A NOTE ON THE APPLICATION OF THE GUERMOND-PASQUETTI MASS LUMPING CORRECTION TECHNIQUE FOR CONVECTION-DIFFUSION PROBLEMS ( ) SERGII V.

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1 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.

2 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

3 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,

4 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 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 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 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

6 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 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

7 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, err C err * L err err err P , P, P C, * 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 err C

8 err L * err err err P , P, P,C * 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 err C err * L err err err P , P, P,C * 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 err C err err err P , P, P C,

9 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 err err err err 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

10 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 / err x y z / err x y z / err x y z / err x y z / err C, ds err , ds err , ds err , ds err , 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

11 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 / err x y z / err x y z / err x y z / err x y z / err C, ds err , ds err , ds err , ds err , ds err C L err L err L err L err L 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

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