Applied Mathematics, 2011, 2, 866873 doi:10.4236/am.2011.27116 Published Online July 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Analysis of an Il’in Scheme for a System of Singularly Perturbed ConvectionDiffusion Equations Mohammad Ghorbanzadeh, Asghar Kerayechian Department of Ap pl i e d M athematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mash had, Iran Email: Ghorbanzadeh_imamreza@yahoo.com, krachian@math.um.ac.ir Received May 8, 2011; revised May 22, 2011; accepted July 22, 2011 Abstract In this paper, a numerical solution for a system of singularly perturbed convectiondiffusion equations is studied. The system is discretized by the Il’in scheme on a uniform mesh. It is proved that the numerical scheme has first order accuracy, which is uniform with respect to the perturbation parameters. We show that the condition number of the discrete linear system obtained from applying the Il’in scheme for a system of singularly perturbed convectiondiffusion equations is O(N) and the relevant coefficient matrix is well con ditioned in comparison with the matrices obtained from applying upwind finite difference schemes on this problem. Numerical results confirm the theory of the method. Keywords: ConvectionDiffusion, Il’in Scheme, Uniform Convergence, Singular Perturbation, Condition Number 1. Introduction Consider the following system of coupled singularly perturbed convectiondiffusion equations: Find such that l =Au =u 2 1,,(0,1)0,1 l l uuC C :=LEB uuu , =0,1, 0= 1=0,x f uu (1.1) With 12 =diag,,,l E , where , i for are known small positive diffusion coefficients, =1,,il ij = ax ll is an matrix, and i = xf for is a vectorvalued right hand side. Furthermore, we shall assume that is diagonal with diagonal elements and define =1,i,l i bx B [0,1] =>0 for =1,,. min kk xbxk l (1.2) Some results for systems of singularly perturbed of differential equations can be found in: Linss and Madden [1], Madden and Stynes [2] and Gracia and Lisbona [3]. Bellew and O’Riordan [4], Cen [5], Amiraliyev [6] and Andreev [7] used the finite difference method for a coupled system of two singularly p erturbed convection – diffusion equations. T. Linss [8] considered an upwind finite difference scheme on special layer adapted Shishkin and Bakhva lov meshes. He showed that the error in the discrete maximum norm is bounded by and 1lnCN N 1 CN for Shishkin and Bakhvalov meshes respectively, where is independent of the perturbation parameters k C for , and is the number of mesh points used. =1, ,klN The discrete linear systems that arise from Shishkin or Bakhvalov meshes do not have a good condition number. H. G. Roos in [9] showed that the condition number of the discrete linear system associated with the upwind schemes on Shishkin meshes for a single equation is 22 2 ln ONN , which is not good when is small. In fact, if is the coefficient matrix of the linear system associated with the upwind schemes on Shishkin meshes, then 2 22 ln N AC N and 1 C , where Cis a constant independent of and the norm is the discrete maximum norm for matrices. Nevertheless, he proposed a precondition which has reduced this condition number to 21 ln ON N . In this paper, we study the Il’in scheme (see [10]), for problem (1.1). We show that for this method the error in the discrete maximum norm is bounded by 1 CN , where is independent of the perturbation parameters k C =1,,kl , and we prove that if is the coefficients
M. GHORBANZADEH ET AL.867 matrix of the linear system associated with the Il’in scheme on uniform meshes, then CN and 1<. C So the condition number of the discrete linear system associated with the Il’in scheme on uniform meshes for single equation is , which is better in comparison with the precondition of upwind schemes on Shishkin meshes. ON The paper is organized as follows: in section 2, we give some properties of the solution of (1.1) and in section 3, we state the difference approximation Il’in scheme. In section 4, we analyze the error of Il’in scheme applied to (1.1), and some numerical examples are presented in section 5. 2. Properties of the Exact Solutions In this section, we analyze the exact solution of (1.1). Assume that 0,1vL 01 =<<<x and , where =0 =1 iN xxx = i ih and =1hN. Consider the following norms: , (0,1) 1 11, 0=0 =,= max max =d,= i xx i N i i vvxv vvxxvhv , , v N where i for Consider the fol lowing discrete norm: = i vvx = 0,,.i 1, 1,1 =1 =, N hi Wi vvhDv where the space 1,1 contains , such that h W1N vR 1,1 <. h W v Difference operators are defined as follows: 0 11 =,=,= 2 ii iiii iii uu uuuu Du Du Du hh 11 . h Let 11 00 == NN N RvRvv =0, and consider the space on We define the following norm on as 0 1,1 h W1 0. N R 0 1,1 h W01,1=. hi WDv =1 N i vh Also for the dual of , which is denoted by 0 1,1 h W1, h W , we define 01,1 1, < ,> =max , h h WW vf fv where 1 =1 ,= N ii i vf hvf . In [11], it has been shown that , 1, = min h WcR Dff c, (2.1) or equivalently , 1, := = min h WFDFf f.F (2.2) For the vectorvalued function 1 =,, T l vx vx v, consider the infinity norm as =1, , =. max i il xv v To estimate the error in our difference approximation, we shall require some bounds for the derivatives of the solution of (1.1), under the assumptions =1 0, ()>0 and () 0 for , [0,1] and ,=1,, l ij iiij j ax axax ijxij l (2.3) and strict inequality hold at least for one i.e., k =1 >0 l kj j ax (2.4) Lemma 2.1 If , in 1 =,, T l yx yxy 0Ly and ,10y 0xy00y then in . Proof. Let i x be minimum at i for , i.e, t =1,2, , il ii yt =min i xyx and also assuming = =1, , min ji il t yt yi (2.5) If 0 jj yt the lemma is proved. So let <0 jj yt . If = jkj tyt y for , then it follows =1,2,,kl that =0 j t y and 0 j t y. By (1.1) := =. jj tj jkj LEtBtAt At ytA yyyy y1 In this case according to (2.4) since , the th component of <0 kj yt k ty is negative, which is a contradiction to the assumption of the lemma. If there is a with 1kkl such that < jkj ytyt then =1 =1 =1 , =1, ,=1, =1 = max j jjjjjjjmj mj m l jjjjm jj j m l jm jm jj j mmj ll mj jjjmjjjjmj ml mmjm ytbtyta tyt yta tyt atyt yt ytyta tyta t l If =1, <0 l jm j mmj at , it is obvious that the right hand side of the above inequality is negative. If =1, =0 jm j mmj at l , then since we have >0 jj a =1 >0 jm j at l m so the right hand side is negative and again we reach a contradiction. So the lemma is proved. Copyright © 2011 SciRes. AM
M. GHORBANZADEH ET AL. Copyright © 2011 SciRes. AM 868 3. Discretization Theorem 2.2 Suppose solves (1.1) and assume that for satisfies (2.3) and (2.4), then u ij a,=1, ,ij l .uC By theorem 2.2, T.Linss [10] has proved the following lemma. f Lemma 2.3 Let be the solution of (1.1) and suppose (1.2), (2.3) and (2.4) hold. Then for u 0,1x and =0,1n 1exp 1 1 exp. nk kk k n k k n kk k k x Cb ux x Cb k ,,, By lemma (2.3) and the application of the technique mentioned in [12], the following lemma can be proved in a similar way. Lemma 2.4 Let satisfy (1.1). If u =, for =1 kkk uxvx zxkl where 1 1 0exp 0 0 =1exp 11, 1 kk kk kk k k kk kkk k k ubxb b vx ubxb b then, for 0,1x and =0,1 n . n k zxC Remark 2.5 By lemma 2.4 and direct computation, without loss of generality, assume that k bk ; hence, we have =1 1 =1 = =00 l kkkkk kmm m l kkmmkkkkk m k Lzbxzxazx avxb bbxvx gx z and for =0,1 n 1exp, =1,, nnk kk k x xC k l In this section, we deal with the discretization of probem (1.1) by the Il’in scheme. We apply the Il’in scheme : Find such that: 1 0 l N UR 0 ;; ;; =1 =, kk kjkjkjkj j l km jmjk j m LUD DUbDU aU f := (3.1) for and , where =1,, kl=1, ,1jN =coth kk k qx qx with =2 k kk hb x qx , kj U is the approximate value for that is obtained by the Il’in scheme, ; bb and kj ux j x= kj k ;= km jkmj aax. Consider the diagonal matrices 1 =diag,,l Qxqx qx and 1 coth()= diagcoth,,coth. l Qxq xq x We define 11 =coth= coth 22 hE BhE B Qx Qx . We rewrite (1.1) in matrix form as follows: 0= for =1,1, jj jjj EDDU BDUAUf jN j (3.2) where 1 =,, T jjlj UU U, 1 =,, T jl DU DUDU j lj , 1 =,, T jj UDUDU 00 0 1 =,, T and jlj UDU DU The corresponding coeffi cient matrix in (3.1) is: 11 121 21 222 12 = l l h ll ll DD D DD D A DD D (3.3) where blocks ij in Dh are and diagonal blocks for are tridiagonal matrices as follws 1NN , ,l 1 kk D=1k ;1;1 ;1 ;1 22 ;2 ;2;2;2 ;2 ;2 22 2 ;1 ;1;1;1 22 20 2 20 =22 2 02 kkkk k kk kk kkkkk k kk kk kkN kNkkN kk N b ah hh bb a Dhh hh h ba hhh
M. GHORBANZADEH ET AL. Copyright © 2011 SciRes. AM 869 b and for where are diagonal matrix as ij D,=1, ,ij lij ;1;1 =diag,, ijijij N Daa . Remark 3.1 When is constant and , the system (1.1) is reduced to 1 =diag,, l Bb 0A =.EB uuf Then its general solution for homogeneous equation, which is also boundary layer function, is =exp for >0 and 1 =exp for <0, k kk k k k kk k k bx ux Cb bx ux Cb where for are arbitrary constants. The difference operator (3.2) is exact for these boundary layer functions. i.e., k C=1,,kl 0= for =1,,1, jj jj j EDDU BDUEB jN uu j (3.4) where index j indicates the evaluation at point , and . 1; ; =,,T jjlj uuu ;= kjk j uux In what follows, we try to estimate the condition number of the discrete linear system associated with the Il’in scheme presented. Then we apply Mcriterion [13], [14] to obtain the condition number. Theorem 3.2 (Mcriterion). Let a matrix satisfy 0 ij a for ij . Then is an Mmatrix if and only if there exists vector such that . Further more, we have >0e>0Ae 1. min k k e A e (3.5) Theorem 3.3 The corresponding coefficient matrix in (3.1) is an Mmatrix and satisfies h CN and 12 h A where is independent of and CN k for and =1,,kl =1, , minkl k = Proof. To evaluate the maximum norm of the matrix h for the ith row of the th row of the block matrix k h , where =1,,1iN and , we have =1,,k l 1;;; ;; ;; =1 =1 ;;;;;;;; 2=1 ;;; ;; 2=1 1 cothcoth1 coth 22 =1coth2coth2coth1 coth =coth11coth = lN l kikiki h ijkikiki kmi j m l kkikiki kiki kikikikmi m l kki kikikikmi m bbb aqq qa hhh qqqqqqq qa h qqqq a h ; ;; ; 2=1 4coth . l kki kikmi m qq a h By 2; ;; ; coth 1, 1 ki ki kiki q qq C q See [14], we conclude that 2 1; ;; ; 22 =1=1 ; 2;; 2=1 ; 22 2 ;; ;; 2 2=1 =1 ; 4coth111 1 22 24 2 =. 2 22 lN lki hkk ijkik ikm i jm ki l ki kkm i m ki k ll kii kkkik km ikm i mm ki k q aqq aC q hh b Ca h bh bh bhbh CaC h bh h a Since in singularly perturbed equations ,h is considered, hence
M. GHORBANZADEH ET AL. 870 1 =1 21=. 2 lN i ij j b aCCCN hh Therefore, 1 =1 =. max N hij ij aC N kx If we choose where 1 =,,T l ee e=1 ii e for and for k, then <0 k bx =1 k ii ex bx>0 h e h for the th row of the th row of the block matrix i k for case we have bx<0 k ;;; 1 ;; ; ;;; 1 ;; =1, =coth 1 222 coth1 2 coth1 222 > kiki ki hi ik kiki kk ii k kiki kii k lm km iik i mmk bhbb Ax hh bhb ax h bhbb x hh ae b e > k and from (3.1), for we have . Hence the Mcriterion yields that ijh0 h ij a is an Mmatrix and 12. min hh k k e AAe A similar proof is for the case . >0 k bx Notation. Throughout the above theorem we let denote positive constant that may take different values in different formulas but that are always independent of both the perturbation parameters C k and of , the number of mesh intervals. By theorem 3.3, the condition number of N is of order ON. We recall that, in theorem 3.2 and theorem 3.3, is the usual maximum norm of matrices. This shows that the matrices arising from the Il’in scheme for discretization of a singularly perturbed differential equation are well conditioned respect to the upwind finite difference method applied on Shiskin or Bakhvalov meshes. For arbitrary , we define 1 10 =,,l N l UU UR ,, =1, , =. max i il UU A single equation of (3.1) can be written as follows: 0 ; ;; ; =1, := =. kkkkjkjkjkj j l kk jkjk jkmjmj mmk LUqD DUbDU aU faU (3.6) Suppose solves (3.1). We propose the following lemma to obtain a bound for U.U Lemma 3.4 Suppose is a matrix such that and >0 ii a 0 ij a for )( ji and Also assume ,=1, ,.ijn that for Then for every arbi trary vector we have =1 1 jk ka l=1,,.j ,T n n 1 =, ,, A Proof. Suppose for the element of j , ,=. Without lose of generality, let = j (otherwise we consider = A ). =1 =1, =1 , == , nn kk jjjjkk jkkkj n jj jjkk kkj Aaa a aa (since 0 jk a for jk ). Therefore =1 , =1, =1 =1, =0 n jjjj jkkj jkkj nn . jjjk kjkj kkjk n jk kj kkj Aa a aaa a So >0. j j A Hence ,, =1, , == max j kj kn AAA Similar to theorem 2.2, we have the following theorem in the discrete case. Theorem 3.5 Suppose solves (3.1). Assume and satisfy (1.2), (2.3) and (2.4). Then Uk b ik a ,, .UCf Proof. Dividing (3.6) by , we have ;kkj a ; ;; ;; =1, ; =:= . kj kkkkkkj j jkkj lkmjmj mmkkk j LULUaa aU a (3.7) The matrix associated with operator is a matrix that satisfy lemma 3.4. We have k L ,,, , =1 , . k kkkkk lkm m mmkkk f ULUa aU a Therefore Copyright © 2011 SciRes. AM
M. GHORBANZADEH ET AL. Copyright © 2011 SciRes. AM 871 1 = ==1,,=1, N ij i j Ghgi ljN ,, =1, . lk kkmm mmkkk f UU a 1, (4.4) where i x for is defined in remark 2.5. By (4.1)(4.4), we can establish the following theorem. =1,,il By (2.3) and after some manipulation, we obtain ,, .UCf Theorem 4.1 Let be the solution of (1.1) and be the solution obtained by the Il’in scheme (3.1). Suppose the data uU 1 ,, ,0,1 kkkmk a fCgb for satisfy (1.2), (2.3) and (2.4). Then ,=km 1, ,l Corollary. According to theorem 3.5, we have ,,.UCLU uu (3.8) ,.U uCh (4.5) 4. Error Analysis Proof. From lemma 2.4, we can split u as = xuvzx, where v is a boundary layer function. Hence, For the error analysis of Il’in scheme applied to the system of singularly perturbed differential equations (1.1), suppose that for and satisfies (1.1). Similar to lemma 2.4, we split >0 i bx =1, ,ilu = u xvz, where 1 =, , T l vx ,()vxv= z 1 So we have ((), , l zx z()). T x k x= kk zxux , and for v = 0 n,1 ,,, =.UZVZV , uzvzv First, consider , Z z. From (4.1)(4.4), we have =(),=0,1 iiii Lgon zz and () , n k zx C =and= for= 0,,. iiij ij jj LZDAZgD G jN and 1 0 =exp0 0 kk kk k u vxb x b . k d , Thus, by = ii Lxgxz and =, ii j LZ gj =, =0, , ii iij j x AZ GjN 0,1 andxx z (4.6) First we analyze the truncation error in a single equation. We introduce the co ntinuous and discrete operators with constants and . 1 1 =1 = d0,1,=1, iiiiiii x l im m xm vxbvbsvss avssxil v (4.1) From (2.1), , 1, =(). min h ii WcR LZ AZc zz Taking =c , from (4.6) we get and , )( ~ =][ ; 1= 1 = ;1 1 = ; mim l m N j ii N j ijjiijiji VahbDhV VbxVDVA (4.2) , 1, . h iiiii i W LZAxG x zzz Furthermore, 1 ; = 1 ;1; = = dd N iiiiiiii jjj N xx NN iiiii xx jj j AG Dzzh where 1 =ii ivv Dv h and =. coth 1 ii h hqq , shzDb bzss zz where We note that . We introduce the continuous and discrete functions >hh =1 =. l iimm m az g i 1 =d=1, ii x, gs sil (4.3) Applying Taylor expansion with the integral remainder form and using lemma 2.4 and remark 2.5, we have and 11 11 ;;1 ; 111 1 d=dd() dd =dddd, xxt xtx iiiiii ii xxxxxx xxx x ii ii xtx t htt sstsdstChhzDbbz btzs stbtzs stCh tt
M. GHORBANZADEH ET AL. Copyright © 2011 SciRes. AM 872 and 1 1 =d dd. xx i iii i xt j xx ii ii xt Dzzhhzz sst h hhz zsst hh d On the other h a nd, we have =1 coth1 =. 2 iii i i ii hhq q hbh qCh Hence 1=1 1dd . iii j l xt iii imm xx m Dz z Chgb zazstCh h Thus ,. ii ii GC zz h In [15] it has been shown that ,1, 2. h i ii ZLZ zz Hence we have ,. Ch z (4.7) To bound the boundary layer function = v ,v , a direct computation gives T 1,, l vx vx 1 LE v =00BBBxA v and, in the grid points we can write 1 2 =si sinh sinh0 . LVQ xB xQ h QxQV AV nh0 By a technique used in [9], we can show that ,.VC vh (4.8) So by (4.7) a n d (4.8) we have ,.UC uh This completes the proof of the theorem. 5. Numerical Experiments In this section, we compare the Il’in scheme with the upwind finite difference scheme (see [8,15]) for the following two examples. Example 1. Consider 11 11211 222 1222 2= 0=1=0, 24=cos 0=1=0, x uuuueuu uuuuxu u In this example, we expect two layers at which behave like =0x 1 exp for and like 1=1b 2 2 exp for 22b . Let 8 1=10 and , which are sufficiently small values to bring out the singularly perturbed nature of the problem. The exact solution to the test problem is not available, so we estimate the accuracy of the numerical solution by comparing it to the numerical solution computed on the finer mesh. Let 6 2=10 U be a numerical solution in grid point. We estimate the error by N 2 , NN UU The rates of convergence rN are computed using the following formula: 2 , 224 , log . NN N NN UU rUU For the upwind scheme we use Shishkin mesh with 1=m and 22 1 =min, log 2N . We divide the inter vals 1 0, and 12 , into 4N subintervals and 2,1 into 2N subintervals of equal length. For Il’in scheme, we use the uniform mesh and divide the interval 0,1 into subintervals of equal length. Numerical results are contained in Table 1. From this Table, we observe that the Il’in scheme is a first order uniformly convergent method. N Example 2. Let 111 1 2 222 2 3 2 3313 23 3= 0.5= cos 52=1 x uuuue uuuu x uxuuu.x In this example, we expect layers 1 exp 3,x Table 1. Numerical results for example 1. upwind scheme Il’in scheme N Error Rate N r Error Rate N r 128 4.62e2 0.8019 5.00e  3 1 256 2.65e2 0.8461 2.50e  3 1 512 1.48e2 0.8945 1.25e  3 1 1024 7.90e3 0.9462 6.30e  4 1 2048 4.10e3   
M. GHORBANZADEH ET AL.873 Table 2. Numerical results for example 2. Upwind scheme Il’in scheme N Error Rate Error Rate 32 1.03e  2 1.0429 9.6e  3 0.9652 64 5.0e  2 0.9685 4.9e  3 0.9826 128 2.55e  2 0.9439 2.5e  3 0.9913 256 1.33e  2 0.9382 1.3e  3 0.9957 512 6.9e  3 0.9384 6.0e  4 1 1024 3.6e  3  3.0e  4  23 0.5 15 exp,exp. x 87 Let , 1=10 2=10 and We use the upwind finite difference scheme on a Shishkin mesh(See [11]). Assume that, 6 3=10. 312 12 3 3 3 =min,ln, =min,1ln, 234 0.5 1 =min, ln, 25 N N with 11 =min,3 and 31 =max, 3 . Then the mesh is obtained by dividing each of the intervals 112 23 , ,,,0, and 3,1 into 4 subintervals. Numerical results are shown in Table 2. From Table 1 and Table 2 we see that the rate of convergence for Il’in scheme is close to 1, which agree with the convergence estimate of theorem 4.1. Numerical results confirm the theoretical results. 11. References [1] T. Linss and N. Madden, “Accurate Solution of a System of Coupled Singularly Perturbed ReactionDiffusion Equations,” Computing, Vol. 73, No. 2, 2004, pp. 121133. doi:10.1007/s0060700400653 [2] N. Madden and M. Stynes, “A Uniformly Convergent Numerical Method for a Coupled System of Two Singu larly Perturbed Linear ReactionDiffusion Problems,” IMA Journal of Numerical Analysis, Vol. 23, No. 4, 2003, pp. 627644. doi:10.1093/imanum/23.4.627 [3] J. L. Gracia and F. J. Lisbona, “A Uniformly Convergent Scheme for a System of ReactionDiffusion Equations,” Journal of Computational and Applied Mathematics, Vol. 206, No. 1, 2007, pp. 116. doi:10.1016/j.cam.2006.06.005 [4] S. Bellew and E. O’Riordan, “A ParameterRobust Nu merical Method for a System of Two Singularly Per turbed ConvectionDiffusion Equations,” Applied Nu merical Mathematics, Vol. 51, No. 23, 2004, pp. 171186. doi:10.1016/j.apnum.2004.05.006 [5] Z. Cen, “ParameterUniform Finite Difference Scheme for a System of Coupled Singularly Perturbed Convec tionDiffusion Equations,” International Journal of Com puter Mathematics, Vol. 82, No. 2, 2005, pp. 177192. doi:10.1080/0020716042000301798 [6] G. M. Amiraliyev, “The Convergence of a Finite Differ ence Method on LayerAdapted Mesh for a Singularly Perturbed System,” Applied Mathematics and Computa tion, Vol. 162 No. 3, 2005, pp. 10231024. doi:10.1016/j.amc.2004.01.015 [7] V. B. Andreev and N. Kopteva, “On the Convergence, Uniform with Respect to a Small Parameter of Monotone ThreePoint FiniteDifference Approximations,” Journal of Difference Equations, Vol. 34, 1998, pp. 921929. [8] T. Linß, “Analysis of an Upwind FiniteDifference Scheme for a System of Coupled Singularly Perturbed ConvectionDiffusion Equations,” Computing, Vol. 79, No. 1, 2007, pp. 2332. doi:10.1007/s006070060215x [9] H. G. Roos, “A Note on the Conditioning of Upwind Schemes on Shishkin Meshes,” IMA Journal of Numeri cal Analysis, Vol. 16, No. 4, 1996, pp. 529538. doi:10.1093/imanum/16.4.529 [10] A. M. Il’in, “A Difference Scheme for a Differential Equation with a Small Parameter Affecting the Highest Derivative,” in Russian, Matematicheskie Zametki, Vol. 6, 1969, pp. 237248. [11] V. B. Andreev, “The Green Function and A Priori Esti mates of Solution of Monotone Three Point Singularly Perturbed FiniteDifference Schemes,” Differrence Equa tions, Vol. 37, No. 7, 2001, pp. 923933. doi:10.1023/A:1011949419389 [12] R. B. Kellogg and A. Tsan, “Analysis of Some Difference Approximations for a Singular Perturbation Problem with out Turning Points,” Mathematics of Computation, Vol. 32, 1978, pp. 10251039. doi:10.1090/S00255718197804834849 [13] O. Axelsson and L.Kolotilina, “Monotonicity and Discre tization Error Estimates,” SIAM Journal on Numerical Analysis, Vol. 27, No. 6, 1990, pp. 15911611. doi:10.1090/S00255718197804834849 [14] H. G. Roos, M. Stynes and L. Tobiska, “Robust Methods for Singularly Perturbed Differential Equations,” 2nd Edition, Springer Series in Computational Mathematics, Springer, Berlin, 2008. [15] T. Linß, “Analysis of a System of Singularly Perturbed CovectionDiffusion Equations with Strong Coupling,” SIAM Journal on Numerical Analysis, Vol. 47, No. 3, 2009, pp. 18471862.doi:10.1137/070683970 Copyright © 2011 SciRes. AM
