The purpose of this study is to elucidate the role of digital innovation in mediating digital capabilities on business performance. The sampling method used is purposive sampling. The number of samples is 75 respondents. Data collection is carried out by distributing questionnaire to SME managers in IT sector in Denpasar City. The data analysis technique used is the Path Analysis and Sobel Test technique. The results of the study show that the digital capability variable has a positive and significant effect on business performance. Likewise, digital innovation has a positive and significant effect on business performance, and digital innovation is able to mediate the effect of digital capabilities on business performance. Therefore, in the future, SMEs in IT sector must always improve their digital capabilities in order to develop digital innovations that have an impact on improving business performance.  Article visualizations

Yasa, Ni Nyoman Kerti

Ekawati, Ni Wayan

Rahmayanti, Putu Laksmita Dewi

Hemker, P.W. (Piet)

Zeeuw, P.M. (Paul) de

CWI's Institutional Repository

ma the ti ch C n trum AFDELING NUMERIEKE WISKUNDE (DEPARTMENT OF NUMERICAL MATHEMATICS) P.W. HEMKER & P.M. DEZEEUW DEFECT CORRECTION FOR THE SOLUTION OF A SINGULAR PERTURBATION PROBLEM Preprint amsterdam NW 132/82 AUGUSTUS 1982 stichting mathematisch centrum AFDELING NUMERIEKE WISKUNDE (DEPARTMENT OF NUMERICAL MATHEMATICS) P.W. HEMKER & P.M. DEZEEUW DEFECT CORRECTION FOR THE SOLUTION OF A SINGULAR PERTURBATION PROBLEM Preprint ~ MC NW 132/82 AUGUSTUS kruislaan 413 1098 SJ amsterdam Bl!:lLIOTHl:EK M/,\"Hi:MA T !SCH Cl:.N lHUM -AMSTERDAM-Pf1inte.d a.t the. Mathematlc.ai Ce.nt'Le, 413 KJt/.L,{,6la.a.n, Am6teJul.am. The. Mathematic.al. Ce.ntite. , 6ounde.d the. 11-th 06 Fe.bJw.aJty 1946, iA a. non-ptto 6-U i..nt,:ti..:tu:ti.,o n a.i.ming a,t, the. pll.Omoilo n o 6 p!.VLe mathemailC-6 an.cl ilA a:ppf.ic.ail.on&. 1:t iA -0po11oo1te.d by the NethvrltJ.nd6 Goven.nme.nt thlwu.gh the Nethelll.and6 0Jtga.nJza.Uon 601t the. Advanc.e.me.nt o 6 PuJte. Rel> eaJte.h ( Z. W. 0. ) • 1980 Mathematics subject classification: 65N05, 65N10 Defect correction for the solution of a singular perturbation problem*) by P.W. Hemker & P.M. de Zeeuw ABSTRACT A method is described for the accurate discretization of a differential equation in which te highest derivative is multiplied by a small parameter. It is well known that for such singular perturbation problems with a strongly asymmetric differential operator almost all discretizations are either unsta-ble or inaccurate or direction dependent. By the combination, in an iterative process, of an inaccurate stable and an accurate unstable scheme we obtain an accurate stable solution, without adapting the scheme to the flow direction. In fact, two approximate solutions are obtained, that -uniformly in£- are both O(h2) accurate in the smooth part of the solution. The difference between both solutions can be used for the detection of the unsmooth parts. KEY WORDS & PHRASES: singular perturbation problem; defeat correction; stiff boundary-value problem *) This report will be submitted for publication elsewherer 1 • INTRODUCTION An iterative process is used to obtain the accu-rate solution of a singular perturbation problem. As a model problem we use the convection diffu-sion equation -+ (la) -E~U +a.Vu= f, z on a bounded domain Qc]R , with either Dirichlet or natural boundary conditions (lb) u = g on an 1, -+ (le) nEVu =hon an2 , where~ is the outward normal on the boundary an= an 1uan2 • The problem is written in symbolic form as (2) LEu = f. It is well known that for such a problem with a strongly asymmetric differential operator, the usual discretizations are either unstable (the usual symmetric discretization methods: central differences, finite elements or Bubnov-Galerkin methods) or inaccurate (artificial viscosity) or direction dependent (various streamline-upwind or Petrov-Galerkin discretizations). By the combination, in an iterative process, of an in-accurate stable and an accurate unstable scheme, we construct a solution which is accurate, uniformly for all E, without adapting the scheme to the subcharacteristic directions (flow-direc-tions) in the problem. In the iterative process two standard discreti-zations of (2) are used: (3) LE,h uh= fh, a standard accurate discretization (e.g. standard central differences or a finite element discreti-zation with piecewise linear test and trial functions on a regular triangularization), which is unstable for e:«h , and an artificial diffusion (artificial viscosity) discretization (4) La,h ¾ = fh, which is the same as (3) but with a= E + O(h), whence (4) is stable. It is well known [6) that both discretizations yield bad results for small ratios E/h. We combine the discretizations (3) and (4) in an iterative process of defect correction type. In case of a linear problem an elementary defect correction process generates a sequence of appro-ximate solutions u< 1 ), i = I ,2,3, .•••• , by the iteration (iterative refinement) u(O)= 0 ' (5) Lu(i+l) = f - Lu(i) + Lu(i). If lim. u(i)= u* and Lis injective, then u* is the §6iution of the "target equation" Lu= f. Lis usually some approximation to L, for which the equation (5) is readily solved. For our problem we study a "mixed" defect correc-tion iteration with two target operators and two approximate operators. They are combined as: (6a) (6b) u(O)= O, 1 u<i+D= f ~1 (i+I) I L2u = f 2 -If this iteration converges, utions" uA = lim. u (i) and 1-+«> With f 1 = f 2 =: f, obviously by we obtain uB = lim. 1--UA is characterized two "sol-u(i+D. ~ ~ ~-I ~ A ~ ~-I (7) [L2-<L2-L2)Ll (LI-Ll)]u = [I-(L2-L2)LI ]f and uB is given by a similar equation. 2, THE DISCRETIZATION To obtain an approximate solution for (2), we use the process (6) with f 1 = f 2 = fh and with the operators L e:,h' 11 = L a,h' L a,h' 12 = 2 diag (La,h) =· D, In this way the second iteration step (6b) is a damped Jacobi-relaxion-step for the solution of (4). The first step (6a) is an order improving defect correction step towards the solution of (3), where the defect defining operator is simply given by ( 9) E := 1 1-1 1 = (a-e:)~h. A B uh and uh are now characterized The two solutions by (IO) (1 h + e:, (II) (L h + e:, By using local mode analysis [4] we can show that, for the smooth parts of ~he solution, {and~ are both accurate of O(h ). Furtherf the opera-tors L h + L hn-lE and L h + ED- L hare e:, o., e:, Ct, stable and the numerical boundary layers extend over a region of only O (h) in the boundary layer region [SJ. Remark. It is not possible to find an accurate approxima-tion for our problem (I) by application of the simple defect correction (5) alone. If we apply (5) with L = L h and i'. = L , for e:«h iteration would not conv~tge to a senitRle solution because of the instability of L . It would converge to the unwanted solution of•£-he "target problem" (3). Theoretically, already a single step (Qr a few steps) of (5) result in a 2nd order accurate method cf. [2] Satz 2.2 , from which follows II uh - ¾ull s: ch2 11 u"II • Here ¾u denotes the restriction of the true so-lution u to the mesh. However this is not a use-ful errorbound in our case, where 11 u"ll may be very la2ge. In fact, for small e:, we do not find the O(h) convergence in practice for any rea-sonable value of h. This is shown in table I. e: = I ~D D A B uh ¾ ¾ h = 1/8 0.630 0.0740 0.0780 0.0693 ratio 2.4? 3.65 3.64 3.46 h = 1/16 0.0255 0.0203 0.0214 0.0201 ratio 1. 71 4.02 4.01 3,89 h = 1/32 0.0149 0.00505 0.00533 0.00516 Table la. 2 10-6 uCD D A B e: = h ¾ ¾ ¾ h = 1/8 0.790 0.634 0.608 0.459 ratio 1.37 1.76 3.82 3.48 h = 1/16 0.578 0.360 0.159 0. 132 ratio 1.52 2.08 4.75 4.54 h = 1/32 0.380 0.173 0.0335 0.0291 Table lb. Table I shows the error in the smooth part of the solution in the max-norm. ~~): the solution after one step (6a); ¾ : the solution after two steps (6a); ut, u~ obtained by iteration of (6). The problem: e:~u + u = f on the unit square; with the Dirichlet b6undary data and the data f such that u (x,y) = sin(nx)sin(ny) + cos(nx)cos(3ny) + + (exp (-x/e:) - exp (-I /e:))/ ( I-exp (-1 /e:)). 3. RELATION WITH MULTIGRID TECHNIQUES The algorithm in section 2 can be considered independent of multigrid techniques. However, it is related to previous work done on multigrid methods. The use of defect correction in combination with a multigrid algorithm is already mentioned by Brandt[!] and is theoretically studied by Hack-busch [2,3]. They consider a multigrid algorithm which, in the elementary form of a two-level algorithm, can be described by a process (6) as well. Then, (6a) is a "coarse grid correction" which is now written in the form (i+I) = (i) + L-1 (f -L (i)) u u I l 1u , because~-! 1 1 is of deficient rank: ~-] -1 1 1 = Ph,2h1a,2h R2h,h' where Ph 2 and R2 hare the grid transfer operator~ Retween he fine and the coarse grid; (6b) denotes a sequence of relaxation sweeps. In the standard two-level algorithm both target operators are the same: 1 1 = 1 2• In combination with defect correction (non-standard), the ope-rator 1 1 corresponds to a more accurate discre-tization then 1 2• In the approach we discuss here, we differ from [1,2,3] by using a full rank operator 11 = L h' We use the multigrid algorithm only to a, solve efficiently the equation (6a). 4. THE DIFFERENCE BETWEEN {AND~ Considering the difference~ - { of the solutions of (IO) and (11), we find the following ( l 2) Proof Because 0f (bb) we have D (a~ - u~) L :1,8 - f t,h n h and hence,by (II), or (1 5) u 8 -· .. , La,h h "1, "' A ." uh . From ( n 3) and ( 1 5 ) the le~ ..a fo.t lows B A From (12) we see that the differ<i'nce uh - uh is lftrge for large ~~u:. In the regions where 1\,- ut is large, the 2nd order diffen,nces ot u~ are large. Here the solution cannot aceurate-ly be represented. Thus,we may use the differ-ence u:-ut as an indicator where the mesh should be refinea for a better approximation of the solution. To study the local behaviour of the difference u;-uA in more detail, we resort to the local rrride~~nalysis [ l }. We consider the problem (la) on R" and form the discrete operator L h over all R 2 . As a forcing function we takes' the umode 11 f, (jh) ,l ijh(;.; e ; 1 j ,,z~ Re(hw) Now the solutions A and uB are studied by Fourier analysis. ife note ~hat the Fourier transform is a norm-preservin~ bijection between the l 2 (a:2) and the L2 ([-r.,+r.]~) functions, i.e. i! u Ii ii G !! i.•here ~ is the Fourier transform of u. From derive (16) u.-u~=FT(Hon)diag- 1(L h)th n n a, A ) = u. ti A a-£( . 2( h''') . 2( h/Z))AA =: a-£ = - 2; sin w1 ,~_+sin w2 J uh 2a s2 '½i' and from (lO) we find (17) f ,AA - =~~s 2 (1+~=£s2) + ~! T (1+~=£s2) h1 uh - h" 2E h 2'.ll ' I (' ' ) . • • where 4iT,h = LO hlW 1s the characcer11t1c trigonometric polynomial for the reduced difference operator. In previous work [4,5] we derived from (17) that ~ ~s bounded, uniform in£, Moreover, u.A is 0th~) accurate in the smooth parts of th~ solution. We also showed that, for sma~l cA at boundary or interior layers the error in~ may be 0(1),uniformly in£. In these regions (where the discrete solution is not able to represent the true solution anyway) the numeric.al approx-imations may show oscillations. However, the discretization (lO) is asymptotic.ally stable and the critical regions near the layers have only O(h) width. 3 T(\ s~·t~ th~-: cf'!ei.:t ,,f b ~.- U er. tb~ .S?pr1.,xi.cu.tiouw \.:.::• ~t~idy ~epar;it~ly tl'i.- "11Jw> 11 .und th~ "high" frequet~·c·it:16 in the tr-:~~ ?So1utirm .. Tbe frequ,;nci«M art.• cdl led 11hig,h 1' i:ir f'low 1~ with rt.:ftl!i:r~ncw: to tht: i:w.~-:t,t:, ua,~d i,si1:.e h_). For ::h.~ 1~1~~1 frequent~'ii:s in th1.~ :.•".11:Jtim, ci,,.7 e,.~ (:orrn~idit?.r •• fixt:-d ,.,., and let h-.,.· O. F•.}r tht' 11hi~hn f r~;qut'!acii~Si ~·;.;: c,.)n~id«r ~h fi-Jl:.ed ,an.d Iet: ~1'-,-(L Regi~·i.1~ whcrtt~ the molutioo ii sr.iouth .an:1 charact.cri1:ed by 11.;w fr~qu€ncieft~ Region;!; w-hcrt: t:he mesh i~ tor: c-oart~ tu repr~-s~nt ~ sol~JtiQn prr,·,tH.~riy,flrli! dot:linated :)y high fr,equencies .. Now '.It• us,~ ( lo l to ~~e how u"-uA b~hnv~:3 it1 th~ '" h 1 h hf . different regions. S2 . ~ ' sin ... (i"'!h/2) + find fl A 1 u -u n•r Le .ow roequen;::1e11 sin2{~~h/2) ~ 0(h2) and we ,, C ~=~ h2 ·:i u,A Ii t'l ll :1 h 0 2 A Ji, C h~~ h t,.h ' Because that in u; is bounded!:t un.ifcrn, in ~ we ttle smooth part of the ~olution corn: l ude we have u~ - u~ !'. C h2 , with C independent of c~ For th~ high frequencies al so S ... ) t1.nd we find hence f~ '¾-i;~ !: ~f;f s2 C h/ (h+c:). Hence, fur h<. r, ,, B I, '½i c fixed, we find - u~ :1 = O(h} n and for s<h we find Ii B A ll "¾ - ¾" for the high frequencies. '.·I A 11 . uh 'I Hence we find that in the smooth part of the solufion (low frequencies) the difference u!l-ui is srn,all, whereas it may be large in those rggions of transition layers, where the mesh is not fit to represent the true solution. ln this way the difference u.B-ut behaves similar to the • • 11 .i A R, error 1n the approximation ~--11u. 5. EXA."!PLES As a ~irzt example to show the behaviour of { and uh -u,, , we use the same equation as Hughes and Brool!s [ 6 J, -Etu + cos(e) u + sin(6\ u ; 0 X ' y , on the unit square, with Dirichlet boundary _6 conditions at the inflow boundary ands= 10 (0, I) flow-direction e (O,A)IL---(0,0)--------Inflow boundary conditions u(x,O) l, 0 s X s I • u(O,y) l, 0 s y s A, u(O,y) - 0, 2c y > A. ( l ,0) A = 3/16. 1.0 o.o Fig. la. Example I. Neumann BCs, e 1.0 0.0 Fig. 2a. Example I. Neumann BCs, e 1 .o o.o . -· Fig. 3a. Example I. Neumann BCs, e 1.0 0.0 Fig. 4a. Example I. Neumann BCs, e 22.5° 22.5 0 A ~--;/ 0 A 45 . ~--:-1 67.5°, A ~-0.05 0.0 --0 .05 Fig. lb. 0.05 0.0 -0.05 Fig. Cl.OS 0.0 -0.05 Fig. 0.05 0.0 -0.05 2b. 3b. Fig. 4b. 4 121 .,11~ B A ISi uh -~ B A ~ - uh 1. G .. O Fig .. Sa. Example . Jirichlet BCs, '·: l.0 o.o Fig. 6a. Example 1. Dirichlet BCs, 8: 45° ] .o 0.0 Fig. 7a. Example !. Dirichlet BCs, l. o.o Fig. 8a. Example 2. "- e / -v/ e = 67.5°, A °r,· 0 .. 0. -0.05 0.05 0.0 -o.os Fig. 7b. 0.05 0.0 -0.05 Fig. 8b. Example 2. At the outflow boundary either natural or homo-geneous Dirichlet boundary conditions are used. For L_ h we use the usual finite element discre-tizatian with piecewise linear functions on a regular triangularization. Note: the way the squares are triangularized (1SI or IZI) makes no essential difference in the results. If the triangle division follows the internal layer, the numerical results are -of course- slightly better. In the figures the triangularization is indicated. In colmlOn with [6] we use as flow directions 8 • 22.5°, 45° and 67.5°. In the figxres (I )-(7) we show tRe ~umerical solution uh and the difference uh-uh for the different cases. As a second example we use the variable coefficient problem -r6u + l.vu = 0 on [-1,+l]x[0,1], ; • ( y(J-x2), - x(l-y2))T, with the Dirichlet boundary conditions u(x,y) • u(x,y) = u(x,y) = u(x,y) = I+ tanh(I0+20x), y=O, -I ,;x,sO, 0 , y= I , - I s x s l , 0 , x= l , 0 s y s I , 0 ,x=-1,0<y,;I; and homogeneous Neumann boundary conditions at the outflow boundary: y=O, 0 < x < I. Asymptotically for E+O, the true solution is constant over the subcharacteristics and the outflow profile is the mirror i111111age of the inflow profile. (-1, I) flow \ (~l (I, I) (-1,0) 1--------t------.... (1,0) Inlet Outlet In the figures (8a) and (8b) we show the numerical solution t{ and the difference ~-{. REFERENCES [I] A.Brandt, Numerical stability and fast sol-utions to boundary problems. In: Boundary and interior layers - Computa-tional and asymptotic methods.(J.J.H.Miller ed.) Boole Press,I>ublin,1980. [2] W.Hackbusch, Bemerkungen zur iterierten Defektkorrektur und zu ihrer Kombination mit Mehrgitterverfahren. Rev.Roum.Math.Pures Appl.26(1981)1319-1329. [3] W.Hackbusch, On multigrid iterations with defect correction. In: Multigrid Methods (W.Hackbusch and 6 U.Trottenber,eds) LNM, Springer, 1982. [4] P.W.Hemker, An accurate method without dir-ectional bias for the n1.1111erical solution of a 2-D elliptic singular perturbation prob-lem. ln: Singular perturbations and applications (W.Eckhaus and E.M. de Jager eds) LNM,Springer,1982. [5] P.W.Hemker, Mixed defect correction iter-ation for the accurate solution of the convection diffusion equation, In: Multigrid Methods (W.Hackbusch and U.Trottenberg, eds) LNM, Springer,1982. [6] T.J.R.Hughes and A.Brooks, A multi-dimensional upwind scheme with no crouwind diffusion. In: Finite Element Methods for Convection Dominated Flows (T.J.R.Hughes ed.}, ASME, New York, 1979. [7] P.Gresho and R.L.Lee, Don't suppress the wiggles - They're telling you something. In: Finite Element Methods for Convection Dominated Flows (T.J.R.Hughes ed.), ASME, New York, 1979. 

English

Defect correction for the solution of a singular perturbation problem

Scientific Computing R. Stepleman et al. (eds.) © IM ACS/North-Holland Publishing Company, 1983 113 DEFECT CORRECTION FOR THE SOLUTION OF A SINGULAR PERTURBATION PROBLEM P. W. Hemker and P.M. de Zeeuw Mathematisch Centrum, Amsterdam The Netherlands A method is described for the accurate discretization of a differential equation in which the highest derivative is multiplied by a small parameter. It is well known that for such singular perturbation problems with a strongly asymmetric differential opera-tor almost all discretizations are either unstable or inaccurate or direction depen-dent. By the combination, in an iterative process, of an inaccurate stable and an accurate unstable scheme we obtain an accurate stable solution, without adapting the scheme to the flow direction. I2 fact, two approximate solutions are obtained, that - uniformly in E - are both O(h ) accurate in the smooth part of the solution. The difference between both solutions can be used for the detection of the unsmooth parts. I . INTRODUCTION An iterative process is used to obtain the accu-rate solution of a singular perturbation problem. As a model problem we use the convection diffu-si.on equation -+ (la) -cLlu + a.Vu= f, 2 on a bounded domain QclR , with either Dirichlet or natural boundary conditions (I b) (I c) u g on J£1. 1 , h on ;H1. 2 , where ri: is the outward normal on the boundary :iu. = 0r1 1uci\l2 • The problem is written in symbolic form as (2) L u = f. E It is well known that for such a problem with a strongly asymmetric differential operator, the usual discretizations are either unstable (the usual symmetric discretization methods: central differences, finite elements or Bubnov-Galerkin methods) or inaccurate (artificial viscosity) or direction dependent (various streamline-upwind or Petrov-Galerkin discretizations). By the combination, in an iterative process, of an in-accurate stable and an accurate unstable scheme, we construct a solution which is accurate, uniformly for all £, without adapting the scheme to the subcharacteristic directions (flow-direc-ti.ons) in the problem. In the iterative process two standard discreti-zations of (2) are used: (3) L c , h uh = f h' a standard accurate discretization (e.g. standard central differences or a finite element discreti-zation with piecewise linear test and trial functions on a regular triangularization), which i.s unstable for t0«h, and an artificial diffusion (artificial viscosity) discretization (4) La,h uh = fh' which is the same as (3) but with a= c + O(h), whence (4) is stable. It is well known [6] that both discretizations yield bad results for small ratios E/h. We combine the discretizations (3) and (4) in an iterative process of defect correction type. In case of a linear problem an elementary defect correction process generates a sequence of appro-ximate solutions uCi), i = 1,2,3, ..... , by the iteration (iterative refinement) u(O)= 0 ' (5) Lu(i+l) = f - Lu(i) + Lu(i). If lim. u(i)= u* and Lis injective, then u* is the §;;J'.ution of the "target equation" Lu= f. L is usually some approximation to L, for which the equation (5) is readily solved. For our problem we study a "mixed" defect correc-tion iteration with two target operators and two approximate operators. They are combined as: (6a) (6b) u(O)= 0, 1u(i+!J= ~ 1 (i+l) L2u = If this iteration converges, utions" uA = lim. u (i) and J_+oo With f 1 = f 2 f, obviously by we obtain two "sol-u8 = lim. u (i+4). A i.+o> u is characterized ~ ~ ~- l - A - ~- I (7) [L2 -(L2-L 2 )L 1 (L 1-L 1 )Ju = [I-(L2 -L2 )L 1 ]f and u8 is given by a similar equation. 114 P. W. Hemker and P.JH. de Zeeuw 2. THE DISCRETIZATION To obtain an approximate solution for (2), we use the process ( 6) with f I J2 = fh and with the operators ( 8) LI L €,h' LI L et' h' L2 L a,h' L2 2 diag (L h) =: D. a, In this way the second iteration step (6b) is a damped Jacobi-relaxion-step for the solution of (4). The first step (6a) is an order improving defect correction step towards the solution of (3), where the defect defining operator is simply given by ( 9) E := L 1-L 1 = (a-s)Llh. A B uh and uh are now characterized The two solutions by (10) (L + E:,h (I I) (L h + E, (I + By using local mode analysis [4] we can show that, for the smooth parts of zhe solution, u~ and u~ are both accurate of O(h ). Further! the opera-tors L h + L hD-IE and L h + ED- L h are E, a, E, a, stable and the numerical boundary layers extend over a region of only 0 (h) in the boundary layer region [5]. Remark. It is not possible to find an accurate approxima-tion for our problem (1) by application of the simple defect correction (5) alone. If we apply (5) with L = L h and L = L , for e«h iteration would not conv~tge to a sen~iRle solution because of the instability of L • It would converge to the unwanted solution of' l'.he "target problem" (3). Theoretically, already a single step (or a few steps) of (5) result in a 2nd order accurate method cf. [2] Satz 2.2 , from which follows II uh - ~ull o:: Ch2 11 u"ll . Here ~u denotes the restriction of the true so-lution u to the mesh. However this is not a use-ful errorbound in our case, where II u" II may be very la2ge. In fact, for small s, we do not find the O(h ) convergence in practice for any rea-sonable value of h. This is shown in table l. £ = l um D A B h uh uh uh h = 1/8 0.630 0.0740 0.0780 0.0693 ratio 2.47 3.65 3.64 3.46 h = 1/16 0.0255 0.0203 0.0214 0.0201 ratio 1. ?1 4.02 4.01 3,89 h = 1/32 0.0149 0.00505 0.00533 0.00516 Table Ja. D A B £ = 10-6 um h uh uh ~ h = 1/8 0. 790 0. 634 0.608 0.459 ratio 1. 3? 1.?6 3.82 3.48 h = I I 16 0.578 0.360 0.159 o. 132 ratio 1. 52 2.08 4.75 4.54 h = 1/32 0.380 0.173 0.0335 0.0291 Table lb. Table J shows the error in the smooth part of the solution in the max-norm. u~~): the solution after one step (6a); uh : the solution after two steps (6a); u~, u~ obtained by iteration of (6). The problem: Elu + u = f on the unit square; with the Dirichlet b6undary data and the data f such that u (x,y) = sin(rrx)sin(~y) + cos(rrx)cos(3ny) + +(exp(-x/£) - exp(-1/€))/(1-exp(-J/£)). 3. RELATION WITH MULTIGRID TECHNIQUES The algorithm in section 2 can be considered independent of multigrid techniques. However, it is related to previous work done on multigrid methods. The use of defect correction in combination with a multigrid algorithm is already mentioned by Brandt[!] and is theoretically studied by Hack-busch [2,3]. They consider a multigrid algorithm which, in the elementary form of a two-level algorithm, can be described by a process (6) as well. Then, (6a) is a "coarse grid correction" which is now written in the form (i+J) = (i) L-1 (f -L (i)) u u + 1 1 1u , because~-! L 1 is of deficient rank: ~-1 -1 1 1 = ph,2h1a,2h R2h,h' where Ph 2 and R2 h are the grid transfer operators £etween l'.fie fine and the coarse grid; (6b) denotes a sequence of relaxation sweeps. In the standard two-level algorithm both target operators are the same: L1 = L2 . In combination with defect correction (non-standard), the ope-rator L 1 corresponds to a more accurate discre-tization then L2 . In the approach we discuss here, we differ [l,2,3] by using a full rank operator 11 We use the multigrid algorithm only to solve efficiently the equation (6a). 4. THE DIFFERENCE BETWEEN u~ AND u~ Considering the difference u~ - ~ of the solutions of (JO) and (11), we find the following from La,h' Defect Correction for the So!u tio11 of a Singular Perturbation Problem 115 Lemma (12) Proof -1 A D Euh Because of (6b) we have B A (13) D (uh - uh) and hence,by (11), B (14) LE,huh - fh (L h E' or B A (15) La,h uh - fh E uh . . -1 ) A diag (L. h 6 uh. Ct, h From (13) and (15) the lemma follows immediately, From (12) we see that the difference uB - uAh is A h l~rgeAfo: large llhuh . In the regions where uh- uh is large, the 2nd order differences of u~ are large. Here the solution cannot accurate-ly be represented. Thus, we may use the differ-ence u~-u~ as an indicator where the mesh should be refinea for a better approximation of the solution. To study the local behaviour of the difference u~-u~ in more detail, we resort to the local mode analysis [l]. We consider the problem (la) on lR 2 and form the discrete operator L h over all lR2. As a forcing function we takes' the 11mode 11 ij hw . .,,2 lR2 e ; J E,,, he + ; 2 Re(hw) E [-n,+n] . Now the solutions u~ and u~ are studied by Fourier analysis. We note that the Fourier transform is a norm-preserving bijection between the f2 Cn:~ and the L2 ([-n,+n]2) functions, i.e. II u II II Q II where Q is the Fourier transform of u. From (12l_we derive /]/\ I -1 ) A ) = (16) uh-uh= FT( 2Ca-s) diag (La,h llh uh = ~~~(sin2 (wlh/2)+sin2 (w2h/2))~~ =: a;~ s2 u~, and from (10) we find A -4E 2 a-s 2 4i a-s 2 (17) ~h;Qh = -h2s (1+-2~s) + -h T c1+-2~s ), where 4iT/h = £0 h(w) is the characteristic trigonometric polynomial for the reduced difference operator. In previous work [4,5] we derived from (17) that uA ~s bounded, uniform ins, Moreover, u~ is o?h ) accurate in the smooth parts of the solution. We also showed that, for sma~l EA at boundary or interior layers the error in uh may be 0(1) ,uniformly in s. In these regions (where the discrete solution is not able to represent the true solution anyway) the numerical approx-imations may show oscillations. However, the discretization (10) is asymptotically stable and the critical regions near the layers have only O(h) width. To see the effect of h -+ 0 on the approximation, we study separately the "low" and the "high" frequencies in the true solution. The frequencies are called "high" or "low" with reference to the mesh used (size h). For the "low" frequencies in the solution we consider a fixed w and let h-+ 0. For the "high" frequencies we consider wh fixed and let h->-0. Regions where the solution is smooth are characterized by low frequencies. Regions where the mesh is too coarse to repre-sent a solution properly,are dominated by high frequencies. Now we use (16) to different regions. B A . h see how uh-uh behaves in t e For the low frequencies ? ? s- = sin~ (w 1h/2) ? 2 + sin~(w2h/2) = O(h ) and we find 2 11 u~ ? 11{ a-e II £='=- II B A -2;; s ,; c h-uh -uh a C _]L h2 11 u~ II h+s Because uA is bounded, uniform in s, we tRe smooth part of the solution conclude that in we have II u~ - u~ II ,; c h2 , with C independent of c. For th7 high frequencies wh is fixed (and hence also S ) and we find II u~-u~ II ~~~ s2 II u~ II ,; a:s ,,; C h/ (h+s) . Hence, for h<s, s fixed, we find 11 c\ - u~ 11 = o Ch) and for e<h we find 11 c\ - u~ 11 oc1) for the high frequencies. Hence we find that in the smooth part of the soluAion (low frequencies) the difference uB-u is small, whereas it may be large in those rggi8ns of transition layers, where the mesh is not fit to represent the true solution. In this way the difference ~B-uAb behaves similar to the . .n AR error in the approximation ui.;--nu. 5. EXAMPLES . f A As a ~irxt example to show the behaviour o uh and uh-u , we use the same equation as Hughes and BrooRs [6], -c~u + cos(8) u + sin(8) u = O, x y on the unit square, with Dirichlet boundary _6 conditions at the inflow boundary and E = 10 (0, I) flow-direction (0 ,A) (O,O) Inflow boundary conditions u (x, 0) 1, 0 ,; x,; 1, u(O,y) 1, 0 ,; y ,; A, u(O,y) o, I <: y > A. (1,0) A 3/16. 116 P. W. Hemker and P.M. de Zeeuw 1.0 0.05 0.0 --0.05 0.0 0 A Fig. lb. B - A IZI Fig. la. Es:ample l. Neumann BCs, 6 22.5 uh. uh uh 1.0 e~ 0.05 ', ' ' 0.0 . ' -·O .05 0.0 Fig. 2a. Example 1. Neumann BCs, 8 22.5° Fig. 2b. 1.0 8.05 o.o -0.05 0.0 Fig. 3a. Example 1. Neumann BCs, 8 Fig. 3b. 1.0 C. GS c.o ·-0.05 0.0 Fig. 4a. Example I. Neumann BCs, 8 67.5°, Fig. 4b. Defect Correction for the Solution ofa Singular Perturbation Problem e v J.0 o.o ! ~ ,,, •I •• , ~.; J Fig. Sa. Example 1. Dirichlet BCs, 6 = 22.5°, u~. 1.0 o.o Fig. 6a. Example I. Dirichlet BCs, 6 = 45°, u~. J, .. 1.0 o.o Fig. 7a. Example I. Dirichlet BCs, 6 = 67.5°, ? ~ ·-•I.' ···, ,', ]. 0 0.0 A Fig. 8a. Example 2. uh A ~-o.os o.o -0.05 Fig. 6b. 0.05 o.o -o.os Fig. 7b. 0.05 o.o ·-0.05 Fig. 8b. Example 2. 117 .. ,. 118 P. W. Hemker and P.M. de Zeeuw At the outflow boundary either natural or homo-geneous Dirichlet boundary conditions are used. For L h we use the usual finite element discre-tizat~6n with piecewise linear functions on a regular triangularization. Note: the way the squares are triangularized (\SI or IZl) makes no essential difference in the results. If the triangle division follows the internal layer, the numerical results are -of course- slightly better. In the figures the triangularization is indicated. In common with [6] we use as flow directions e = 22.5°, 45° and 67.5°. In the figures ( 1 )-(7) we show the numerical s~lution u~ and the difference u~-u~ for the different cases. As a second example we use the variable coefficient problem -s6u + !.9u = 0 on [-1,+l]x[0,1], ! = ( y(l-x2 ), - x(l-y2))T, with the Dirichlet boundary conditions u(x,y) u(x,y) u(x,y) u(x,y) I + tanh(I0+20x), 0 0 0 y=O, -1 ,; x ,; 0, y=l, -I,; x,; I, x=l, 0,; y,; I, x=- 1 , 0 < y ,; 1 ; and homogeneous Neumann boundary conditions at the outflow boundary: y=O, 0 < x < 1 . Asymptotically for s+O, the true solution is constant over the subcharacteristics and the outflow profile is the mirror immage of the inflow profile. (-1, I) (~l (1, I) (-I ,0) .__ __________ __, ( 1, 0) Inlet Outlet In the figures (Sa) and (Sb) we show the numerical solution u,A and the difference B A n uh-uh. REFERENCES [I] A.Brandt, Numerical stability and fast sol-utions to boundary problems. In: Boundary and interior layers - Computa-tional and asymptotic methods.(J.J.H.Miller ed.) Boole Press,Dublin, 1980. [2] W.Hackbusch, Bemerkungen zur iterierten Defektkorrektur und zu ihrer Kombination mit Mehrgitterverfahren. Rev.Roum.Math.Pures Appl.26(1981)1319-1329. [3] W.Hackbusch, On rnultigrid iterations with defect correction. In: Multigrid Methods (W.Hackbusch and U.Trottenber,eds) LNM, Springer, 1982. [4] P.W.Hemker, An accurate method without dir-ectional bias for the numerical solution of a 2-D elliptic singular perturbation prob-lem. In: Singular perturbations and applications (W.Eckhaus and E.M. de Jager eds) LNM,Springer, 1982. [5] P.W.Hemker, Mixed defect correction iter-ation for the accurate solution of the convection diffusion equation. In: Multigrid Methods (W.Hackbusch and U.Trottenberg, eds) LNM, Springer,1982. [6] T.J.R.Hughes and A.Brooks, A multi-dimensional upwind scheme with no crosswind diffusion. In: Finite Element Methods for Convection Dominated Flows (T.J.R.Hughes ed.), ASME, New York, 1979. [7] P.Gresho and R.L.Lee, Don't suppress the wiggles - They're telling you something. In: Finite Element Methods for Convection Dominated Flows (T.J.R.Hughes ed.), ASME, New York, 1979. 

THE ROLE OF DIGITAL INNOVATION IN MEDIATING DIGITAL CAPABILITY ON BUSINESS PERFORMANCE

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