..., p 2, l ) = l l+ p 2, l+ ) A l,l+ ed ed Set s, = ) = fo = K,..., fo l = 2...M s, l ) = l l s, l ) + A l,l ed ed Set s 2, M = ) = fo = K,..., fo l = M ),..., s 2, l ) = l+ l s 2, l+ ) + B l,l+ ed ed Set t, = ) =, fo l = 2,..., M )t, l ) + A l,l t, l ) = l l ed Set t 2, M = ) = fo l = M ),..., t 2, l ) = t 2, l+ ) + B l,l+ ed Step 4. Compute v l ) fo [ K +, K], l [, M ]usig2.27). Step 5. Compute u l ), [ K +, K], l [, M ]usig2.4). Step 6. Fially, compute ω l e 2πik/N ) by ivetig v l ) + u l )) usig FFT. The sequetial algoithm fo the P2) poblem is simila. 4.2 Algoithmic complexity The computatioal complexity of the above fast algoithm fo the complex Poisso equatio i the uit disc is peseted below.

13 836 A. GHOSH AND P. DARIPA Step Opeatio cout Computatio of γ ad f by M + ) times applicatio of FFT, each time to N data poits cotibutes OMN log N) 2 Computatio of A i,i+, ad B i,i+,, i [, M ] cotibutes OMN) 3 Computatio of p ), l), p ) 2, l), s ), l), s ) 2, l), t ), l), t ) 2, l), l [, M ] cotibutes OMN) 4 Computatio v l ), l [, M ] cotibutes OMN) 5 Computatio u l ), l [, M ] cotibutes OMN) 6 Computatio of ω l e 2πik N ), k [, N] usig FFT cotibutes OMN log N) We see fom above that the algoithm has complexity of the ode OMN log N) fo a total of M N degees of feedom i.e., umbe of gid poits). Thus, it has computatioal complexity of the ode Olog N) pe degee of feedom. Fo the Neuma poblem also, it is easy to see that the complexity emais the same. 4.3 Fast algoithms fo the complex ihomogeeous bihamoic poblems Fo D2) ad D4) bihamoic poblems, we apply the above fast Poisso algoithm fo the P) poblem twice i successio. Sice the algoithms fo bihamoic poblems D) ad D3) ae simila, below we outlie the steps of the fast algoithm fo solvig D) bihamoic poblem ad skip the details fo D3) bihamoic poblem. Iitializatio: Choose M ad N = 2 whee is a itege. Defie K = N/2. Iputs: M, N, h e 2πik/N ), f l e 2πik/N ), l [, M ], k [, N] ad [ K +, K]. Step. Compute the Fouie coefficiets h, l ), f l ), [ K +, K], l [, M ]. Step 2. Compute u l ), [ K +, K], l [, M ] usig the Poisso algoithm. Step 3. Compute the Fouie coefficiets u l ), [ K +, K], l [, M ] usig FFT. Step 4. Compute ω, l ), [ K +, K], l [, M ] usig the Poisso algoithm. Step 5. Compute the ω 2, l ), [ K +, K], l [, M ] usig Theoem 3.. Step 6. Fially, compute ω l e 2πik N ) by ivetig ω, l ) + ω 2, l )) usig FFT. It is easy to see that this algoithm has the same computatioal complexity, sice it uses ou fast complex Poisso solve twice i successio. 5. Numeical esults I this sectio, umeical esults obtaied usig above fast Poisso ad bihamoic algoithms o seveal poblems ae peseted ad discussed. Numeical implemetatio of these fast algoithms wee doe i MATLAB, ad computatios wee pefomed usig double pecisio 6 digit) aithmetics. We have implemeted the tapezoidal, Simpso s ad Eule Maclaui fomulas Sidi & Isaeli, 988) fo umeical itegatio, ad detemied the accuacy of the fast solves. Eule Maclaui fomulas ae used with two poits-based umeical itegatio of oe-dimesioal itegals that appea i the fast algoithms discussed i Sectios 4. ad 4.3. Computatios have bee pefomed fo seveal values of M ad N. I all the examples to be peseted below, the souce tems f ad all bouday data associated with the Poisso ad bihamoic poblems have bee take to be C fuctios, so that the solutios of these poblems ae also C. Sice the fast algoithms use Fouie expasios of the souce tem ad of

14 THE FFTRR-BASED FAST DECOMPOSITION METHODS 837 the solutios of these poblems i the azimuthal diectio with gid size 2π/N, the solves have expoetial ate of covegece i the gid size 2π/N. It has bee foud moe tha sufficiet to use N = 32 as the solutios of these poblems usig ou algoithms have bee foud to emai the same withi machie oud-off eo fo N > 32 with M fixed. Theefoe, the ode of accuacy of the solutios has bee detemied fo seveal iceasig values of M by evaluatig elative eos i L om. The eo ad ode of accuacy fo seveal values of M usig diffeet umeical itegatio schemes will be show below. We fist show the pefomace of ou algoithms o complex Poisso poblems followed by complex bihamoic poblems. 5. The complex Poisso poblem We coside Diichlet P) ad Neuma P2) complex Poisso poblems see Sectio 2) with appopiate ihomogeeous tems f ad bouday coditios, so that the solutios of both the poblems ae give by ωz) = z 3/2 z 5/2 + iz 5/2 z 3/2, 5.) which is efeeed to as Example i the est of this pape. Example 2 which has bee take fom Boges &Daipa2), the solutio is give by ω, θ)= 3 e cos θ+si θ) cos θ cos θ) 2 ) si θ si θ) 2 ) ) Fo Example, Table shows elative eo ad ode of accuacy i azimuthal gid size /M )) fo seveal iceasig values of M with the value of N fixed at 64. Simila esults ae show i Table 2 fo Example 2. These tables show the fouth-ode accuacy i the azimuthal gid size of the algoithms. This is a maked impovemet ove secod-ode accuacy of the fast Poisso solve of Boges & Daipa 2), whee tapezoidal ad Simpso s itegatio fomula wee used. Theefoe, i these tables we do ot show the esults with tapezoidal ad Simpso s ules. 5.2 The complex bihamoic poblem Now we coside complex ihomogeeous bihamoic poblems with diffeet bouday coditios ad apply the Algoithm 4.3 to solve them. We should ecall fom Sectio 3 that fou poblems D), D2), D3) ad D4) discussed thee ca be gouped ito two classes fom the view poit of thei solutio pocedue: Goup cosistig of poblems D) ad D3), ad Goup 2 cosistig of poblems D2) Table Relative eos ad odes of accuacy of the Poisso solves fo Example see solutio 5.)) usig Eule Maclaui fomula ad N = 64 Diichlet Poisso poblem P) Neuma Poisso poblem P2) M Ode of accuacy Ode of accuacy

15 838 A. GHOSH AND P. DARIPA Table 2 Relative eos ad odes of accuacy of the algoithms fo Example 2see solutio 5.2)) usig Eule Maclaui fomula ad N = 64 Diichlet Poisso poblem P) M Ode of accuacy Table 3 Relative eo fo f z) = 2 z + iz) usig Eule Maclaui fomula fo the bihamoic poblem D) with N = 64 Tapezoidal Eule Maclaui M Ode Ode ad D4). Each of the goup poblems equie solutio of a homogeeous bihamoic poblem i thei solutio pocedue by decompositio method, wheeas each of the Goup 2 poblems do ot equie this see Sectio 3). Theefoe, we show esults fo oly oe poblem fom each of these two classes. Fist, we coside the bihamoic poblems D) ad D2) with f z) = 2 z + iz), ad thei espective bouday coditios h = z + i z, h = 3 + 2i z 2 fo D) poblem) ad h = z + iz, h 2 = 6 z + iz) fo D2) poblem). The tapezoidal ad Eule Maclaui fomulas have bee used fo umeical itegatio i the adial diectio fo seveal values of M with N = 64. A secod-ode covegece usig tapezoidal ule ad a fouth-ode covegece usig Eule Maclaui ule ae obseved i both of these tables Table 3 ad 4). Computatios have bee pefomed with may moe souce ad bouday tems o all fou poblem types, ad simila esults o the eo ad ode of accuacy have bee obseved. We just show oe moe set of esults obtaied with a diffeet set of souce f ) ad bouday data, but with bihamoic poblems D3) ad D4). The souce tem take fo both the poblems is give by f z, z) = 72z 2 z + z z 2 ), ad thei espective bouday coditios ae give by h = z + i z, h = 7z + 7 z fo D3) ad h = z + z, h 2 = 6 z + 6z fo D4). The souce tem used hee is symmetic: f z, z) = f z, z), as opposed to the oe used befoe whee it was ot symmetic. Table 5 shows the elative eos ad ode of accuacy obtaied with Eule Maclaui fomulas tapezoidal esults ae simila as befoe) fo both of these poblems which suppot that these bihamoic algoithms give fouth-ode accuacy esults. Next we compae umeical asymptotic complexity with ou theoetical complexity ON 2 log N), whee N 2 is the total umbe of gid poits. Numeical complexity has bee calculated based o CPU

16 THE FFTRR-BASED FAST DECOMPOSITION METHODS 839 Table 4 Relative eo with f z) = 2 z + iz) usig Eule Maclaui fomula fo the bihamoic poblem D2) see D2)) with N = 64 Tapezoidal Eule Maclaui M Ode Ode Table 5 Relative eo fo f z) = 72z z 2 + z 2 z) usig Eule Maclaui fomula fo the D3) ad D4) poblem with N = 64 Fo the bihamoic poblem D3) fo the bihamoic poblem D4) M Ode Ode Table 6 CPU timigs ad estimates fo the costat c usig Eule Maclaui example fo D2) ad D4) poblems Fo the bihamoic poblem D2) Fo the bihamoic poblem D4) M = N CPU time c CPU time c time equied to solve two bechmak poblems, amely bihamoic poblems D2) ad D3),fo diffeet values of N usig Eule Maclaui itegatio scheme. The umbe of gid poits i the adial ad azimuthal diectios have bee take to be same, N, which is the coect way to estimate of the complexity umeically fo the pupose of compaiso with the theoetical oe. Fom this, we also compute the costat, c hidde i the ode estimate by dividig the total computatio time by N 2 log N. The CPU timigs ad the estimate of the costat c fo these two poblems, D2) ad D3), have bee tabulated i Table 6 fo seveal values of N. The CPU timigs vesus N 2 log N is plotted i Fig. a), which

17 84 A. GHOSH AND P. DARIPA a) 6 x 6 5 b) 3.5 x 4 3 Tx N 2 logn x 5 T/N 2 logn) N Fig.. Plots of the CPU timigs ad the costat c = T/N 2 log N) fo the fast algoithms applied to the bihamoic poblems D2) add3) poblems usig the Eule Maclaui fomula. Plots fo D2) add3) poblems i each of the a) ad b) ae idistiguishable fom each othe. a) Plot of CPU timigs vesus N 2 log N, wheen 2 is the total umbe of poits i the discetizatio of the domai. b) Plot of T/N 2 log N) vesus N,wheeN 2 is the total umbe of poits i the discetizatio of the domai. is cosistet with the theoetical complexity. Moe impotatly, this figue shows that the costat c hidde behid the complexity estimate is vey small ote that the abscissa of the plot is scaled up by 5 ), which is futhe cofimed by the plot of computed values of c vesus N i the Fig. b). This is aothe sigificat advatage that these fast algoithms offe. 6. Eo estimate fo the sigula itegals Fo a give poblem ad choice of itegatio fomula fo the oe-dimesioal itegals i the fast algoithm, fee paametes available to cotol the umeical eo ae M ad N. The goal hee is to show how M is detemied apioifom pescibed values of N ad maximum desied eo ɛ p i the solutio of the poblem. Fo this, we choose the picipal solutio vz) defied by the sigula itegal 2.4). We fist show below the estimate fo the eo ɛ i its umeical evaluatio of vz) by the fast algoithm. Lemma 6. Fo ozeo ad ifiitely diffeetiable fuctio f i D, if each itegal i step 2 of the fast algoithm see Sectio 4.) is computed with a eo boud δ, the the eo ɛ i the umeical evaluatio of the picipal solutio vz), defied by the sigula itegal 2.4), is bouded by 2δM )K + 2)/K, whee N is the total umbe of poits i the azimuthal diectio, M is the total umbe of poits i the adial diectio ad K = N/2. We povide some otatios hee befoe goig to the poof. Fo i < i+, the eo i computig the itegal /2)A i,i+, /2)B i,i+, A i,i+, B i,i+ ae give by ɛ i,i+,, ɛi,i+ 2,, ɛi,i+,, ɛi,i+ 2, ad A i,i+ ) app, B i,i+ ) app ae the appoximate values of A i,i+, B i,i+, espectively. It follows fom the RRs 2.26) i

18 THE FFTRR-BASED FAST DECOMPOSITION METHODS 84 Sectio 2.3. p, l ) = s, l ) = l i i=2 l l l i=2 M t, l ) = i=l Next we give the poof of the lemma. i M B i,i, p 2, l ) = A i,i, s 2, l ) = M A i,i, t 2, l ) = i=l i=l M i i=l B i,i+. l l i A i,i+, >, B i,i+, <, Poof. Recall fom Coollay 2.4 that p 2, M = ) = s 2, M = ) = t 2, M = ) =. It follows fom Theoem 2. that fo z = l e 2πij/N, l = 2, 3,..., M, j =, 2,..., N v l e 2πij/N ) = whee ɛ = K = K+ + = K+ K = i=2 + + = K+ l i=2 v l )) e 2πij/N = K p, l ) + p 2, l )) e 2πij/N = s, l ) + s 2, l )) e 2πij/N + l log l )t, l ) + t 2, l ) l i l l i=2 i l B i,i ) app ) e 2πij/N + A i,i ) app ) e 2πij/N + K M M l log l [A i,i ) app ] + [B i,i+ ) app ] + ɛ, K 2 l i = i=2 l M 2 i + = K+ i=l l ɛ i,i 2, + K = i=l 2 M l i=l i i = i=l l M i = K+ i=l l ɛ i,i+, + = K+ l l ɛ i,i+ 2, + l log l ɛ i,i, + i=2 i=2 ) ) A i,i+ ) app e 2πij/N 2 ɛ i,i+ 2,. ) ) B i,i+ ) app e 2πij/N l l i=2 i ɛ i,i, We have fo 2 < i < l, i < l ad fo l < i < M, l < i. If each itegal is computed withi a eo bouded by δ, whee { δ = max i,,ρ ɛ i,i,, ɛi,i+ 2,, ɛi,i 2,, ɛi,i+,, ɛi,i+ 2,, ɛi,i },. 6.)

19 842 A. GHOSH AND P. DARIPA The ɛ K = 2δ M ) + K = 2δ M ) + = = K+ K ) K = 2δM ) + 4δM ) = = 2δ M ) + = K+ 2δ M ) + 2δM ) 2δM )K + 2). 6.2) K This cocludes the poof of the lemma. Theefoe, fo ɛ <ɛ p we have 2δM )K + 2) K <ɛ p, 6.3) fom which a estimate fo miimum value of M ca be obtaied as show below though two examples. I the case of tapezoidal ule, it follows fom 6.) that δ i 6.3) is give by Δ) 3 { 2 ) i+ 2 ) ρ ) 2 } δ = max ρf i,,ρ 2 ρ 2 ρ), ρf ρ ρ 2 ρ), ρ 4ρ log ρf ρ)). 6.4) 2 6. Examples The above estimate has bee validated usig may examples. We give below oly two simple case studies. Poblem 6.2 We coside the fuctio f z) = 2 z. Hece, { 2 if =, f ρ) = if =. Theefoe, it follows fom 6.4) Δ) 3 { 2 2ρ 3 δ = max i,,ρ 2 ρ 2 It follows fom 6.3) ad 6.5) i+ M ) 2 > ), Poblem 6.3 We coside the fuctio f z) = 6z z 2. Hece, { 6ρ 3 if =, f ρ) = if =. i 2 } ρ 2 iρ) = 6Δ) ) 2 2K + 2). 6.6) ɛ p K Theefoe, it follows fom 6.) Δ) 3 { 2 2ρ 3 δ = max i,,ρ 2 ρ 2 i+ ), 2 ρ 2 6ρ 5 i )} = Δ) )

20 THE FFTRR-BASED FAST DECOMPOSITION METHODS 843 a) b) Fig. 2. The poit-wise eo iside the disk fo two poblems. This is a colou plot o scee. a) Eo plot fo with f z) = 2 z, N = 64, M = 256. b) Eo plot with f z) = 6z 2 z, N = 64, M = 28. It follows fom 6.3) ad 6.7) M ) 2 > 2K + 2). 6.8) ɛ p K Fo the above two poblems, we have umeically evaluated vz) see 2.4)) iside the disk usig the fast algoithm usig miimum values of M detemied by the above estimates 6.6) fo poblem ad 6.8) fo poblem 2) fo seveal choices of ɛ p, ad also evaluated vz) exactly sice exact itegatio of the sigula itegal 2.4) is possible fo both the poblems, theeby obtaied umeical eos i the computed values of vz) fo both the poblems. We have foud that the umeical eo i the computed values of vz) iside the disk at all gid poits is always less tha the pescibed eo ɛ p fo the poblems. Fo bevity, we show oly two figues. Fo a pescibed eo ɛ p =. ad N = 64 fo poblem, we obtai M = 358 fom the theoetical estimate 6.6). Figue 2 shows poit-wise eo plot iside the disk fo this poblem whe N = 64, M = 256. We see that the eo does ot exceed this pescibed eo. Similaly fo poblem 2, fo a pescibed eo ɛ p =. ad N = 64, we obtai M = 47 fom the theoetical estimate 6.8). Figue 2b) shows poit-wise eo plot iside the disk fo this poblem whe N = 64, M = 28. We see that the eo is less tha the pescibed eo. Note fo both poblems, the umeical eo is withi pescibed eos with values of M less tha the estimates. 7. Applicatio to low ad modeate Reyolds umbe flow We apply ou fast algoithms fo the bihamoic poblems to study steady ad viscous icompessible flows iside a cylide. Such flows have bee studied by may see Mills, 977, fo example). The pactical applicatio of these types of poblems aises i eciculatio of fluids i cavities ad i cofied vetilatio see Mills, 977). The goveig equatios fo such flows ae give by u )u = p + R u), u =, 7.)

21 844 A. GHOSH AND P. DARIPA whee p is the pessue, R is the Reyolds umbe ad u is the velocity. The velocity compoets u, u θ ) i tems of steam fuctio ψ ae give by u = ψ θ, u θ = ψ. 7.2) The scala voticity ϕ satisfies the Poisso equatio ϕ = Δψ. Takig cul of both sides of 7.) gives Δ 2 ψ = RJ[ψ, Δψ], 7.3) whee the Jacobia J[ψ, Δψ] = /) ψ θ Δψ Δψ θ ψ). Fo R, equatio 7.3) is the homogeeous bihamoic equatio. I tems of geealized deivatives, Δ 2 = z z ) 2 /6. Theefoe, a typical flow poblem iside the cylide fo pescibed bouday values of ψ ad ψ/ is the give by see Kuwahaa & Imai, 969; Mills, 977) z z ) 2 ψ = R 6 J[ψ, ψ z z] i <, ψ = f θ) o =, ψ = f 2θ) o =, whee the Reyolds umbe R = U/ν based adius of the cylide, U is the speed of otatio of the pat of the cylide wall ad ν is the kiematic viscosity of the fluid. With slight abuse of otatio, we use above the same otatio ψ whe ψx, y) is witte as a fuctio of z = x + i y) ad z = x i y). The poblem 7.4) above is the bihamoic poblem D3) see Sectio 3), except that the ihomogeeous tem i the bihamoic equatio 7.4) depeds o the solutio itself. Theefoe, the poblem 7.4) is solved usig a iteatio scheme i which the fast algoithm discussed i Sectio 4.3 is used to solve the D3) bihamoic poblem at evey iteatio util some suitable covegece citeio is met. We apply this iteative method ad the fast algoithm developed i pevious sectios to specific fluid flow poblems which will validate the applicatio potetial of ou algoithms. We fist coside slow ceepig Stokes flows whe R. I this case, the above poblem 7.4) educes to the D3) homogeeous bihamoic poblem. This is fist solved usig ou fast algoithm discussed i Sectio 4.3. The solutio of this zeo Reyolds umbe poblem seves as a iitial guess to solve the poblem 7.4) fo flows with ozeo Reyolds umbe. I paticula, we ae fist iteested i flows with low Reyolds umbe, fo which we follow a simple iteatio method descibed i Geegad & Kopiski 998) ad Mills 977). We stat with a iitial guess ψ ) obtaied fom the solutio of Stokes flow, ad the at each k + )th stage we solve ψ k+) z zz z = R 6 J[ψ k), ψ k) z z ] i <, ψ k+ = f θ) o =, 7.5) ψ k+) = f 2 θ) o =, 7.4)

22 THE FFTRR-BASED FAST DECOMPOSITION METHODS 845 usig the fast algoithm fo the ihomogeeous bihamoic poblem D3) as metioed above. The voticity ϕ is obtaied fom ϕ = Δψ usig umeical appoximatio of the Laplacia. The Jacobia is obtaied usig the cetal diffeece fomula o mesh poits iside the disk ad backwad diffeece, fowad diffeece fo poits o the bouday. We cotiue the iteatio util the covegece citeio ψ k+ ψ k )/ ψ k+ < tol is met fo some suitable choice fo the value of the toleace. I ou case, we used tol = 3 4. I ou umeical expeimets, the above iteatio method failed fo Reyolds umbe beyod 4. This is well kow as othes have also expeieced the same poblem with this iteatio method Mills, 977; Geegad & Kopiski, 998) i the past. Theefoe, we modify ou iteatio method fo R > 4. We use a elaxatio facto as i the Gauss Seidel SOR method see Mills, 977). We stat with ou iitial guess as befoe to obtai ψ ) z) ad use the fast algoithm to compute the iteate ψ k+) z). Fo covegece, we use two elaxatio factos α ad β fo fields ϕ ad ψ. To update the values of ϕ ad ψ, weuse ϕ k+),l = αϕ k+),l + α)ϕ k),l, ψ k+),l = βψ k+),l + β)ψ k),l. The staed quatities deote the values obtaied at each iteative step. The elaxatio facto helps i covegece, ad suitable choices fo α ad β ae take to be.3 ad.5, espectively. The iteatio is cotiued util toleace is met. This iteatio method woks fo poblems with modeate Reyolds umbe, but becomes ustable fo poblems with high Reyolds umbe. Simila pefomace of this iteatio method has bee obseved i Mills 977). Results o seveal flow poblems ae discussed below. The flow poblem we solve has bouday data f 2 θ) = cos θ si θ ad f θ) =, take fom Geegad & Kopiski 998). The flow poblem is solved fo two values of R: ad 5. The umbe of iteatios equied to obtai coveged solutio withi a toleace = 3 4 is 4 at R = ad 26 at R = 5, espectively. Steamlie plots at R = ad R = 5 ae show i Fig. 3a,b), espectively. Simila plots fo voticity ae show i Fig. 4a,b), espectively. The flow patte is symmetic about x- ad y-axes, as it should be due to the same symmety i the bouday data f 2 θ). The plots agee with those obtaied by Geegad & Kopiski 998). This flow was ivestigated fo R 5. Sigificat chage i the voticity patte is obseved with iceasig Reyolds umbe. Now we coside the movig wall poblem with bouday coditio ψ = ) =, ad discotiuous bouday coditio ψ =, θ<π, 7.6), π θ<2π. This flow poblem was solved fo R 2. The computed steamlie pattes at R = ad R = ae show i Fig. 5a,b), espectively. Nosymmetic flow pattes ae obseved hee, ad the cete of the votex is see shifted to the diectio of the flow. The umbe of iteatios equied at R = is 27. Fo computatio of this flow poblem, the umbe of gid poits used is take moe see the figue captio) tha fo the othe poblems due to the pesece of discotiuity i the bouday coditio of this poblem see 7.6)). This poblem has also bee studied by Mills 977) ad Mabey 957), ad ou esults ae i excellet ageemet with thei esults.

23 846 A. GHOSH AND P. DARIPA a) b) Fig. 3. Steamlie pattes fo flows with f θ) =, ad f 2 θ) = cos θ si θ. Computatios have bee pefomed with paamete values N = 64, M = 64 usig the fast Algoithm 4.3 i a D3) bihamoic poblem. This is a colou plot o scee. a) Steamlie pattes fo R =. b) Steamlie pattes fo R = 5. a) b) Fig. 4. Level sets of voticity fo flows with f θ) =, ad f 2 θ) = cos θ si θ. Computatios have bee pefomed with paamete values N = 64, M = 64 usig the fast Algoithm 4.3 i a D3) bihamoic poblem. This is a colou plot o scee. a) Level sets of voticity fo R =. b) Level sets of voticity fo R = 5. Next we show ou esults o the followig outflow iflow poblem take fom Mills 977)), i which ψ is pescibed o the bouday as a piecewise cotiuous data. ψ =, <θ<2π, 7.7)

24 THE FFTRR-BASED FAST DECOMPOSITION METHODS 847 a) b) Fig. 5. Steamlies fo flows with f θ) = ad discotiuous f 2 θ) give by 7.6). Computatios have bee pefomed with paamete values N = 28, M = 29 usig the fast Algoithm 4.3 i a D3) bihamoic poblem. This is a colou plot o scee. a) Steamlie pattes fo R =. b) Steamlie pattes fo R =. a) b) Fig. 6. Steamlies fo flows with f θ) = ad discotiuous f 2 θ) give by 7.8). Computatios have bee pefomed with paamete values of N = 28, M = 65 usig the fast algoithm 4.3 i a D3) bihamoic poblem. This is a colou plot o scee. a) steamlie pattes fo R =.9. b) Steamlie pattes fo R =.2. θ α) +, α ɛ<θ<α+ ɛ, ɛ 2, α + ɛ<θ<β ɛ, ψ = β θ) +, β ɛ<θ<β+ ɛ, ɛ, β + ɛ<θ<2π + α ɛ. 7.8)

25 848 A. GHOSH AND P. DARIPA I ou computatio, α =, ɛ = π/32, β = π ad R = Uɛ/ν is the Reyolds umbe, whee U is the speed ad Uɛ the flow acoss the ac itecepted by ɛ. The umbe of iteatios equied to compute this flow at R =.9 ad R =.2 is 3 ad 2, espectively. Computed steamlie pattes at R =.9 ad R =.2 ae show i Fig. 6a,b), espectively, which agee vey well with that obtaied by Deis 975). 8. Coclusios We have developed FFTRR-based fast ad high-ode accuate algoithms fo solvig the complex Poisso ad the complex ihomogeeous bihamoic poblems. Algoithms based o these same piciples have bee developed befoe by Boges & Daipa 2, 2), Daipa 992, 993) ad Daipa & Mashat 998) fo othe types of elliptic poblems. Fo ease of efeece i the futue fo algoithms based o these piciples, we have amed these algoithms fo the fist time i this pape as FFTRR-based algoithms. These algoithms have may desiable featues which ae listed i the Sectio. These algoithms fo the complex Poisso poblems have bee applied hee to solve fou classes of ihomogeeous complex bihamoic poblems usig the decompositio methods, esultig i efficiet ad accuate fast bihamoic solves. These bihamoic solves have bee applied to solve steady, icompessible slow viscous flow poblems fo low to modeate Reyolds umbe withi cicula cylides. The computatioal esults agee vey well with the existig esults o these poblems. These algoithms ca also be applied to solvig simila poblems i abitay domais usig domai embeddig techique Badea & Daipa, 2, 22, 23, 24), which is a topic of futue eseach. I closig, we metio the fast method fo ectagula domais developed by Be-Atzi et al. 28). Thee ae eve moe fouth-ode accuate methods fo solvig Poisso s equatio o egula egios with the same Olog ) opeatio cout pe mesh poit, whee the costat behid the complexity estimate is small. Pefomace ad vaious othe featues of the algoithms poposed hee eed to be compaed with that of a vaiety of othe fouth-ode accuate methods that have bee developed by may i the past fo solvig the bihamoic equatio o a disk with uifom meshes with the same asymptotic Olog ) opeatio cout pe mesh poit. This is a topic of futue eseach ad falls outside the scope of this pape. O theoetical goud, oe ca easily see that these algoithms, simila to othe algoithms developed by Boges & Daipa 2, 2), Daipa 992, 993) ad Daipa & Mashat 998), offe seveal distict advatages ove the existig oes with simila accuacy ad complexity: i) aalytical methods i.e., exact RRs) i cojuctio with FFT povides accuate ad computatioally efficiet algoithms; ii) these algoithms allow apioiselectio of the uifom mesh sizes fom pescibed desied eo i the solutio see Sectio 6); iii) compaed with methods based o discetizatio of the PDE ove egula mesh i pola coodiates, thee is o eed fo ay special teatmet ea coodiate sigulaity at the oigi; iv) solutios locally ea ay ig i the disk ca be computed without computig solutios eveywhee else. This popety is built ito the algoithms ad will be exemplified i the futue; v) the algoithms automatically simplifies temedously whe solvig the eal bihamoic equatio with bouday data eal i.e., whe ihomogeeous tem f ad the bouday coditios ae eal). This ca be easily woked out by usig the fact that f = f fo eal f ; vi) these FFTRR-based algoithms by thei vey costuctio allow costuctio of exact explicit solutios of a cetai class of poblems, i paticula whe Fouie coefficiets of the souce tem f ca be witte explicitly as a fuctio of the adial coodiate. This ca be easily see by aalyzig the algoithms give i this pape. This will be exemplified i the futue.

26 THE FFTRR-BASED FAST DECOMPOSITION METHODS 849 Ackowledgemets A.G. thaks the Depatmet of Mathematics at Texas A&M Uivesity fo teachig assistatship duig he gaduate study whe this wok was completed. The authos ae gateful to the eviewes fo thei costuctive citicisms which have helped us to impove the pape. Refeeces Badea, L.& Daipa, P. 2) O a bouday cotol appoach to domai embeddig methods. SIAM J. Cotol Optim., 4, Badea, L.& Daipa, P. 22) A domai embeddig/bouday cotol method to solve elliptic poblems i abitay domais. IEEE Cofeece o Decisio ad Cotol, vol. 3, pp Badea, L.& Daipa, P. 23) O a Fouie method of embeddig domais usig a optimal distibuted cotol. Nume. Algoithms, 32, Badea, L.& Daipa, P. 24) A domai embeddig method usig the optimal distibuted cotol ad a fast algoithm. Nume. Algoithms, 36, Begeh, H. 27a) Bihamoic Gee fuctios. Matematiche Cataia), 6, Begeh, H. 27b) Basic bouday value poblems i complex aalysis. J. Appl. Fuct. Aal., 2, Begeh, H. 28) Six bihamoic Diichlet poblems i complex aalysis. Fuctio Spaces i Complex ad Cliffod Aalysis L. H. So & W. Tutschke eds). Haoi: Natioal Uivesity Publ. Haoi, pp Be-Atzi, M.,Coisille, J.-P.&Fishelov, D. 28) A fast diect solve fo the bihamoic poblem i a ectagula gid. SIAM J. Sci. Comput., 3, Bjostad, P. 983) Fast umeical solutio of the bihamoic Diichlet poblem o ectagles. SIAM. J. Nume. Aal., 2, Boges, L.& Daipa, P. 2) A paallel vesio of a fast algoithm fo sigula itegal tasfoms. Nume. Algoithms, 23, Boges,L.&Daipa, P. 2) A fast paallel algoithm fo the Poisso equatio o a disk. J. Comput. Phys., 69, Baess, D.&Peiske, P. 986) O the umeical solutio of the bihamoic equatio ad the ole of squaig matices fo pecoditioig. IMA J. Nume. Aal., 6, Cheg, X.-L., Ha, W.& Huag, H.-C. 2) Some mixed fiite elemet methods fo bihamoic equatio. J. Comput. Appl. Math., 26, 9 9. Daipa, P. 992) A fast algoithm to solve ohomogeeous Cauchy Riema equatios i the complex plae. SIAM J. Sci. Statist. Comput., 3, Daipa, P. 993) A fast algoithm to solve the Beltami equatio with applicatios to quasicofomal mappigs. J. Comput. Phys., 6, Daipa, P.& Mashat, D. 998) Sigula itegal tasfoms ad fast umeical algoithms. Nume. Algoithms, 8, Deis, S. C. 975) Applicatio of the seies tucatio method to two-dimesioal iteal flows. Poceedigs of the Fouth Iteatioal Cofeece o Numeical Methods i Fluid Dyamics, vol. 4, pp Geebaum, A., Geegad, L.& Mayo, A. 992) O the umeical solutio of the bihamoic equatio i the plae. Physics D, 6, Expeimetal mathematics: computatioal issues i oliea sciece Los Alamos, NM, 99). Geebeg, M. D. 97) Applicatio of Gee s Fuctios i Sciece ad Egieeig. Beli: Petice-Hall. Gee s fuctio. Geegad, L.& Kopiski, M. C. 998) A itegal equatio appoach to the icompessible Navie Stokes equatios i two dimesios. SIAM J. Sci. Comput., 2, Geegad, L., Kopiski, M. C.& Mayo, A. 996) Itegal equatio methods fo Stokes flow ad isotopic elasticity i the plae. J. Comput. Phys., 25,