Research Article Smoothing Analysis of Distributive Red-Black Jacobi Relaxation for Solving 2D Stokes Flow by Multigrid Method

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1 Matematical Problems in Engineering Volume 205, Article ID 57298, 7 pages ttp://dx.doi.org/0.55/205/57298 Researc Article Smooting Analysis of Distributive Red-Black Jacobi Relaxation for Solving 2D Stokes Flow by Multigrid Metod Xingwen Zu,2 and Lixiang Zang Department of Engineering Mecanics, Kunming University of Science and Tecnology, Kunming, Yunnan , Cina 2 Scool of Matematics and Computer, Dali University, Dali, Yunnan 67003, Cina Correspondence sould be addressed to Lixiang Zang; zlxzcc@26.com Received 5 September 204; Revised 7 Marc 205; Accepted 8 Marc 205 Academic Editor: Vassilios C. Loukopoulos Copyrigt 205 X. Zu and L. Zang. Tis is an open access article distributed under te Creative Commons Attribution License, wic permits unrestricted use, distribution, and reproduction in any medium, provided te original work is properly cited. Smooting analysis process of distributive red-black Jacobi relaxation in multigrid metod for solving 2D Stokes flow is mainly investigated on te nonstaggered grid by using local Fourier analysis (LFA). For multigrid relaxation, te nonstaggered discretizing sceme of Stokes flow is generally stabilized by adding an artificial pressure term. Terefore, an important problem is ow to determine te zone of parameter in adding artificial pressure term in order to make stabilization of te algoritm for multigrid relaxation. To end tat, a distributive red-black Jacobi relaxation tecnique for te 2D Stokes flow is establised. According to te 2-armonics invariant subspaces in LFA, te Fourier representation of te distributive red-black Jacobi relaxation for discretizing Stokes flow is given by te form of square matrix, wose eigenvalues are meanwile analytically computed. Based on optimal onestage relaxation, a matematical relation of te parameter in artificial pressure term between te optimal relaxation parameter and related smooting factor is well yielded. Te analysis results sow tat te numerical scemes for solving 2D Stokes flow by multigrid metod on te distributive red-black Jacobi relaxation ave a specified convergence parameter zone of te added artificial pressure term.. Introduction Multigrid metods [ 7] are generally considered as one oftefastestnumericalmetodswicaveanoptimally computational complexity for solving partial differential equations (PDEs), especially for 3D steady incompressible Newtonian flow governed by Navier-Stokes equations. In multigrid metods, smooting relaxations play an important role. Several multigrid relaxation metods were developed for solving PDEs, wic are rougly classified into two categories, collective and decoupled relaxations [8]. Te collective relaxations are considered as a straigtforward generalization of te scalar case [2]. Te early decoupled relaxation is on a distributive Gauss-Seidel relaxation [9]. Gradually, it is generalized to an incomplete LU factorization relaxation [0]. Recently, Stokes system wit distributive Gauss-Seidel relaxation based on te least squares commutator as been researced []. Muc of te relaxations for Stokes system is seen in [2, 3]. For multigrid metods, LFA is a very useful tool to design efficient algoritms and to predict convergence factors for solving PDEs wit ig order accuracy [ 7]. Distributive relaxation for poroelasticity equations is optimized by LFA [4]. Using LFA, textbook efficiency multigrid solver for compressible Navier-Stokes equations is designed [5]. Allat-once multigrid approac for optimality systems wit LFA is discussed in detail, and an analytical expression of te convergence factors is given by using symbolic computation [6 8]. Te smooting analysis of te distributive relaxations forsolving2dstokesflowisinvestigatedwitlfa.aswe know, te discretizing Stokes flow in computational domain is not stable by means of standard central differencing on nonstaggered grid. Tus, in order to overcome te stability

2 2 Matematical Problems in Engineering problem, an artificial pressure term is generally added by te metod in [, 2]. Te optimal one-stage relaxation parameter and related smooting factor of te distributive relaxation wit te red-black Jacobi point relaxation need to be developed. In deriving an explicit formulation of te smooting factor for te multigrid metod, te symbolic operation process is carried out by using te MATLAB and Matematica software, especially, by te cylindrical algebraic decomposition (CAD) function in te Matematica build-in command [9]. 2. Discretizing Stokes Flow and LFA 2.. Discrete Stokes Flow. 3D steady incompressible Newtonian flow governed by Navier-Stokes equations is given as Δ u+ p = f (x,y,z) Ω, u=0 (x,y,z) Ω, () u= g (x,y,z) Ω, were u = (u(x, y, z), V(x, y, z), w(x, y, z)) is te velocity field, p = p(x,y,z) represents te pressure, f = (f (x, y, z), f 2 (x, y, z), f 3 (x,y,z))is te external force field, (x,y,z) Ω R 3,and Ω is te Diriclet boundary of te computing domain. From (), 2D Stokes operator is written as on nonstaggered grid Δ 0 x L=( 0 Δ y ) (2) x y 0 G ={ x=(x,y):=(k, k 2 ) (k,k 2 ) Z 2 }. (3) From [], te above nonstaggered scemes (4) are not stable. Stabilization may be acieved by adding an artificial elliptic pressure term c 2 Δ to te continuity equation in (2) [, 2, 6]. Wit discrete operator in (5)and parameter c, te discrete Stokes operator is canged to Δ 0 x L =( 0 Δ y ). (6) x y c 2 Δ 2.2. Elements of LFA in Multigrid. In LFA, a current approximation and its corresponding error and residual are represented by a linear combination of certain exponential functions, for example, Fourier modes, wic form a unitary basis in space on a bounded infinite grid functions [ 7]. From [, 2], on nonstaggered grid (3), a unitary basis of te Fourier modes is defined by φ ( θ, x) := exp ( i θ x ), (7) in wic θ = (θ,θ 2 ) Θ := ( π,π] 2 is called Fourier frequency, x G, and complex unit i =. Tus,a Fourier space is yielded as F( θ):=span {φ ( θ, x) θ Θ}. (8) From [ 7], applying (3) and (7), for 2D scalar discrete operator D wit discrete stencil D =[l k ], (9) were l k R and k J Z 2 containing (0, 0);teFourier mode of (9) is defined by Discretizing Stokes operator (2) by means of standard central differencing is given as D ( θ):= k J l k exp (i θ k), (0) Δ 0 x L =( 0 Δ y), (4) x y 0 were denotes te uniform mes size and Δ, x,and y are te second-order difference operator wit te following discrete stencils: Δ = [ 4 2 ], [ ] x = 2 [ 0 ], y = [ 0 ]. 2 [ ] (5) wit θ k=θ k +θ 2 k 2,subjectedto D φ ( θ, x) = D ( θ)φ ( θ, x). () A main idea of LFA is to analyze relaxation properties in multigrid for (6) by evaluating teir effects on te Fourier components. From [2, 4, 6], if standard coarsening in 2D is selected, eac low frequency θ= θ 00 Θ 2 low = ( π/2, π/2]2 is coupled wit tree ig frequencies { θ, θ 0, θ 0 } Θ 2 ig in te transition from G to G 2,wereΘ 2 ig = Θ\Θ2 low, and θ α = θ (α sign (θ ),α 2 sign (θ 2 )) π, (2) were α Λ = {00,, 0, 0} and α=(α,α 2 ) are denoted by (α,α 2 ):=α α 2.Intispaper,standardcoarseningisto

3 Matematical Problems in Engineering 3 be assumed. Ten, te Fourier space (8) is subdivided into 2-armonics subspaces were Δ ( θ)= 2 (4 exp ( iθ ) exp (iθ ) F 2 ( θ):= span {φ ( θ 00, x),φ ( θ, x), φ ( θ 0, x),φ ( θ 0, x)}. (3) exp ( iθ 2 ) exp (iθ 2 )) = 2 (4 2 cos θ 2cos θ 2 ), (8) 3. Smooting Process 3.. Distributive Relaxation of System (6). From [, 2, 7], a distributive operator for te discrete system (6) is constructed as I 0 x C =( 0 I ), (4) y 0 0 Δ were I is te unit operator wit discrete stencil [].From (4), te discrete system (6) is transformed as Δ 0 0 L C =( 0 Δ 0 ), (5) x y c 2 Δ 2 Δ 2 were te discrete stencils of Δ 2 and Δ 2 are Δ 2 = , [ ] [ Δ 2 = 4 2 ] [ ] [ = [ ]. [ ] 2 ] (6) From (9) (), te Fourier modes of te scalar discrete operators of (6) are x ( θ)= 2 (exp (iθ ) exp ( iθ )) = i sin θ, (9) y ( θ)= 2 (exp (iθ 2) exp ( iθ 2 )) = i sin θ 2. (20) 3.2. Optimal One-Stage Relaxation. For te discrete scalar operator of (5), standard coarsening and an ideal coarse grid correction operator [2]are applied as Q 2 F 2 ( =: Q _ 2 θ) = diag (0,,, ) C4 4, (2) were Q _ 2 is te Fourier representation of te operator Q 2 wit subspace (3), wic suppresses te low frequency error components and makes te ig frequency components uncanged. Ten, from [2], te smooting factor for discrete operator (9) is defined by ρ (n, D ) = sup (ρ ( Q _ 2 (_ S ( θ,ω)) n /n )). (22) θ Θlow It implies tat te asymptotic error reduction of te ig frequency error components is given by n sweeps of te relaxation metod, were _ S ( θ,ω)is te Fourier representation of te relaxation operator S (ω) on subspace (3) and ω is te relaxation parameter. From [2, 4], a good smooting factor is obtained by using one-stage parameter ω in te relaxation operator S (ω); te optimal smooting factor and related smooting parameter are given by ω opt = ρ opt = 2 2 S max S min, S max S min 2 S max S min, (23) were S max and S min aretemaxandmineigenvaluesofte product matrix Q _ 2 _ S ( θ,) wit te relaxation parameter ω = for θ Θ 2 low.from[2, 9], te smooting factor of (6) wit te distributive relaxation (4) is determined by te diagonal blocks of te transformed system (5),wicisgiven by Δ 2 ( θ)=( Δ ( θ)) 2, Δ 2 ( θ)= [ x ( θ)] 2 [ y ( θ)] 2, (7) ρ(n,l )=max {ρ (n, Δ ),ρ(n,c 2 Δ 2 Δ 2)}. (24) 3.3. Optimal Smooting for Stokes Flow. Te red-black Jacobi point relaxation S RB is applied to (5) to discuss te optimal

4 4 Matematical Problems in Engineering smooting problems for Stokes flow. From [, 2, 4], te operator S RB makes te 2-armonics subspace (3) invariant; tat is, S RB F 2 ( =: _ S RB θ) ( θ) C 4 4, (25) in wic A α = D ( θ α ) (27) α D 0 ( θ ) were _ S RB ( θ) is te Fourier representation of S RB (ω) wit relaxation parameter ω=and is given as _ RB S ( θ,) = _ S RB ( θ) = _ S B ( θ) _ S R ( θ) A 00 + A = 2 ( A 00 A ) 0 0 A 0 + A A 0 + A 0 + A 00 + A ( A 00 A ), 0 0 A 0 + A A 0 A 0 + (26) denotes te Fourier mode of te point Jacobi relaxation for te discrete operator (9) on subspace (3) and D 0 ( θ α ) is te Fouriermodeoftediscreteoperatorwittestencil[l (0,0) ] in (9).Fortesakeofconvenientdiscussionintefollowing, two variables are introduced as s = sin 2 θ 0 2 = sin2 θ 2, s 2 = sin 2 θ = sin2 θ 2 2. (28) Tus, θ = (θ,θ 2 ) Θ 2 low = ( π/2, π/2]2 is transformed to s=(s,s 2 ) S low = [0, /2] 2. Teorem. For te Poisson operator Δ,teoptimalonestage relaxation parameter and related smooting factor of te red-black Jacobi point relaxation are stated as ω opt = 6 5, ρ opt = 5. (29) Proof. For te red-black Jacobi point relaxation for te Poisson operator D = Δ,substituting(2), (8), and(28) into (26) and (27),andfrom(5),teproductof(2) and (25) is written as _ Q 2 _ S ( θ,)= Q _ 2 _ S ( θ) = 2 ( (s +s 2 )( s s 2 ) (s +s 2 )(s +s 2 ) 0 0 ). 0 0 (s s 2 )(s s 2 +) (s 2 s )(s s 2 +) 0 0 (s s 2 )(s 2 s +) (s 2 s )(s 2 s +) (30) Tus, eigenvalues of (30) are obtained as λ =0, λ 2 =(s s 2 ) 2, λ 3 =0, λ 4 = (s +s 2 )(s +s 2 ). 2 (3) From (3),te max and min eigenvalues of(30) are yielded as S max = max {λ,λ 2,λ 3,λ 4 } = max (s,s 2 ) [0,/2] 2 (s,s 2 ) [0,/2] 2λ 2 = 4, S min = min {λ,λ 2,λ 3,λ 4 }= min (s,s 2 ) [0,/2] 2 (s,s 2 ) [0,/2] 2λ 4 = 8. From (23) and (32), (29) is obtained. Teorem olds. (32) Next, ρ(n, c 2 Δ 2 Δ 2) for te red-black Jacobi point relaxation need to be computed. Meanwile, te smooting factor of distributive relaxation (5) is given as follows. Teorem 2. For te discrete operator c 2 Δ 2 Δ 2 wit c> 0, te optimal one-stage relaxation parameter and related

5 Matematical Problems in Engineering 5 smooting factor of te red-black Jacobi point relaxation are given by + 20c { 0<c ω opt = + 6c 32 { 2 ( + 20c) 2 { + 56c + 744c 2 32 <c 2, { 0<c + 6c 32 ρ opt = { + 24c + 04c 2 { + 56c + 744c 2 32 <c 2. Proof. For te discrete operator (33) D =c 2 Δ 2 Δ 2, (34) from (7) (20),teFouriermodeof(34) is given by D ( θ)= 2 [4c (2 cos θ cos θ 2 ) 2 + sin 2 θ + sin 2 θ 2 ]. (35) Tus, wen te red-black point relaxation is applied to (34), from (6),substituting(2), (28),and(35) into (26) and (27), te product of (2) and (25) is _ Q 2 _ RB S ( θ,)= _ Q 2 _ RB S ( θ)= 4 diag (R,R 22 ), (36) were bot R and R 22 are 2 2 square matrices, wose expressions are below: R =( ) (AB 00 + AB AR A B 00 AB +) (A R 00 AR +) = 4 ( + 20c) 2 ( + 4 [ 36c + 6c (s +s 2 )] [s s 2 +s 2 s c(s +s 2 ) 2 ] R c ( + s +s 2 )[s s 2 +s 2 s c( 2+s +s 2 ) 2 ] ), R 22 =( 0 0 ) (AB 0 + AB AR 0 A B 0 + AB 0 +) (A R 0 AR 0 +) = 4 ( + 20c) 2 R + 24c + 80c 2 (48c + 4) (s +s 2 ) +384c 2 (s s 2 ) + (92c 2 +4)(s 2 +s2 2 ) [ (256c 2 64c) (s 3 s3 2 ) 64cs 64c (s s 2 )[ 4c (s s 2 ) 2 +8c(s s 2 ) 2 ] +4c + s s 2 +s ] 2 s 2 2 [ +28cs cs s 2 (2s 2 2s )( 2c) ] + 24c + 80c 2. (48c + 4) (s +s 2 ) ( 64c (s s 2 )[ 4c (s s 2 ) 2 8c(s s 2 ) 384c 2 (s +4c + s s 2 +s ] s 2 ) + (4 + 92c 2 )(s 2 +s2 2 ) ) 2 s 2 2 [ + (256c 2 64c) (s 3 s3 2 ) 64cs ] ( [ +28cs cs s 2 (2s 2s 2 +)( 2c) ]) (37) Tus, te eigenvalues of matrix (36) are obtained as λ 3,4 = λ 2 = ( + 20c) 2 λ =0, 64c ( + 20c) 2 ( + s +s 2 )[s s 2 +s 2 s c( 2+s +s 2 ) 2 ], + 24c + 80c 2 (4 + 80c) (s +s 2 ) + (4 + 64c + 92c 2 )(s 2 +s2 2 ) + (32c 384c2 )s s 2 [ ±32c (s s 2 ) + 80c2 + 24c + ( 64c c + 4) (s 2 +s2 2 ). (39) ] [ (80c + 4) (s +s 2 ) + (28c c) s s 2 ] (38)

6 6 Matematical Problems in Engineering By using te MATLAB and Matematica software wit cylindrical algebraic decomposition function [9], for s = (s,s 2 ) (0, /2) 2, tere is no extreme value for (39); wen 0 < c /32, one of extreme values of (38) is obtained as s = 64c c 3, 48c 6 s 2 = 64c c 3. 48c 6 (40) Tus, for s S low = [0, /2] 2, besides (40), tepossible extreme values of te eigenvalues of matrix (36) are placed on te boundary of S low.from λ k wit k=,...,4, ten 0 < c /2.Notingtat(40) exists wit 0 < c /32. From (38) (40), wen0 < c /2, for s S low,temax and min eigenvalues of (36) are yielded as S max =λ 3,4 (0, 0) = +4c + 20c, { λ 3,4 ( S min = 2, 4c )= 0<c c 32 { λ { 2 (0, 0) = ( 32c 2 20c + ) 32 <c 2. (4) Substituting (4) into (23), (33) is obtained. Teorem 2 olds. From (33), /2 ρ opt (c 2 Δ 2 Δ 2) <olds wit 0<c /2. Terefore, from Teorems and 2,wen0 < c /2, te smooting factor of (6) wit te distributive relaxation (4) is as 2 ρ opt (L ) 4. Conclusions = max {ρ opt ( Δ ),ρ opt (c 2 Δ 2 Δ 2)} =ρ opt (c 2 Δ 2 Δ 2) <. (42) Te smooting analysis process of te distributive redblack Jacobi point relaxation for solving 2D Stokes flow is analytically presented. Applying (28), te Fourier modes wit te trigonometric functions for te discrete operator and relaxation are mapped to rational functions. So, it is possible to apply te cylindrical algebraic decomposition function in tematematicasoftwaretorealizecomplexsmootinganalysis, and te computation process is simplified. Te analytical expressions of te smooting factor for te distributive redblack Jacobi point relaxation are obtained, wic is an upper bound for te smooting rates and is independent of te mes size wit te parameter c. Obviously,itisvaluableto understand numerical experiments in multigrid metod. Conflict of Interests Te autors declare tat tere is no conflict of interests regarding te publication of tis paper. Acknowledgments Te autors were supported by te National Natural Science Foundation of Cina (NSFC) (Grant no ) and te Doctoral Foundation of Ministry of Education of Cina (Grant no ). References [] U. Trottenberg, C. W. Oosterlee, and A. Sculler, Multigrid, Academic Press, San Diego, Calif, USA, 200. [2] R. Wienands and W. Joppic, Practical Fourier Analysis for Multigrid Metods,Capman&Hall,CRCPress,2005. [3]W.L.Briggs,V.E.Henson,andS.McCormick,AMultigrid Tutorial, Society for Industrial and Applied Matematics, 2nd edition, [4] W. Hackbusc, MultigridMetodsandApplications,Springer, Berlin, Germany, 985. [5] P. Wesseling, An Introduction to Multigrid Metods,JonWiley & Sons, Cicester, UK, 992. [6] K. Stuben and U. Trottenberg, Multigrid metods: fundamental algoritms, model problem analysis and applications, in Multigrid Metods,W.HackbuscandU.Trottenberg,Eds.,vol. 960 of Lectwe Notes in Matematics, pp. 76, Springer, Berlin, Germany, 982. [7] A. Brandt and O. E. Livne, 984 Guide to Multigrid Development in Multigrid Metods, Society for Industrial and Applied Matematics, 20, ttp:// aci/ classics.pdf. [8] C.W.OosterleeandF.J.G.Lorenz, Multigridmetodsforte stokes system, ComputinginScienceandEngineering, vol. 8, no. 6, Article ID 7733, pp , [9] A. Brandt and N. Dinar, Multigrid Solutions to Elliptic Llow Problems, Institute for Computer Applications in Science and Engineering, NASA Langley Researc Center, 979. [0] G. Wittum, Multi-grid metods for stokes and navier-stokes equations, Numerisce Matematik,vol.54,no.5,pp , 989. [] M. Wang and L. Cen, Multigrid metods for te Stokes equations using distributive Gauss-Seidel relaxations based on te least squares commutator, Scientific Computing, vol.56,no.2,pp ,203. [2] M. ur Reman, T. Geenen, C. Vuik, G. Segal, and S. P. MacLaclan, On iterative metods for te incompressible Stokes problem, International Journal for Numerical Metods in Fluids,vol.65,no.0,pp ,20. [3] C. Bacuta, P. S. Vassilevski, and S. Zang, A new approac for solving Stokes systems arising from a distributive relaxation metod, Numerical Metods for Partial Differential Equations, vol. 27, no. 4, pp , 20. [4]R.Wienands,F.J.Gaspar,F.J.Lisbona,andC.W.Oosterlee, An efficient multigrid solver based on distributive smooting for poroelasticity equations, Computing,vol.73,no.2,pp.99 9, [5] W.Liao,B.Diskin,Y.Peng,andL.-S.Luo, Textbook-efficiency multigrid solver for tree-dimensional unsteady compressible Navier-Stokes equations, Computational Pysics,vol. 227,no.5,pp ,2008. [6] V. Pillwein and S. Takacs, A local Fourier convergence analysis of a multigrid metod using symbolic computation, Symbolic Computation,vol.63,pp. 20,204.

7 Matematical Problems in Engineering 7 [7] S. Takacs, All-at-once multigrid metods for optimality systems arising from optimal control problems [P.D. tesis], Joannes Kepler University Linz, Doctoral Program Computational Matematics, 202. [8] V. Pillwein and S. Takacs, Smooting analysis of an all-at-once multigrid approac for optimal control problems using symboli ccomputation, in Numerical and Symbolic Scientific Computing: Progress and Prospects, U. Langer and P. Paule, Eds., Springer, Wien,Austria,20. [9] M. Kauers, How to use cylindrical algebraic decomposition, SeminaireLotaringiendeCombinatoire,vol.65,articleB65a, pp. 6, 20.

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