NTE N THE PPLICTIN F THE UERMND-PSQUETTI MSS LUMPIN CRRECTIN TECHNIQUE FR CNVECTIN-DIFFUSIN PRLEMS Prer submed o Elsever jourl 0 My 0 SERII V. SIRYK Nol Teccl Uversy of Ure "Igor Sorsy Kyv Polyecc Isue", Kev, Ure Prosec Peremogy 7, cdemc buldg 5, Kyv 005, Ure sv.sry@.u bsrc: We rovde creful Fourer lyss of e uermod-psque mss lumg correco ecque [uermod J.-L., Psque R. correco ecque for e dsersve effecs of mss lumg for rsor roblems // Comuer Meods led Meccs d Egeerg. 0. Vol. 5. P. -9] led o ure rsor d coveco-dffuso roblems. I rculr, s foud cresg e umber of correcos reduces e ccurcy for roblems w dffuso; owever ll e correced scemes re more ccure e cosse ler formulo s cse. For e ure rsor roblems e suo s e oose. We lso vesge e dffereces bewee wo umercl soluos e cosse soluo d e correced oes, d sow cresg e umber of correcos mes soluos of e correced scemes closer o e cosse soluo ll cses. Keywords: fe-eleme meod, ler meod, coveco-dffuso equo, mss lumg, rfcl dsso / dserso, uermod-psque ecque.. Iroduco lyg e fe eleme meods for e sl roxmo of o-sory covecodffuso equo usully yelds sem-dscree roblem [ 7],.e., sysem of ordry dfferel equos SDE of e form M F,, were s e vecor of uow coeffces of e exso of e umercl soluo w resec o e rl fucos, F s some vecor fuco, M s e so-clled cosse mss mrx [ ], wc s srse, o-dgol d eve o-symmerc e geerl cse. I e subseque ssge from SDE o dfferece scemes we me dervves re relced by dffereces e resulg dfferece scemes become mlc due o e o-dgoly of e mrx M. I ddo, e mrx M urs ou o be me-deede some semes of umercl roblems, wc c led o e ecessy o verse/fcorze e mrx M ec me se of egro of e SDE obed [ 7]. Te mss lumg ecque [, 7 ] s ofe used comuol rcce o fcle comuol effors d vod e ecessy for sosced d comuolly g-cos lgebr. Te essece of s ecque s o relce e mrx M by dgol mrx we deoe by M. Tere re severl oos for mlemeg s ecque [, 9 ]. Sce w follows we lyze oly e cse w couous ecewse-ler Lgrge rl fucos, we smly use e sums of elemes e corresodg rows of e mrx M o ob e dgol elemes of e mrx M, wc s e sdrd d commoly used rocedure e lerure see [, 7 ], lso equvle o coosg e erolo os s qudrure os for umercl egro s cse []. I sould be oed s oero my roduce o-osve defe w zero or egve dgol elemes mss mrces for ger-order elemes [, 0, ], erefore secl eo d secl cosrucos le qus-lumg roduced [] my be eeded s cse. fer erformg mss lumg we ob e "lumed" SDE M F, sed of e orgl "cosse" SDE. Te use of mss lumg mes ossble o re e rl dervve w resec o me e fe-eleme meod FEM scemes e sme wy s s doe e fe dfferece meods. I s cler ere s o eed o erform me-cosumg verso oeros fer crryg ou s dgolzo of e mrx M. Noe lso mss lumg lys mor role e cosruco of mxmum rcle reservg meods see [ ] d e refereces ere.
However, s ow see [, 7 ] e lco of mss lumg c roduce dserso d dsso errors o umercl scemes ledg o sgfc ccurces e umercl soluo deled revew of e "ro" d "cor" ers of usg mss lumg umercl scemes s gve []; see lso [] for revew of vodg e verso of mss mrx. Te fudmel er of uermod J.-L. d Psque R. [] s devoed o e vesgo d overcomg of s essel drwbc of mss lumg. Ter ecque s bsed o usg mrx seres o roxme e mrx M : e mrx M s rereseed e form M M I were I s e dey mrx, e mrx M M M, from wc we ge M I M Neum seres. I sould be oed oly e ure dffusoless rsor equo ws cosdered [], d e m focus ere ws o sem-dscree roxmos of e sdrd clsscl ler meod wou sblzo rculr, e covergece of e corresodg Neum mrx seres ws rgorously roved for e clsscl ler FEM w ler elemes. Te uors oe e use of eve oe e frs correco erm.e., I M sed of M c sgfcly mrove e ccurcy of e umercl soluo, d correcg e lumed mss mrx four mes M I M mes umercl resuls rcclly dsgusble from ose obed by e cosse formulo. However, e uors dd o me y deled eorecl esmes of e quly d ccurcy of e soluo deedg o e umber of erms of e mrx seres e. Dese e comuol rcveess d wde lco of s owerful ecque e.g., cosrucg mxmum rcle reservg meods [ ], level se meods for wo-se flows [5], egeerg lcos [] ec., d e overll g quobly of e er [], ese mor ssues rem sll uexlored. Terefore, our er c be cosdered s e frs em o clrfy ese ssues. ur er rovdes creful Fourer lyss of s ecque lco o ure rsor d coveco-dffuso roblems. We sow cresg e umber of correcos.e., e umber of correcg erms e Neum seres leds o error crese e resece of dffuso erms us, cors o ure rsor roblems, s dvsble o use oly oe e frs correco for roblems w dffuso. We lso sow ll e correced scemes re more ccure e cosse ler formulo for roblems w dffuso. Tese resuls seem o be ew, que uexeced, d uoced erler e lerure. For e ure dffusoless rsor roblems e suo s comleely oose.e., cresg e umber of correcos mroves e ccurcy of e umercl soluo, d e cosse ler formulo roduces more ccure resuls ll e correced scemes. We lso vesge e dffereces bewee wo umercl soluos e cosse soluo d e correced oes, d sow cresg e umber of correcos mes soluos of e correced scemes closer o e cosse soluo ll cses. Dese e oe-dmesoly of e reseed Fourer lyss bu c be exeded o esor roduc meses ger dmesos, umercl exmles cofrmed e corresodg eorecl resuls lso D d D cses.. Fourer lyss of sem-dscree roxmos.. Couous d umercl roblems seg We cosder e oe-dmesol coveco-dffuso equo [, 5, 7, 7, ] u u u 0, x x
were e coeffces d re gve, 0, 0, d u u, x s e uow soluo. To fcle e use of Fourer lyss [,, 7, ] we ssume e sl mes s uform w mes sze d e coeffces d re cos. Usg e sdrd couous ecewse-ler rl fucos Lgrge fucos, or "" fucos e clsscl ler fe-eleme formulo for, we ob e SDE were e ycl - equo corresodg o e ycl mes ode x s e form see [,, 7, 7, ] d d d 0. d d d Here { } re e coeffces of e exso of e roxme soluo w resec o e corresodg rl fucos. lyg mss lumg o, we ob e followg equo: d d Tus, mss mrces M d M ve e followg obvous rereseos: /, M j /, 0, for e ycl mes ode x. f f j, j, oerwse, M j 0., f j, 0, oerwse Equos d form e cosse d lumed sem-dscree ler FEM formulos w ler elemes, resecvely. Iroducg fe-dfferece oerors C d D of e cerl frs d secod dervves w e mrx rereseos /, f j, /, f j, C j D j /, f j, 0, oerwse, 0, oerwse for e ycl mes ode x, sysems d c be rewre s M C D 0 d M C D 0, resecvely. For e sdrd ssues of seg d dlg e l/boudry codos ler FEM formulos oe c see for exmle [,, 7, ]... Mss lumg correcos Te geerc form of e cosse SDE M F, c be rewre s M F,. roxmg e mrx M e mer descrbed bove we rrve o e followg defo. Defo. Te sysem I M F, s clled e - correced sceme. Noe from s defo we ob e sdrd o-correced lumed sem-dscree sceme M F, for 0. y y Le us roduce e followg oo see [9]: y y y, y y y, x, / x, / y~ y y / y y /, y y y y y y /, x, x, x, y~ y~ y y y y /, xxx, x xx, xx, x x,
y y y 5 y y y xxxx, ~ xxxxx, xxxxxx, y y y y y y / xx y ~ x xx, xxxx, xxxx, y y 5y 5y y, y / y y y 5y 0y 5y y y / xx ec. W usge of e oerors C d D we c lso rerese ese dervves s D Cy 5,,, D y y, y, 0, for rbrry mes fuco y see [9]. Noe for e cse uder cosdero we ve F D C, d corresodg correced scemes re crcerzed by e followg. Prooso. Te ycl - equo corresodg o e ycl mes ode x of e - correced sceme s e form d d C D m or, usg e roduced oo, c be rewre s d d m m m m m D C D 0 5 0. Proof. drec clculo sows e mrx s e followg mrx rereseo:,. /, f j, j /, f j, 0, oerwse, ereby y / yxx, / Dy d y / D y for rbrry mes fuco y. Te ls equo d e rereseo F D C mly.-.... Fourer lyss Due o e Fourer roc [,, 7, ] we re loog for rculr soluos of e semdscree roxmo e form of rmocs. e e, were s some comlex umber o be deermed,, s e rel umber e sl wve umber of e rmoc. Subsug s o, we ob e followg exresso for we deoe by for dsgusg: s s. s Le us deoe e umber for e - correced sceme equos.-. by Prooso. Te rel d mgry rs of c be exressed s follows: Re s s s s, 9 Im s s s s s s s. 9.
5 Proof. Subsug e e o e equo., usg Euler formuls d, rculr, e relo / s e e d e rereseo of e coeffces of cerl dffereces v boml coeffces e.g., see [0], fer rmec rsformos we ob, s s s s s s s s wc s equvle o e bove exressos. Subsug e sz e x e x u, o e equo we ob. Noe 0 0 lm lm,.e., e soluos soug e clss of rmocs of ll cosdered umercl roblems ed o e soluo of e roblem f eds o zero. Prooso. Le 0,, d le be rbrry. Te e equly s rue for ll suffcely smll. Proof. Le us deoe s, s s, 0, 0 for. Exdg o e Tylor seres w resec o eds o zero, we ob e followg:. s s s Im Re Im Re Im Re 5 Prooso. Le 0,, d 0. Te e equly s rue for ll suffcely smll. Proof. Le. Followg e roof of e revous rooso we c sow. 5 s s 0 Im 5 For we ve 0 7 0 0 e ls exresso dffers from e revous oe, sce we, us d corbues o e rcl erm of e exso. Prooso 5. Le 0,, d le be rbrry. Te e equly s rue for ll suffcely smll. Proof. s s s 0 e
geomerc rogresso w e commo ro / /s, were d re defed s e Prooso, 0, 0. Te d we ob:. s s s Im Im Im Re Re Re Im Re Im Im Re Re 5 Prooso. Le 0,, d 0. Te e equly s rue for ll suffcely smll. Proof. Le. Followg e roof of e revous rooso we c sow. 5 s s 0 5 Flly, for we ve 0 7 0 0. Prooso 7. Le, d rmeers d be rbrry. Te e equly s rue for ll suffcely smll. Proof. Usg see e roof of Prooso 5, s esy o esbls. Ts exresso comlees e roof.. Numercl exmles Le us comre e ccurcy of e correced d cosse scemes for e clsscl ler FEM o cofrm e eorecl resuls obed. s e er [], e me seg s doe w e sdrd exlc for-order Ruge-Ku meod RK w me se 0, 0 7, 0 9, 0 for exmles,, d -, resecvely. Ts meod d very smll me ses were used o mmze e effec of dscrezo o e me vrble, us o esure e me error corbued by e me roxmo s eglgble comrso w e error duced by sl roxmos see []. I sould be oed o surous oscllos ered ll e exmles cosdered below. Te l codo for 0 d e boudry codos re deermed from ow lycl soluos by er couous exesos o e corresodg boudg yerles bu for ure rsor
roblems Exmles d we use erodc boudry codos. I ll oe-dmesol exmles we use uform meses w e se o vlde e eorecl esmes of Fourer lyss Subseco.. I rculr, we deerme e emrcl orders s of covergece esmes of e form for e dffereces of squres for e correced, cosse d lycl soluos logues of d see e roofs of Proosos -, d revel ese emrcl orders coverge o e corresodg eorecl oes, esblsed Subseco.. I ll wo-dmesol ree-dmesol exmles we use ler rgulr eredrl - oded -oded Lgrge-ye elemes obed w Deluy rgulo lgorm. Clculos ll e exmles D, D d D cosdered below sow e resece of dffuso erms, e frs correced sceme gves e mos ccure resuls d cresg e order of correco oly worses e ccurcy; flly, e les ccure resuls re gve by e orgl ler formulo w e cosse mss mrx s cse. e-dmesol exmles Exmle. Cosder e l-boudry vlue roblem for e equo w e ow soluo u, x e cos x exc, were, 0, s 5 0, 0, x [0;0]. Te errors ogeer w e corresodg emrcl orders of covergece re gve Tble. We deoe e error uumercl, x uexc, x by err, err C d err L for e - correced sceme, cosse sceme Eq. for D cse d lumed sceme, resecvely. We lso deoe by P, j e vlue of l, j/, j/l / ble,, j s e vlue of, j err errj ere s e se for e - colum of e for e se gvg emrcl order of covergece w decresg e sl se, o cofrm e resuls of Fourer lyss Subseco.. Vlue Te umber of sl odes Tble. Vlues for x 5, 50 0 70 0 90 00 0 0. err C.09 5.9000.7.5599.7050.97.0 err * 5 05 97 0 0 L err.77.059.09.77.997.977.7 err.05 5.997.75.90.750..09 err.0090 5.99999.5.555.705.90.057 P 5.959 5.5 5.79 5.755 5.090 5., P, 7.5 7.79 7.905 7.975 7.95 7.95 P C, 9.55 9.7 9.9099 9.905 9.9 9.9550 * e row cos oly eger rs of umbers Le us ow cosder e sme roblem bu w 0 d. For s cse e errors ogeer w e corresodg emrcl orders of covergece re gve Tble. Vlue Tble. Vlues for x 5, Te umber of sl odes 50 0 70 0 90 00 0 0. err C 0.5707 0.59 0.950 0.07 0.050079 0.050 0.0 7
err L * 50 0 7 9 9 err.99.5795 0. 0.70 0.059 0.007 0.9 err 0.5570 0.99 0.5970 0.07 0.057 0.0709 0.0975 err 0.570 0.5707 0.9500 0.075 0.0500 0.05 0.0 P 7.90 7.97 7.990 7.99 7.99 7.995, P, 0.00 0.007 0.00 0.00 0.00 0.009 P,C.97.990.99.997.995.99 * e row cos oly eger rs of umbers Exmle. Le us cosder e roblem w erodc boudry codos d l d 0 s0x for e equo w 0, o e ervl x [0;]. Te exc soluo s u exc, x 0 s0 x. For s cse e errors ogeer w e corresodg emrcl orders of covergece re gve Tble. Tble. Vlues for x /,. Te umber of sl odes Vlue 50 0 70 0 90 00 0 err C 7.099.0.07.9.975 5.907.777 err * 5 5 7 5 500 5 L err 5.9 5.59 5.75 79.5 9.755.7.5 err.50.5.5.00.00 5.07.777 err 7..055.0.99.979 5.99.77 P 7.99 7.99 7.995 7.997 7.997 7.997, P, 0.000 0.00 0.00 0.00 0.000 0.000 P,C.99.995.9970.997.99.99 * e row cos oly eger rs of umbers Exmle. Le us cosder e l-boudry vlue roblem for e equo w e ow lycl soluo x u exc, x ex, were 5, 0, 0, x [0;0]. For s cse e errors ogeer w e corresodg emrcl orders of covergece re gve Tble. Tble. Vlues for x 5, / 5. Te umber of sl odes Vlue 50 0 70 0 90 00 err C.777099 9.07.5.055.777 7.090 err.99 9.5990077.05595.0509.79790 7.00 err.775709 9.0975.59.0575.7777 7.09050 err.77770 9.079.09.05.779 7.090 P 5.99 5.99 5.995 5.995 5.997, P, 7.99 7.99 7.995 7.9957 7.99 P C, 9.99 9.995 9.9970 9.995 9.999
I c be see e emrcl orders P, relve o of e dffereces of squred resduls j dced Tbles - fully corresod coverge w decresg o e resuls of e Fourer lyss crred ou Subseco. d e eorecl esmes obed e roofs of Proosos - for e corresodg dffereces of e ques. Tus, e exmles cosdered cofrm e coclusos of Fourer-lyss regrdg e ccurcy of ll e cosdered sem-dscree scemes w decresg. Two-dmesol d ree-dmesol exmles u Exmle. Le us cosder e roblem for e wo-dmesol coveco-dffuso equo u u u u, x ; y [0;] [0;], w e ow exc soluo x y xx yy x y ex x y u exc, x, y ex, were 00,,,, /, /. Te mes s ssumed o be uform log ec dreco. For s cse e errors re gve Tble 5. u Vlue Tble 5. Error for x y /, /. Cous of sl odes N ; N y 5; 5 9; 9 5; 5 9; 9 5; 5 9; 9 err C.59.799.959.07 0.97 0.79 err.5.707.9.09 0.959 0.757 err.97.777.9.005 0.99 0.77 err.505.795.950.00 0.979 0.755 err.50.7975.955.07 0.970 0.70 Exmle 5. Le us cosder e roblem for e ree-dmesol coveco-dffuso equo u u u u u u, x y z xx yy zz x x ; y; z [0;], w e exc soluo x y z ex x y z u, x, y, z ex exc were 0,, /,,, /,,. frs we cosder wo dffere grds w e sme umber of odes uform 5 -oded grd Fg. d usrucured o-uform grd were e erl odes.e., odes sde e dom re rdomly dsrbued Fg. b. Subsequely, we led e Deluy rgulo lgorm o ob feeleme decomosos bo cses. Flly, le us lso cosder s roblem o 57 -oded grds s e revous cse, o uform grd Fg. d o grd w rdomly dsrbued erl odes Fg. b. Corresodg vlues of e locl error e cube ceer s well s e verged dscree Eucld orm of e error err j, ds N errj x N ere N s e ol umber of odes,.e., e sum s e over ll grd odes { x } for e me / re gve Tble., 9
b Fgure. xx5-oded grds: uform grd; b grd w rdomly dsrbued odes. Vlue b Fgure. x5x7-oded grds: uform grd; b grd w rdomly dsrbued odes. Tble. Vlues of bsolue locl error d 5 -oded uform grd 5 -oded rdom grd -orm of error. /,ds 57 -oded uform grd 57 -oded rdom grd err C x y z /.5990 9.5.70.070 err x y z /.95 90.7.050 7.797 err x y z /.579 9.90.557.0 err x y z /.570 9.9977.09.595 err x y z /.5950 9.597.975.9 err C, ds 0.95 7.05 0.75.9 err 0. 7.9 0.0.5, ds err 0.95 7.005 0.5.7, ds err 0.9 7.075 0.00.5, ds err 0.99 7.0995 0.., ds Exmle. Le us cosder e roblem for e ree-dmesol coveco-dffuso equo u u u u u u u x y z, x y z x y z x ; y; z [0;], w e exc soluo 0
u exc, x, y, z x y z, were 0,, 0. Le us cosder s roblem o 5 -oded d 57 -oded grds from e revous Exmle 5 see Fgures. Corresodg errors for s cse for / re reored Tble 7. For furer delg we lso comued e globl L -orm errors over e wole cube for e me /, wc re reseed Tble 7 s well. C Vlue Tble 7. Vlues of bsolue locl error, 5 -oded uform grd,ds 5 -oded rdom grd -orm d -orm of error. / L 57 -oded uform grd 57 -oded rdom grd err x y z / 0.59 0.9 0.979 0.9 err x y z / 0.77 0.09 0.05 0.09 err x y z / 0.57 0.55977 0. 0.5 err x y z / 0.559 0.005 0.7 0.775 err x y z / 0.577 0.7 0.7 0.0 err C, ds 0.005 0.00 0.0009 0.00 err 0.00 0.007 0.0000 0.0055, ds err 0.00 0.00 0.000 0.00, ds err 0.0050 0.005 0.0009 0.00, ds err 0.005 0.00 0.0009 0.00, ds err C L 0.79 0.9099 0.705 0.007 err L 0.99 0.090 0.59 0.5 err L 0.7 0.5750 0.95 0.5799 err L 0.5 0.7 0.995 0.595 err L 0.07 0.00 0.709 0.59. Coclusos Te er rovdes Fourer lyss of e uermod-psque ecque lco o ure rsor d coveco-dffuso roblems. We sow cresg e umber of correcos leds o error crese e resece of dffuso erms Prooso. We lso sow ll e correced scemes re more ccure e cosse ler formulo for roblems w dffuso Prooso 5. For e ure dffusoless rsor roblems e suo s comleely oose.e., cresg e umber of correcos sould mrove e ccurcy of e umercl soluo Prooso, d e cosse ler formulo roduces more ccure resuls ll e correced scemes Prooso. We lso vesge e dffereces bewee e cosse soluo d e correced oes, d sow cresg e umber of correcos mes soluos of e correced scemes closer o e cosse soluo ll cses Prooso 7. Numercl exmles cosdered for D, D d D cses cofrmed e corresodg eorecl resuls. Declro of eres Declro of eres: oe
cowledgemes. Te uor ws rlly suored by H00-MSC-RISE-0 Projec MMDIT Projec umber 57. Te uor exresses s dee grude o Prof. lbero Redell d Dr. Flo P Polecco d Mlo, Ily for fruful dscussos d er wrm welcome durg severl uor's secodmes. Refereces. Flyso.. Numercl meods for roblems w movg fros. Sele, Wsgo US: Rve Pr Publsg, Ic., 99... Roos H.-., Syes M., Tobs L. Robus umercl meods for sgulrly erurbed dfferel equos. erl, Hedelberg: Srger-Verlg, 00. 0.. Zeewcz.Z., Tylor R.L., Zu J.Z. Te Fe Eleme Meod: Is ss d Fudmels. 7 ed. xford: Elsever, 0. 7.. bgrll R. Hg order scemes for yerbolc roblems usg globlly couous roxmo d vodg mss mrces // Jourl of Scefc Comug 07. Vol. 7 -. P. -9. 5. Jo V., Kobloc P., Novo J. Fe elemes for sclr coveco-domed equos d comressble flow roblems ever edg sory? // erl: Weersrss Isue for led lyss d Socscs, 07, Prer No 0.. Sr S.V. Esmo of e ccurcy of fe-eleme Perov ler meod egrg e oedmesol sory coveco-dffuso-reco equo // Ur Memcl Jourl 05. Vol. 7 7. P. 0-090. 7. Sry S.V. lyss of Lumed roxmos e Fe-Eleme Meod for Coveco-Dffuso Problems // Cyberecs d Sysems lyss 0. Vol. 9 5. P. 77-7.. Wedld E., Sculz H.E. Numercl exermes o mss lumg for e dveco-dffuso equo // Revs Merv 005. Vol. No.. P. 7-. 9. Hsbo P. secs of coservo fe eleme flow comuos // Comuer Meods led Meccs d Egeerg 99. Vol. 7. P. -7. 0. Yg Y., Zeg H., Svselv M.V. rgorous d ufed mss lumg sceme for ger-order elemes // Comuer Meods led Meccs d Egeerg 07. Vol. 9. P. 9-5.. uermod J.-L., Psque R. correco ecque for e dsersve effecs of mss lumg for rsor roblems // Comuer Meods led Meccs d Egeerg 0. Vol. 5. P. - 9.. uermod J.-L., Nzrov M., Poov., Yg Y. secod-order mxmum rcle reservg Lgrge fe eleme ecque for oler sclr coservo equos // SIM Jourl o Numercl lyss 0. Vol. 5. P. -.. uermod J.-L., Poov., Yg Y. Te effec of e cosse mss mrx o e mxmum-rcle for sclr coservo equos // Jourl of Scefc Comug 07. Vol. 70. P. 5-.. d S., oll J. Mooocy-reservg fe eleme scemes bsed o dffereble oler sblzo // Comuer Meods led Meccs d Egeerg 07. Vol.. P. -5. 5. uermod J.-L., de Lu M.Q., Tomso T. coservve -dffuso ecque for e level se meod // Jourl of Comuol d led Memcs 07. Vol.. P. -.. Yoso H., Um K., Fujr M. - fe eleme/volume meod model of e de verged orzolly D sllow wer equos // Ierol Jourl for Numercl Meods Fluds 0. Vol. 75. P. -. 7. Hue., Dowedr M.H. Fourer lyss of sem-dscree d sce-me sblzed meods for e dvecve-dffusve-recve equo: I. SUP // Comuer Meods led Meccs d Egeerg 005. Vol. 9. P. 5-.. Sb F., Huges T.J.R. ew fe eleme formulo for comuol flud dymcs: IX. Fourer lyss of sce-me ler/les-squres lgorms // Comuer Meods led Meccs d Egeerg 99. Vol. 7 P. 5-5. 9. Smrsj.. Teory of dfferece scemes. New Yor: Mrcel Deer, 00. 7. 0. Jord C. Clculus of Fe Dffereces: rd edo. New Yor: MS Celse Publsg, 95. 5.