ON INTERPOLATION of FUNCTIONS with a BOUNDARY LAYER BY CUBIC SPLINES

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1 Mathematical Modelig O ITERPOLATIO of FUCTIOS with a BOUDARY LAYER BY CUBIC SPLIES IA Blatov, EVKitaeva, AI Zadori 2 Volga Regio State Uiversity of Telecommicatios ad Iformatics, Samara, Rssia 2 Sobolev Istitte of Mathematics of Siberia Brach of Rssia Academy of Scieces, ovosibirsk, Rssia Abstract The problem of article is cbic splie-iterpolatio of fctios havig high gradiet regios It is show that iform grids are iefficiet to be sed I case of piecewise-iform grids, cocetrated i the bodary layer, for cbic splie iterpolatio are aoced asymptotically exact estimates o a class of fctios with a expoetial bodary layer There are obtaied reslts showig diverget i small parameter estimates ad divergece of iterpolatio processes The modified cbic splie with iform i small parameter iterpolatio error is offered The reslts of merical experimets cofirmig theoretical estimates are give Keywords: bodary layer, siglar pertrbatio, cbic splie-iterpolatio, error estimates Citatio: Blatov IA, Kitaeva EV, Zadori AI Abot iterpolatio by cbic splies of the fctios with a bodary layers CEUR Workshop Proceedigs, 206; 68: DOI: 08287/ Itrodctio Cbic splies are widely applied to smooth iterpolatio of fctios Sch splies are ivestigated i [], [2] ad i may other works However, accordig to [], [], applicatio of the polyomial splie-iterpolatio to fctios with large gradiets i a bodary layer leads to essetial errors of O() type I [] there was costrcted a o-polyomial aaloge of cbic splie, which is exact for the bodary layer compoet merical experimets have show the advatage i the accracy of the costrcted splie However, the bodary layer compoet is't always kow, ths, i sch case, there is o reasoable alterative for codesig a grid i the bodary layer I this work a traditioal cbic splie iterpolatio [2] o the piecewise iform grid codesig i the bodary layer is ivestigated There were obtaied error estimates of iterpolatio which, however, are't iform i small parameter It is show that a iterpolatio error of a bodary layer compoet might icrease withot limits while 0, ths, we eed to develop special methods of iterpolatio for sch fctios Sch a method of iterpolatio is also offered ad ivestigated i Iformatio Techology ad aotechology (ITT-206) 55

2 Mathematical Modelig Blatov IA, Kitaeva EV, Zadori AI this work We shall pass it otig that the divergece of iterpolatio processes by cbic ad parabolic splies o oiform grids was regarded i works [2], [5], [6] ad some other works However, the examples of divergece provided there had a artificial character or were implied with the help of Baach-Steihas theorem I the preset article we have show divergece for fctios describig soltios of a variety of applied problems These reslts testify the eed of developmet of iversal high-order methods of smooth splie iterpolatio of fctios o oiform grids ad developmet of projective-grid methods of a high order for siglar pertrbed bodary vale problems, becase there is o eed for applicatio of the grid soltio iterpolatio while sig projective-grid methods Statemet of the problem Let s itrodce the followig otatios Let : 0 x x x partitio of a segmet [ 0, ] Let (, k,) 0 S be a space of polyomial splies [2] of degree k ad defect o the grid We meas that C, C j are positive costats idepedet from ad a mber of grid odes We write f O(g) if f C g ad f O * ( g), if f O(g) ad g O( f ), C [ a, b] is a space of cotios fctios with orm Let a fctio ( have a form, ( q( (, x[0,] C[ a, b] ( ) q j ( j) x / j C, ( C e /, 0 j () ( Let s ivestigate a problem of cbic splie-iterpolatio of the fctio () Mai reslts First, let s look ito the case of a iform grid Let be a atral mber, is iform grid with a step of H / ad odes, 0,,,, partitioig of the iterval [ 0, ] Let g ( x, S(,, ) be a iterpolatio cbic splie o the grid, defied by coditios: g x, ( x ), 0, g '(0, '(0), g '(, '() (2) ( x Theorem I case of a iform grid there shall be sch a costat C for which the ext estimate is correct: ( g If i () ( x, ( e x / C( ), the the followig estimate holds: Iformatio Techology ad aotechology (ITT-206) 56

3 Mathematical Modelig Blatov IA, Kitaeva EV, Zadori AI ( g ( x, C mi ( ),( ) ext, accordig to [7] let s defie the grid with odes steps h h,,,, h H,,, / 2 2 / 2 2 I accordace with [7] let s defie x, 0,,,, ad mi, l () 2 De to [2] for iterpolatio cbic splie g ( x, S(,, ) the followig error estimate is valid: g ( x, 5 ( 8 max h () () ote that g x, g ( x, q) g ( x, ), ad de to (),() ( ( g ( x, q) C [0,] 2 max h C C 2 q (x Heceforth, i order to bild a splie iterpolatio to ) with the order of O( l ), it is eeded to satisfy the ieqality: (5) ( g ( x, ) C l [0,] 2 C I case whe i (5) / 2, a ieqality () shall be valid i accot of Theorem * ad de to the relatio O ( / l ) Ths, we shall propose below that / 2 Yet, to keep it short we shall assig g g ( x, ), g ( S(,,) ( Theorem 2 There are sch costats C 2,C, which satisfy the relatio (5) if C Theorem There are sch costats C, C5, ad 0 that if C, the, idepedet from,, l,0 / 2 5 g( x, ) ( C C[ x, x ] 5 ( / 2) (6) e, / 2 ext Theorem shows s that estimates i (6) ca ot be improved Iformatio Techology ad aotechology (ITT-206) 57

4 Mathematical Modelig Blatov IA, Kitaeva EV, Zadori AI Theorem Let idepedet from g ( x, ) ( x / ( e The there are sch costats, C6, 0, that if C[ x, x ] C 6 C 5 e, the ( / 2), 2 C, ow we shall costrct a modified iterpolatio splie Let x ( x x ) / 2, x, [ 0, / 2 ] [ / 2, ] Let / 2 / 2 / 2 ( x, x gm be the iterpolatio cbic splie defied by coditios gm ( x, ( x ), [ 0, ], gm '(0, '(0), gm '(, '() The oly differece betwee gm ( x, ) ad g ( x, ) is that the iterpolatio ode x / 2 is set as x / 2 The splie odes whereas are ot sbject to ay chages ad coicide with the odes Theorem 5 There are sch idepedet from, costats, amely 0 0, C that if l 0, it shall satisfy the followig ieqality : ( (7) gm ( x, C l Commet The coditio l 0 will be satisfied if Therefore, accordig to theorems 2,5 gm ( x, ) if O( ) C applicatio of the iterpolatio splie g ( x, if ad the iterpolatio splie ) O( ) let as to obtai estimates (), (7) iformly i, Reslts of merical Experimets Let s defie the followig fctio: x x ( cos e, x[0,] 2 (8) Reslts of calclatios are provided i the three followig tables Give i the tables below are the maximm errors of splie iterpolatio, calclated at odes of the codesed grid, which is obtaied from the iitial grid by splittig every sigle grid iterval ito 0 parts Table cotais iterpolatio errors for the traditioal cbic splie o the iform grid Reslts cofirm estimates of the Theorem ad a iadeqacy of applicatio of the iform grid for small vales of Table 2 provides errors of a traditioal cbic splie i Shishki meshes It follows from the tables that errors icrease while decreases ad is fixed Reslts of Table describig the Iformatio Techology ad aotechology (ITT-206) 58

5 Mathematical Modelig Blatov IA, Kitaeva EV, Zadori AI errors of modified cbic splie, o the cotrary, show iform covergece, ths, theoretical coclsios are cofirmed ɛ ɴ Table Errors of cbic splie o the iform grid e-7 76e-8 6e-9 02e-0 0e- 268e- 0e- e- 2e-5 5e-6 958e-8 60e-9 e-0 0e e-2 972e-2 800e- 559e-5 65e-6 0e e-2 0e- 05e+2 52e+ 258e+ 25e e-5 06e+ 52e+2 26e+2 2e+2 656e+ 2e+ 0e-6 06e+ 50e+ 265e+ e+ 662e+2 0e+2 0e-7 06e+5 50e+ 265e+ e+ 66e+ 0e+ 0e-8 06e+6 50e+5 265e+5 e+5 66e+ e+ Table 2 Errors of cbic splie o piecewise-iform grid ɴ ɛ 282e-7 76e-8 6e-9 02e-0 0e- 268e- 0e- e- 2e-5 5e-6 958e-8 60e-9 e-0 0e-2 6e- 8e- 70e- 207e-5 227e-6 2e-7 0e- 6e- 8e- 70e- 207e-5 227e-6 2e-7 0e- 6e- 8e- 70e- 207e-5 227e-6 2e-7 0e-5 7e-2 25e- 70e- 207e-5 227e-6 2e-7 0e-6 7e- 25e-2 62e- 207e-5 227e-6 2e-7 0e e- 62e- 07e- 2e-6 2e-7 0e e-2 07e- 2e-5 992e-7 Table Errors of modified cbic splie o piecewise-iform grid ɴ ɛ e-7 20e-8 e-9 e-0 9e-2 e- 0e- e- 2e-5 5e-6 96e-8 60e-9 e-0 0e-2 6e- 2e- 7e- 2e-5 2e-6 2e-7 0e- 6e- 2e- 70e- 207e-5 2e-6 2e-7 0e- 6e- 2e- 70e- 207e-5 2e-6 2e-7 0e-5 6e- 2e- 70e- 207e-5 2e-6 2e-7 0e-6 6e- 2e- 70e- 207e-5 2e-6 2e-7 0e-7 6e- 2e- 70e- 207e-5 2e-6 2e-7 0e-8 6e- 2e- 70e- 207e-5 2e-6 2e-7 Iformatio Techology ad aotechology (ITT-206) 59

6 Mathematical Modelig Blatov IA, Kitaeva EV, Zadori AI Fig Graphs of fctio (8) ad of its iterpolatio splie o the piecewise iform grid at the [0,] iterval Ackowledgemet The work is spported by Rssia Fodatio of Basic Researches der Grat Refereces Ahlberg JH The theory of splies ad their applicatios ew York: Academic Press, 967; 28 p 2 Zav'yalov YS, Kvasov BI, Miroshicheko VL Methods of Splie Fctios Moscow: aka, 980; 52 p [i Rssia] Zadori AI Method of iterpolatio for a bodary layer problem Sib J of mer Math, 2007; 0(): [i Rssia] Zadori AI, Gryaova MV Aaloge of a cbic splie for a fctio with a bodary layer compoet Proceedigs of the fifth coferece o fiite differece methods: Theory ad Applicatios, 200 Rosse: Rosse Uiversity, 20: Zmatrakov L Covergece of a iterpolatio process for parabolic ad cbic splies Trdy Mat Ist Steklov, 975; 8: Zmatrakov L A ecessary coditio for covergece of iterpolatig parabolic ad cbic splies Mat Zametki, 976; 9(2): DOI: 0007/BF Shishki GI Discrete Approximatios of Siglarly Pertrbed Elliptic ad Parabolic Eqatios Ekateribrg: Rssia Academy of Scieces, Ural Brach, 992; 2 p [i Rssia] Iformatio Techology ad aotechology (ITT-206) 520