Fully Discrete Analysis of a Discontinuous Finite Element Method for the Keller-Segel Chemotaxis Model

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1 Flly Dsrt Analyss of a Dsontnos Fnt lmnt Mtod for t Kllr-Sgl Cmotaxs Modl Ykatrna pstyn and Amt Izmrlogl Abstrat Ts papr formlats and analyzs flly dsrt sms for t two-dmnsonal Kllr-Sgl motaxs modl. T spatal dsrtzaton of t modl s basd on t dsontnos Galrkn mtods and t tmporal dsrtzaton s basd tr on Forward lr or t sond ordr xplt total varaton dmnsng (TVD Rng-Ktta mtods. W onsdr Cartsan grds and prov optmal flly dsrt rror stmats for t proposd mtods. Or proof s vald for pr-blow-p tms sn w assm bonddnss of t xat solton. AMS sbjt lassfaton: 65M60, 65M1, 65M15, 9C17, 35K57 Ky words: Kllr-Sgl motaxs modl, onvton-dffson-raton systms, dsontnos Galrkn mtods, Forward lr, Rng-Ktta, NIPG, IIPG, and SIPG mtods, Cartsan mss. 1 Introdton T goal of ts work s to formlat and analyz flly dsrt dsontnos Galrkn (DG mtods for t solton of t two-dmnsonal (-D Kllr-Sgl motaxs modl, [11, 6, 7, 8, 33, 34]. T ndrlyng spatal dsrtzaton of t modl s basd on t mtods proposd rntly n [18] and t tmporal dsrtzaton s basd tr on Forward lr or t sond ordr xplt total varaton dmnsng (TVD Rng-Ktta mtods. In ts papr, w onsdr t lassal formlaton of t Kllr-Sgl systm [11], w an b wrttn n t dmnsonlss form as { t (χ =, (x, y, t > 0, (1.1 t =, sbjt to t Nmann bondary ondtons: n = n = 0, (x, y. Hr, (x, y, t s t ll dnsty, (x, y, t s t moattratant onntraton, χ s a motat snstvty onstant, s a bondd doman n R, s ts bondary, and n s a nt normal Dpartmnt of Matmatal Sns, Carng Mllon Unvrsty, Pttsbrg, PA, 1513, rna10@andrw.m.d Dpartmnt of Matmats, Unvrsty of Pttsbrg, Pttsbrg, PA 1560; ast1@ptt.d

2 Y. pstyn and A. Izmrlogl vtor. T systm (1.1 s t bas stp n t modlng of many bologal prosss. T Kllr-Sgl modl (1.1 an b gnralzd to bttr dsrb t ralty by takng nto aont som otr fators s as growt and dat of lls, prsn of t food and otr mals n t systm, t. It s wll-known tat soltons of t lassal Kllr-Sgl systm may blow p n fnt tm, s,.g., [4, 5] and rfrns trn. Ts blow-p rprsnts a matmatal dsrpton of a ll onntraton pnomnon tat ors n ral bologal systms, s,.g., [1, 6, 8, 9, 15, 35]. Captrng blowng p soltons nmrally s a allngng problm. A fnt-volm, [1], and a fnt-lmnt, [3], mtods av bn proposd for a smplr vrson of t Kllr-Sgl modl, { t (χ =, = 0, n w t qaton for onntraton as bn rplad by an llpt qaton sng an assmpton tat t moattratant onntraton angs ovr m smallr tm sals tan t dnsty. A fratonal stp nmral mtod for a flly tm-dpndnt motaxs systm from [39] as bn proposd n [40]. Howvr, t oprator splttng approa may not b applabl wn a onvtv part of t motaxs systm s not yprbol, w s a gnr staton for t orgnal Kllr-Sgl modl as t was sown n [10], wr t fntvolm Godnov-typ ntral-pwnd sm was drvd for (1.1 and xtndd to som otr motaxs and aptotaxs modls. T g-ordr dsontnos Galrkn mtod tat s nvstgatd r s basd on t mtod proposd n [18]. T DG mtods av rntly bom nrasngly poplar tanks to tr flxblty for adaptv smlatons, stablty for paralll omptatons, applablty to problms wt dsontnos offnts and/or soltons, and ompatblty wt otr nmral mtods. Ts mtods av bn sssflly appld to a wd varty of problms, rangng from t sold mans to t fld mans (s,.g., [13, 14, 1, 19, 0,, 38] and rfrns trn. Frtrmor, t DG mtods ar among t mtods tat an b sd for ral bomdal problms, w ar oftn onsdrd n omplx domans, av dsontnty n t offnts, and norporat PDs of dffrnt matmatal natr. In ordr to dvlop g-ordr DG mtods for (1.1 n [18], t Kllr-Sgl systm s rwrttn as a systm of t nonlnar onvton-dffson-raton qatons by ntrodng nw varabls (, v := : kq t F(Q x G(Q y = k Q R(Q, (1. wr Q := (,,, v T, t flxs ar F(Q := (χ, 0,, 0 T and G(Q := (χv, 0, 0, T, t raton trm s R(Q := (0,,, v, t onstant k = 1 n t frst two qatons n (1., and k = 0 n t trd and t fort qatons tr. T mtods proposd n [18] ar basd on tr prmal DG mtods: t Nonsymmtr Intror Pnalty Galrkn (NIPG, t Symmtr Intror Pnalty Galrkn (SIPG, and t Inomplt Intror Pnalty Galrkn (IIPG mtods, [3, 16, 36]. T nmral flxs n t proposd DG mtods ar t flxs dvlopd for t smdsrt fnt-volm ntral-pwnd sms n [30] (s also [9, 31]. Ts sms blong to t famly of non-osllatory ntral sms, w ar gly arat and ffnt mtods applabl to gnral mltdmnsonal systms of onsrvaton laws and rlatd problms. Lk otr ntral flxs, t ntral-pwnd ons ar obtand wtot sng (approxmat Rmann problm solvr, w s navalabl for t systm ndr onsdraton. At t sam tm, a rtan pwndng nformaton on-sdd

3 Analyss of a DG Mtod for t Cmotaxs Modl 3 spds of propagaton s norporatd nto t ntral-pwnd flxs. In [18], Cartsan grds ar onsdrd and t ontnos n tm rror stmats ar provd for t proposd g-ordr DG mtods ndr t assmpton of bonddnss of t xat solton. Som nmral tsts tat valdat t mtods ar onsdrd n [18] and [17]. In t followng ston, w ntrod or notatons, assmptons, and stat som standard rslts. In 3-4 w rall som rslts for t ontnos n tm sm. In 5-7 w formlat t xplt sms and drv t rror stmats ndr assmpton of bonddnss of t xat solton (som proof dtals ar postpond to Appndx 9. T proof of t rror stmats s basd on t ndton argmnt w smplfs t analyss sgnfantly sn w onsdr t opld systm of t nonlnar qatons. Assmptons, Notatons, and Standard Rslts W dnot by a nondgnrat qas-nform rtanglar sbdvson of t doman (t qas-nformty rqrmnt wll only b sd for stablsng t rat of onvrgn wt rspt to t polynomals dgr. T maxmm damtr ovr all ms lmnts s dnotd by and t st of t ntror dgs s dnotd by Γ. To a dg n Γ, w assoat a nt normal vtor n = (n x, n y. W assm tat n s drtd from t lmnt 1 to, wr 1 dnots a rtan lmnt and dnots an lmnt tat as a ommon dg wt t lmnt 1 and a largr ndx (ts smplfd lmnt notaton wll b sd trogot t papr. For a bondary dg, n s osn so tat t onds wt t otward normal. T dsrt spa of dsontnos pws polynomals of dgr r s dnotd by W r, ( = { w L ( :, w P r ( }, wr P r ( s a spa of polynomals of dgr r ovr t lmnt. For any fnton w W r,, w dnot t jmp and avrag oprators ovr a gvn dg by [w] and {w}, rsptvly: for an ntror dg = 1, for a bondary dg = 1, [w] := w 1 [w] := w 1, w, {w} := 0.5w1 0.5w, {w} := w 1, wr w 1 and w ar t orrspondng polynomal approxmatons from t lmnts 1 and. W also rall tat t followng dntty btwn t jmp and t avrag oprators s satsfd: [w 1 w ] = {w 1 }[w ] {w }[w 1 ]. (.1 For t fnt-lmnt sbdvson, w dfn t brokn Sobolv spa H s ( = { w L ( : w j H s ( j, j = 1,...,N } wt t norms w 0, = ( w 0, 1 and w s, = ( w s, 1, s > 0, wr s, dnots t Sobolv s-norm ovr t lmnt. W now rall som wll-known fats tat wll b sd n t rror analyss n 6-9. Frst, lt s stat som approxmatons proprts and nqalts for t fnt-lmnt spa.

4 4 Y. pstyn and A. Izmrlogl Lmma.1 (p Approxmaton, [4, 5] Lt and ψ H s (. Tn tr xst a postv onstant C ndpndnt of ψ, r, and, and a sqn ψ r P r(, r = 1,,..., s tat for any q [0, s] wr µ := mn(r 1, s and s t dg on. ψ ψ r C µ q q, r ψ s q s,, s 0, (. ψ ψ r C µ 1 ψ s,, s > 1 0,, (.3 r s 1 Lmma. (Tra Inqalts, [] Lt. Tn for t tra oprators γ 0 : H 1 ( H (, 1 γ 0 v = v and γ 1 : H ( H 1 (, γ 1 v = v n, tr xsts a onstant C t ndpndnt of s tat w H s (, s 1, γ 0 w 0, C t 1 ( w 0, w 0,, (.4 w H s (, s, γ 1 w 0, C t 1 ( w 0, w 0,, (.5 wr s t dg on. Lmma.3 ([36] Lt b a ms lmnt wt an dg. Tn tr s a onstant C t ndpndnt of and r s tat w 0, C t 1 r w 0,, (.6 w n 0, C t 1 r w 0,,, w P r ( (.7 Lmma.4 ([3, 7] Tr xsts a onstant C ndpndnt of and r s tat ( w W r, (, w 0, C w 0, 1 1 [w] 0,, Γ wr dnots t masr of. Lmma.5 (Invrs Inqalts, [37] Lt and w P r (. Tn tr xsts a onstant C ndpndnt of and r s tat w L ( C 1 r w 0,, (.8 w 1, C 1 r w 0,. (.9 W also rall t followng form of t dsrt Gronwall s lmma: Lmma.6 (dsrt Gronwall Lt t, H, and a n, b n, n, d n (for ntgrs n 0 b nonngatv nmbrs s tat a l t l n=0 b n t l n=0 d na n t l n=0 n H for l 0. Sppos tat td n < 1, n. Tn a l t ( ln=0 b n xp t ( l d n n=0 1 td n t ln=0 n H for l 0.

5 Analyss of a DG Mtod for t Cmotaxs Modl 5 In t analyss blow w also mak t followng assmptons: s a rtanglar doman wt t bondary = vr or, wr vr and or dnot t vrtal and orzontal ps of t bondary, rsptvly. W also splt t st of ntror dgs, Γ, nto two sts of vrtal, Γ vr, and orzontal, Γor, dgs, rsptvly; T dgr of bass polynomals s r and t maxmm damtr of t lmnts s < 1 (t lattr assmpton s only ndd for smplfaton of t rror analyss. 3 Dsrpton of t Contnos n Tm Nmral Sm for t Kllr-Sgl Modl W onsdr t Kllr-Sgl systm (1.. Frst, not tat t Jaobans of F and G ar χ 0 χ 0 χv 0 0 χ F Q = and G Q = , and tr gnvals ar λ F 1 = χ, λ F = λ F 3 = λ F 4 = 0 and λ G 1 = χv, λ G = λ G 3 = λ G 4 = 0, (3.1 rsptvly. Hn, t onvtv part of (1. s yprbol. W now dsgn smdsrt ntror pnalty Galrkn mtods for ts systm. W assm tat at any tm lvl t [0, T] t solton, (,,, v T s approxmatd by (dsontnos pws polynomals of t orrspondng dgrs r, r, r, and r v, w satsfy t followng rlaton: r max r mn a, r max := max{r, r, r, r v }, r mn := mn{r, r, r, r v }, (3. wr a s a onstant ndpndnt of r, r, r, and r v. DG mtods ar formlatd as follows. Fnd a ontnos n tm solton ( DG (, t, DG (, t, DG (, t, v DG (, t W r, W r, W r, W v r v,, w satsfs t followng wak formlaton for t motaxs systm (1.: DG t w DG w { DG n }[w ] ε { w n }[ DG ] Γ σ r [ DG ][w ] χ DG v DG (w y Γ or χ DG DG (w x Γ vr (χ DG DG n x [w ] (χ DG v DG n y [w ] = 0, (3.3

6 6 Y. pstyn and A. Izmrlogl DG t w σ r DG w vr v DG w v or DG w Γ [ DG ][w ] DG (w x DG n x w σ DG n y w v σ v DG w Γ vr DG (w v y and t ntal ondtons: DG (, 0w = DG (, 0w = { DG n }[w ] ε Γ r Γ vr Γ or r v Γ or (, 0w, (, 0w, { w n }[ DG ] DG w = 0, (3.4 ( DG n x[w ] [ DG ][w ] = 0, (3.5 ( DG v n y[w v ] [v DG ][w v ] = 0, (3.6 DG (, 0w = v DG (, 0w v = (, 0w, v(, 0w v. (3.7 Hr, (w, w, w, w v W r, W r, W r, Wv r v, ar t tst fntons, σ, σ, σ and σ v ar ral postv pnalty paramtrs. T paramtr ε s qal to tr 1, 0, or 1: ts vals of ε orrspondng to t SIPG, IIPG, or NIPG mtod, rsptvly. To approxmat t onvtv trms n (3.3 and (3.5 (3.6, w s t ntral-pwnd flxs from [30]: (χ DG DG = aot (χ DG DG 1 a n (χ DG DG aot a n ], a ot a n a ot a n[dg (χ DG v DG = bot (χ DG v DG 1 b n (χ DG v DG bot b n ], b ot b n b ot b n[dg ( DG = ( DG 1 aot a n ( DG aot a n ], a ot a n a ot a n[dg ( DG v = ( DG 1 bot b n ( DG bot b n ]. b ot b n b ot b n[vdg Hr, a ot, a n, b ot, and b n ar t on-sdd loal spds n t x- and y- drtons. Sn t onvtv part of t systm (1. s yprbol, ts spds an b stmatd sng t largst and t smallst gnvals of t Jaoban F G and (s (3.1: Q Q ( ( a ot = max (χ DG 1, (χdg, 0, a n = mn (χ DG 1, (χdg, 0, ( ( b ot = max (χv DG 1, (χvdg, 0, b n = mn (χv DG 1, (χvdg, 0 (3.9. (3.8

7 Analyss of a DG Mtod for t Cmotaxs Modl 7 Rmark. If a ot a n = 0 at a rtan lmnt dg, w st (χ DG DG = (χdg DG 1 ( DG = 1 (DG (χ DG DG ( DG, ( DG v = 1 (DG tr. Not tat n any as, t followng nqalts,, (χ DG v DG = (χdg v DG 1 ( DG, (χ DG v DG, a ot a 1, a ot an n b 1, a ot an ot b ot b 1, and b n 1, (3.10 n b ot bn ar satsfd. From now on w wll assm tat a ot a n > 0 and b ot b n > 0 trogot t omptatonal doman. 4 Rslts for t Contnos n Tm Sm Lt s rall r t rslts tat wr obtand n [18] for t ontnos n tm sm (3.3- (3.6. Lmma 4.1 (Consstny Lmma If t solton of t systm (1. blongs to H (, tn t satsfs t formlaton (3.3 (3.6. Torm 4. (L (H 1 and L (L rror stmats. Lt t solton,, and v of t Kllr-Sgl systm (1. b sffntly rglar. Frtrmor, w assm tat pnalty paramtrs σ, σ, σ, σ v ar sffntly larg. Tn tr xsts at last on DG solton to (3.3-(3.6 and tr xsts onstants C and C, ndpndnt of and r, s tat DG ( T L ([0,T];L ( (DG L ([0,T];L ( 0 r Γ [ DG ] 1 0, C ( mn(r1,s 1 r s mn(r1,s 1 r s mn(r1,s 1 r s mn(rv1,sv 1 rv sv DG ( T L ([0,T];L ( (DG L ([0,T];L ( 0 r Γ [ DG ] 1 0, C ( mn(r1,s 1 r s mn(r1,s 1 r s mn(r1,s 1 r s mn(rv1,sv 1, rv sv wr (r, r, r, r v.

8 8 Y. pstyn and A. Izmrlogl 5 Flly Dsrt Sms and Analyss In t followng stons, w formlat two xplt sms and stabls t onvrgn of t nmral soltons sng an ndton ypotss. xstn of t dsrt solton s trval sn t sm s xplt n tm. In t analyss blow, w wll assm tat t xat solton of t systm (1. s sffntly rglar for t T, wr T s a pr-blow-p tm. In partlar, w wll assm tat (,,, v H s 1 ([0, T] H s (, s 1 > 3/, s 3, (5.1 w s ndd for t -analyss (onvrgn rat wt rspt to t ms sz, or (,,, v H s 1 ([0, T] H s (, s 1 > 3/, s 5.5, (5. w s ndd for t r-analyss (onvrgn rat wt rspt to t polynomal dgr. Not tat ts assmptons ar rasonabl sn lassal soltons of t Kllr-Sgl systm (1.1 ar rglar (bfor t blow-p tm provdd t ntal data ar sffntly smoot, s [4] and rfrns trn. Lt t b a postv tm stp and lt t = t dnot t tm stp at t t stp. W dnot by v t fnton v valatd at tm t. 6 Forward lr Tm Dsrtzaton Fnd a dsrt n tm solton ( 1 DG, 1 DG, 1 DG, v1 DG W r, W r, W r, W v r v,, w satsfs t followng wak formlaton for t motaxs systm (1.: 1 DG DG t σ r 1 DG DG t σ w DG w Γ [ DG ][w ] χ DGv DG(w y w r 1 DG w [ DG ][w ] Γ or DG w Γ 1 DG (w x { DG n }[w ] ε Γ χ DG DG (w x DG w Γ vr Γ vr { w n }[ DG ] (χ DG DG n x [w ] (χ DGv DG n y [w ] = 0, (6.3 { DG n }[w ] ε Γ { w n }[ DG ] DG w = 0, (6.4 ( 1 DG n x[w ]

9 Analyss of a DG Mtod for t Cmotaxs Modl 9 vr v 1 DG wv or 1 DG n xw σ 1 DG n yw v σ v 1 DG (wv y r Γ vr Γ or r v Γ or [ 1 DG ][w ] = 0, (6.5 ( 1 DG vn y [w v ] [v 1 DG ][wv ] = 0. (6.6 To approxmat t onvtv trms n (6.3, (6.5-(6.6 w s t sam ntral-pwnd flxs (3.8 as for t ontnos n tm sm, wt t on-sdd loal spds gvn by: a ot b ot ( := max := max ( (χ DG 1, (χ DG, 0, a n := mn ( (χvdg 1, (χv DG, 0, b n := mn Not tat t nqalts smlar to (3.10, (χ DG 1, (χ DG, 0, ( (χvdg 1, (χv DG, 0. (6.7 a ot a ot a n 1, a ot a n a n 1, b ot b ot b n 1, and b ot b n b n 1, (6.8 w ar ndd n or onvrgn proof, ar satsfd for t loal spds dfnd n (6.7 as wll (for smplty, w assm tat a ot a n 0 and b ot b n 0 trogot t omptatonal doman, s Rmark n Ston 3. Also, t ntal ondtons ar: 0 DGw = 0 w, 0 DGw = 0 w, 0 DG w = 0 w, vdg 0 wv = v 0 w v. (6.9 W dnot by,, ũ, and ṽ t pws polynomal ntrpolants of t xat solton omponnts,,, and v of t Kllr-Sgl systm (1. and assm tat ts ntrpolants satsfy t approxmaton proprty (.. W tn mak t followng ndton ypotss: assm tat : 0 n, w av { S = ( DG, v DG W r, Wv r : v, n t DG ũ 0, =0 ( mn(r 1,s C r s 3 C t t, n mn(r1,s r s 3 mn(r1,s r s 3 mn(rv1,sv rv sv 3 (6.10 t v DG ṽ 0, =0

10 10 Y. pstyn and A. Izmrlogl ( mn(r 1,s C v r s 3 Cv t t, sp 0 n DG ũ 0, C mn(r1,s r s 3 r mn mn(r1,s r s 3, sp vdg ṽ 0, Cv 0 n mn(rv1,sv rv sv 3 (6.11 }, (6.1 wr C, C v,, C t, C t v, C, and C v ar postv onstants (w wll b dfnd latr ndpndnt of, t polynomal dgrs (r, r, r, r v, and n. T paramtrs s, s, s, and s v dnot t rglarty of t orrspondng omponnts of t xat solton. Clarly, t ndton ypotss abov olds tr for = 0. W nd now to sow tat assmng tat S olds tr : 0 n, t wll follow tat t wll b tr for 1 = n 1. Lt s frst sow tat from t ndton ypotss S, t follows tat fntons DG, and v DG, 1 n ar bondd. Lmma 6.1 For ( DG, v DG S, tr xst postv onstants M, and M v ndpndnt of, r, and r v, s tat sp DG, M, sp vdg, M v. ( n 0 n Proof: T rslt s drvd from t dfnton of t sbst S and t nvrs nqalty. Torm 6. (l (H 1 and l (L Forward lr rror stmats. Lt t solton,, and v of t Kllr-Sgl systm (1. b sffntly rglar. Frtrmor, w assm tat pnalty paramtrs σ, σ, σ, σ v ar sffntly larg. Tn t ndton ypotss olds tr for n1. Frtrmor, tr xsts onstants C and C, ndpndnt of and r, s tat ( DG l ([0,T];L ( N t1 (DG l ([0,T];L ( t r [ DG ] 1 0, Γ =0 r mn C ( mn(r1,s 1 r s 3 mn(r1,s 1 r s 3 mn(r1,s 1 r s 3 mn(rv1,sv 1 r sv 3 v t ( DG l ([0,T];L ( N t1 (DG l ([0,T];L ( t r [ DG ] 1 0, Γ =0 C ( mn(r1,s 1 r s 3 mn(r1,s 1 r s 3 mn(r1,s 1 r s 3 mn(rv1,sv 1 r sv 3 v t, wr (r, r, r, r v. Proof: W ntrod t followng notaton: τ := DG, ξ :=, τ := DG, ξ :=, τ := DG ũ, ξ = ũ, τv := vdg ṽ, ξv := v ṽ. (6.14

11 Analyss of a DG Mtod for t Cmotaxs Modl 11 It follows from t onsstny Lmma 4.1(s [18] for t dtals tat t xat solton of (1. satsfs t followng wak formlaton: t w w { n }[w ] ε { w n }[] σ r [][w ] χ(w x (χ n x [w ] χv(w y (χv n y [w ] = 0,(6.15 wr Γ vr (χ := aot (χv := bot (χ 1 a ot (χv 1 b ot a n (χ a n b n (χv b n and t loal spds a ot, a n, bot, and b n ar gvn by (6.7. Usng (6.14, qaton (6.15 an b rwrttn as: t (t w w { n }[w ] ε { w n }[ ] σ r [ ][w ] χ DG (w x χ τ (w x χ ξ (w x (χ n x [w ] = χ v DG(w y ξ t (t w χ τ v(w x ξ w Γ Γ or aot a ot bot b ot a n a n b n b n [], [], Γ vr χ ξ v(w x { ξ n }[w ] ε Γ Γ or (χ v n y [w ] { w n }[ξ ] σ r [ξ ][w ] χξ (w x χξ v (w y. (6.16 In ordr to obtan t stmat on t tm stp t = O( w wll prod as follows, sbtrat r 4 qaton (6.16 from (6.3 and oos w = τ, to obtan τ 1 0, τ 0, t τ 0, tσ r [τ ] = 0, τ 1 τ 0, Γ (1 ε t { τ n }[τ] t χτ DG(τ x t χ τ(τ x t Γ t Γ vr ((χ DG DG (χ n x [τ ] t t χ ξ v (τ y t Γ or χτ v DG (τ y t ((χ DG vdg (χ v n y [τ ] χ τ v (τ y χ ξ (τ x

12 1 Y. pstyn and A. Izmrlogl t ξ t (t τ t ε t Γ { τ n }[ξ ] tσ Lt s rmark tat trm τ 1 ( t (t 1 τ t τ t t r τ 1 τ τ t [ξ ][τ ] t =: T 1 T... T 18. ξ τ t Γ χξ (τ x t on t LHS of (6.17 was rwrttn as: τ 1 τ 0, τ 0, τ 1 τ 0, = t t t { ξ n }[τ ] χξ v (τ y (6.17 (6.18 Nxt, w bond a trm on t RHS of (6.17 startng from trm T sng standard DG tnqs, t trm T 1 wll b bondd last. T qantts ε n t stmats blow ar postv ral nmbrs, w wll b dfnd latr. Consdr now, sond trm on t RHS T : T 4 t Γ { τ } 0, [τ ] 0,. As bfor, w dnot by 1 and t two lmnts sarng t dg. Tn, sng t nqalty (.6, w obtan t { τ } 0, [τ ] 4 t ( 1 ( τ 0, 1 ( τ [τ ] 0, 0, 0, Γ tc tr ( τ τ [τ 0,1 0, ], 0, Γ vr Γor and n, sng t fat tat and sng Cay-Swarz nqalty, w nd p wt t followng bond on T : T ε t τ 0, R t r [τ ] 0, Γ (6.19 T trm T 3 s bondd sng Lmma 6.1 for DG and sng Cay-Swarz and Yong s nqalty: T 3 ε 3 t τ 0, R 3 t τ (6.0 T trm T 4 s bondd sng Cay-Swarz and Yong s nqalts and assmpton (3.: T 4 t K τ 0, τ 0, ε 4 t τ 0, R 4 t τ (6.1 0, T trm T 5 s bondd sng Cay-Swarz, Yong s nqalty and approxmaton rslts (.: T 5 ε 5 t τ mn(r1,s 0, R 5 t (6. r s 0,

13 Analyss of a DG Mtod for t Cmotaxs Modl 13 Nxt, w bond T 6 on t RHS of (6.17 as T 6 t ( a ot Γ vr a ot a ot a n a n a n a n a ot a ot a n ( (χ DG DG 1 ( (χ DG DG (χ 1 n x [τ ] (χ n x [τ ] [ DG ]n x [τ ] =: I II III. (6.3 Usng (6.8 and (6.14, t frst trm on t RHS of (6.3 an b stmatd by I tχ ( (χ DG DG 1 (χ 1 n x [τ ] Γ vr tχ Γ vr ( (τ DG 1 n x[τ ] ( τ 1 n x[τ ] (ξ DG 1 n x[τ ] (ξ 1 n x[τ ] =: Ĩ. W now s t Cay-Swarz nqalty, t tra nqalty (.4, t nqalty (.6, t assmpton (3., t assmpton tat < 1, r > 1, t < 1, t approxmaton nqalty (., t bond on DG from Lmma 6.1: Ĩ ε 1 t τ (K 0, 1K t r [τ ] ( K 0, 3 t τ mn(r 1,s 0, K 4 t Γ r s mn(r1,s. r s A smlar bond an b drvd for t sond trm II on t RHS of (6.3. To stmat t last trm on t RHS of (6.3, w frst s (6.14 and t dfnton of t on-sdd loal spds (6.7 to obtan III C t ( [τ ] [ξ 0, ][τ ] := ĨII. Γ vr Tn, sng t Cay-Swarz nqalty, t assmpton tat < 1 (and oosng small nog, r > 1, t < 1, t approxmaton nqalty (., and nqalty (.4, w bond ĨII as follows: ĨII Γ vr tr [τ ] 0, Combnng t abov bonds on I, II, and III, w arrv at mn(r1,s 1 t. r s T 6 ε 6 t τ R 0, 6 t r [τ ] ( R 0, 7 t τ mn(r 1,s 1 0, R 8 t Γ r s mn(r1,s r s. (6.4

14 14 Y. pstyn and A. Izmrlogl T trms T 7 - T 10 ar bondd n t sam way as t trms T 3 - T 6, rsptvly, and t bonds ar: T 7 ε 7 t τ 0, R 9 t τ (6.5 T 8 ε 8 t τ 0, R 10 t τ v 0, (6.6 T mn(rv1,sv 9 ε 9 t τ 0, R 11 t rv sv T10 ε 10 t τ R 0, 1 t r [τ ] R 0, 13 t τv 0, Γ ( mn(r 1,s R 14 t r s For t trm T 11 w obtan t followng bond: 0, (6.7 mn(rv1,sv. (6.8 rv sv T 11 ε 11 t τ 0, R 15 t mn(r1,s r s (6.9 Nxt, w bond trm T 1. Frst sng a Taylor xpanson wt ntgral rmandr w an wrt Hn, 1 = t t (t 1 t 1 t (s t tt (sds, and w an bond 1 t 1 (s t tt (sds 1 ( t 1 1 ( t 1 1 (s t ( tt (s ds t t t 4 t t(t 1 t 1 0, t t 1 t (s t t 1 Hn, sng t abov stmats w obtan t followng bond for T 1: t tt (s 0, ds, T1 ε 1 t τ t1 R 0, 16 t tt (s 0, ds. (6.30 t Nxt, w bond T 13 sng Cay-Swarz and t approxmaton nqalty (.: T 13 t τ 0, ξ 0, ε mn(r1,s 13 t τ 0, R 17 t r s (6.31 Consdr now trm T 14 : T 14 t Γ { ξ n }[τ ] t { ξ n } 0, [τ ] 0, = t 1 { ξ r n r } 0, [τ Γ 1 ] 0,

15 Analyss of a DG Mtod for t Cmotaxs Modl 15 Nxt, sng Cay-Swarz and tra nqalty (.5 w obtan: ε 14 t r [τ ] C t 0, r Γ Fnally, sng t approxmaton rslt (. w drv: ( 1 ξ 0, ξ 0, T14 ε 14 t r [τ ] R 0, 18 t mn(r1,s Γ r s (6.3 Nxt w bond trm T 15 : T15 ε t { τ n }[ξ ] Γ Now sng Lmma.3 and approxmaton rslt (.3 w obtan: ε t τ 0, C 1 r mn(r1,s 1 r s 1 Nxt, applyng Cay-Swarz nqalty w gt t fnal stmat: T 15 ε 15 t τ 0, R 19 t mn(r1,s r s 3 (6.33 Now, w bond trm T16: T 16 tσ [ξ ][τ ] r tσ [τ 1 ] 0, [ξ ] r 0, 1 Γ r Hn, w obtan t followng bond for T 16 by sng approxmaton rslt (.3: Γ T16 ε 16 t r [τ ] R 0, 0 t mn(r1,s Γ r s 3 (6.34 Trms T 17 and T 18 ar bondd rsptvly sng t approxmaton nqalty (. : T17 ε 17 t τ mn(r1,s 0, R 1 t r s (6.35 T 18 ε 18 t τ 0, R t mn(r1,s r s (6.36 T last trm tat nds to b bond s t trm T 1. To bond ts trm w agan sbtrat (6.16 from (6.3 and oos w = τ 1 τ. Usng smlar tnqs as for t stmaton of

16 16 Y. pstyn and A. Izmrlogl (6.17, xpt now s nvrs nqalty to stmat (τ 1 τ] 0,, w obtan t followng stmat: [τ 1 τ 1 τ t r 0, Ck 4 1 τ 0, Ck τ and nqalty (.6 to stmat t r 4 r Γ [τ ] 0, t r 4 C3 k τ t r 0, Ck 4 4 τ t r 0, Ck 4 5 τ v 0, t r 4 ( C6 k mn(r 1,s mn(r1,s mn(rv1,sv t 1 r s 3 C k7 r s r t3 tt (s v sv 0, ds t (6.37 Som dtals of t drvaton of t stmat (6.37 an b fond n Appndx 9. Nxt, w ombn stmats (6.19-(6.36 togtr wt (6.37, plg t n (6.17, and sng agan t assmptons < 1 and r > 1 to obtan: τ 1 0, τ t( C 0, 1 tr 4 τ 0, t(σ tr 4 C r [τ ] 0, Γ C 6 t r 4 t r 4 C 3 τ ( mn(r 1,s r s 3 Now sm (6.38 for = 0,..., n: C t r 4 0, 4 mn(r1,s r s =0 τ 0, C 5 mn(rv1,sv r sv v t r 4 τ v 0, τ n1 0, τ 0 ( C tr 4 n 0, 1 tr 4 n t τ 0, (σ C C 6 tr 4 C 3 tr 4 n =0 n =0 ( mn(r 1,s t r s 3 t τ C tr 4 0, 4 n =0 mn(r1,s r s t τ C tr 4 0, 5 mn(rv1,sv r sv v t1 C 7 t tt (s 0, ds t =0 n =0 n C 7 t Now sng ndton ypotss S for n =0 t τ 0, and n =0 t τ v 0, obtan: τ n1 ( C 0, 1 tr 4 n t τ 0, (σ tr 4 n C =0 =0 (6.38 t r [τ ] 0, Γ t τ v 0, =0 t1 t tt (s 0, ds (6.39, (6.10-(6.11, w t r [τ ] 0, Γ C 3 tr 4 n =0 t τ 0, τ 0 0,

17 Analyss of a DG Mtod for t Cmotaxs Modl 17 ( (C 4 C C 5 C v C6 tr4 mn(r1,s r s 3 mn(r1,s r s 3 n (C 4 C t C 5 C t v tr4 t C 7 t =0 mn(r1,s r s 3 t1 mn(rv1,sv rv sv 3 tt (s 0, ds (6.40 t Lt s now dfn C max := max(c 3, C 4 C C 5 C v C6, C 4 C t C 5 Cv t, wr onstant C max dpnds on t proprts of t xat soltons,, v, T and doman (t follows from t standard DG tnqs and ndton ypotss for = n. Hn, w obtan: τ n1 ( C 0, 1 tr 4 n t τ 0, (σ tr 4 n C =0 C max tr 4 ( mn(r1,s r s 3 Nxt oosng t mn C max tr 4 n =0 mn(r1,s r s 3 n tr 4 C max t C 7 t ( C maxr 4, C 1, r 4 C r 4 ( 1, C 1 tr 4, t τ 0, τ 0 0, =0 =0 mn(r1,s r s 3 t1 t r [τ ] 0, Γ mn(rv1,sv rv sv 3 tt (s 0, ds (6.41 t, sttng pnalty paramtr σ to b larg tr nog, dfnng C mn := mn σ C 4 = 1, w an b don by oosng pnalty paramtr σ larg nog, and sttng C mm := max ( R 1 1C T 7 0 C mn, tt(s 0, ds C mn = 1 C 7 T 0 tt (s 0, ds. Not tat, now, t onstant C mm s ndpndnt of t ndton ypotss. Hn, w drv: C mm n =0 t τ 0, C mm n τ n1 t( τ 0, 0, r [τ ] 0, Γ =0 ( mn(r 1,s r s 3 mn(r1,s r s 3 mn(r1,s r s 3 mn(rv1,sv rv sv 3 C mm τ 0 0, C mm t, (6.4 wr C mm dpnds only on t proprts of t xat soltons, T and doman. Now, applyng dsrt Gronwall s lmma to (6.4, w obtan t fnal stmat for τ n1 : n τ n1 t( τ 0, 0, r Γ =0 [τ ] 0, C ( mn(r 1,s r s 3 mn(r1,s r s 3 mn(r1,s r s 3 mn(rv1,sv C r t t, (6.43 v sv 3

18 18 Y. pstyn and A. Izmrlogl wr onstants C, C t dpnd only on t proprts of t xat soltons, doman, T and ndpndnt of n. W now prod n a smlar way to t drvaton of (6.43 for τ 1 = τ n1 and w obtan: n τ n1 t( τ 0, 0, =0 Γ mn(r1,s r s 3 r [τ ] 0, C ( mn(r 1,s r s 3 mn(r1,s r s 3 mn(rv1,sv C t rv sv 3 t (6.44 For t som dtals of t drvaton of t abov stmat (6.44 s Appndx 9. Nxt, w prod by provng tat t ndton ypotss S for n1 DG. On agan, by t onsstny Lmma (4.1, t xat solton satsfs t followng qaton (ompar t wt (3.5: ũ(t 1 w 1 (w x ( 1 n x [w ] 1 n x w wr σ r Γ vr vr Γ vr [ũ 1 ][w ] = ξ 1 n x w σ ( 1 := aot r Γ vr 1 1 a ot ξ 1 w a n 1 a n vr ξ 1 (w x [ξ 1 ][w ], (6.45 aot a ot a n a n [ 1 ]. Sbtratng qaton (6.45 from (6.5 and oosng w = τ 1, w obtan τ 1 0, σ = r [τ1 ] 0, Γ vr τ 1 (τ 1 x Γ vr ξ 1 (τ 1 x vr (( 1 DG ( 1 n x[τ 1 ] ξ 1 n x [τ 1 ] σ r Γ vr vr τ 1 n x [τ 1 ] ξ 1 τ 1 [ξ 1 ][τ 1 ] =: T 1... T 7, (6.46 and bond a trm on t RHS of (6.46. Lt s now bond trm by trm on t rgt-and sd of (6.46. Consdr t trm T1. Usng ntgraton by parts, w gt: τ 1 (τ 1 x = ( τ 1 (τ 1 x τ 1 τ 1 n x, wr n x dnots t x omponnt of t otward normal vtor to lmnt.

19 Analyss of a DG Mtod for t Cmotaxs Modl 19 Smmng ovr all t lmnts w av: ( (τ 1 x τ 1 = (τ 1 x τ 1 Γ vr τ 1 τ 1 n x [τ 1 τ 1 Rallng t formla for t jmp and avrag (.1 w gt: = (τ 1 x τ 1 [τ 1 ]{τ 1 }n x [τ 1 ]{τ 1 }n x Γ vr Γ vr ]n x vr τ 1 [τ 1 Hn, sng t nqalty (.6, Cay-Swarz s and Yong s nqalts, sng lmma.4 for τ 1 and applyng assmpton (3., w av t followng bond for T 1 T1 ε 1 τ 1 U 0, 1 r Γ vr [τ 1 ] U 0, ( τ 1 0, r Γ A bond for T an b obtand n a way smlar to t bond on T 8 : T ( a ot ( ( 1 a ot Γ a n DG 1 ( 1 1 n x [τ 1 ] a n a ot a n a n aot a ot a n [ 1 DG 1 ]n x [τ 1 ] ]n x [τ 1 ] 0, (6.47 ( ( 1 DG ( 1 n x [τ 1 ] := I II III. (6.48 From (6.8 and (6.14, t frst trm on t RHS of (6.48 an b stmatd by I ( (τ 1 1 n x[τ 1 ] (ξ 1 1 n x[τ 1 ] := Ĩ. Γ Usng tn t Cay-Swarz nqalty, t tra nqalty (.4, t nqalty (.6, and t assmpton (3., w stmat Ĩ as follows: Ĩ KK 1 τ 1 0, KK r [τ1 Γ ] 0, KK 3 mn(r1,s A smlar bond an b drvd for t sond trm on t RHS of (6.48. T trd trm on t RHS of (6.48 s smlar to t trd trm on t RHS of (6.3, n t an b bondd by ( KK4 III 1 [τ1 ] mn(r1,s 1 0, KK 6. By oosng small nog, w obtan: r III Γ r Γ r [τ1 ] 0, mn(r1,s 1. r s r s r s.

20 0 Y. pstyn and A. Izmrlogl Combnng t abov bonds on I, II, and III, and sng lmma.4 w arrv at T U 3 ( τ 1 0, r [τ 1 ] U 0, 4 Γ U 5 ( mn(r 1,s 1 r s r [τ1 ] 0, Γ vr mn(r1,s. (6.49 r s To bond t trm T3, w s t Cay-Swarz nqalty, Yong s nqalty, t nqalty (.6, and lmma.4 w yld T3 U 6 ( τ 1 0, r [τ 1 ] U 0, 7 Γ r [τ1 vr ] 0,. (6.50 T trm T4 s bondd wt t lp of Cay-Swarz nqalty, Yong s nqalty, and t approxmaton nqalty (.: T 4 ε τ1 0, U 8 mn(r1,s r s. (6.51 Usng smlar da as for t trm T1, w an rwrt T 5 n t form T5 = (ξ 1 x τ 1 [ξ 1 ]{τ 1 }n x [τ 1 ]{ξ 1 }n x Γ vr Γ vr ξ 1 [τ 1 ]n x := T 5. (6.5 vr W tn s Lmma.1 and Lmma.3 to obtan T 5 ε 3 τ 1 0, Ũ9 r Γ vr [τ1 ] mn(r1,s 0, U 9. (6.53 r s 3 T trm T6 s bondd sng t Cay-Swarz nqalty, t tra nqalty (.4, and t approxmaton nqalty (.: T 6 U 10 r [τ1 vr ] 0, U 11 mn(r1,s T last trm T 7 s bondd sng t approxmaton rslt (.3. Hn, T 7 U 1 r [τ1 Γ vr ] 0, U 13 r s mn(r1,s r s 3. (6.54. (6.55 Aftr obtanng t stmats (6.47 (6.55, w plg tm nto (6.46 and s t assmpton < 1, r > 1, assmpton (3., oos t pnalty paramtrs larg nog and an approprat salng to obtan

21 Analyss of a DG Mtod for t Cmotaxs Modl 1 τ 1 0, r [τ1 ] 0, Γ vr C 1 ( τ1 0, r [τ 1 ] ( mn(r 1,s 0, C Γ r s 3 mn(r1,s. (6.56 r s 3 Nxt mltply bot sds of (6.56 by t and sm for = 1,..., n to gt n =0 n = 1 n t τ 1 0, t = 1 r [τ1 ] 0, Γ vr C 3 ( t τn1 0, t r [τ n1 ] 0, Γ C 4 t( τ 0, r n [τ ] 0, Γ = 1 ( C 5 mn(r 1,s t r s 3 Nxt, sng t nvrs nqalty, nqalty (.6 w obtan: n =0 n = 1 n t τ 1 0, t = 1 r [τ1 ] 0, Γ vr C 6 ( tr4 τ n1 tr4 τ n1 0, 0, C t( τ 7 0, r [τ ] n C 0, t 8 Γ = 1 ( mn(r 1,s r s 3 mn(r1,s r s 3 mn(r1,s r s 3. (6.57. (6.58 Fnal stmat s obtand by oosng t C r 4 and by sng t stmat (6.44: C ( mn(r 1,s r s 3 n = 1 n t τ 1 0, t = 1 mn(r1,s r s 3 r [τ1 ] 0, Γ vr mn(r1,s r s 3 T stmat (6.59 provs t ndton ypotss (6.10 for = n 1: mn(rv1,sv C r t t. (6.59 v sv 3 n1 ( t τ mn(r 1,s 0, C =0 r s 3 mn(r1,s r s 3 mn(r1,s r s 3 mn(rv1,sv C r t t v sv 3

22 Y. pstyn and A. Izmrlogl Indton ypotss S, (6.1 an b sown sng smlar tnqs as n (6.10, s dtals n Appndx 9. In t sam way, w prov t ndton ypotss S for τ v and sow tat: n1 t τv n1 0, t =0 =0 r v Γ or [τ v] 0, C v ( mn(r 1,s r s 3 mn(r1,s r s 3 mn(r1,s r s 3 mn(rv1,sv C r v t t. (6.60 v sv 3 7 Rng-Ktta Sond Ordr Tm Dsrtzaton Fnd a dsrt n tm solton ( 1 DG, 1 DG, 1 DG, v1 DG W r, W r, W r, W v r v,, w satsfs t followng wak formlaton for t motaxs systm (1.: zdg, w = σ ( DG w t r zdg, w = σ [ DG ][w ] χ DG v DG (w y r z DG,w σ Γ or ( DG w t [ DG][w ] r Γ vr z DG, n xw, DG w Γ χ DG DG (w x { DG n }[w ] ε Γ Γ vr (χ DG DG n x [w ] { w n }[ DG ] (χ DG v DG n y [w ], (7.1 DG w Γ DGw ( [zdg,][w ] = { DG n }[w ] ε Γ { w n }[ DG ] DGw, (7. z DG,(w x Γ or ( z DG, n x [w ] (7.3 vr z DG,v wv σ v or 1 DG w = 1 r v Γ or z DG,n y w v, DG w 1 ( [zdg,v ][wv ] = ( zdg, w 0.5 t z DG, (wv y Γ or ( z DG, v n y[w v ] zdg, w { zdg, n }[w ] Γ (7.4

23 ε Γ Γ vr 1 DG w = 1 ε Γ Analyss of a DG Mtod for t Cmotaxs Modl 3 { w n }[z DG, ] σ r (χz DG,z DG, n x [w ] 1 DG w σ DG w 1 { w n }[z DG, ] vr v 1 DG wv σ v or r Γ vr 1 DG n xw, r v Γ or 1 DG n yw v, [z DG, ][w ] χz DG,z DG,v(w y ( zdg, w 0.5 σ r [ 1 DG ][w ] = χz DG, z DG, (w x Γ or z DG, w Γ [zdg, ][w ] ( ( [v 1 DG ][wv ] = (χzdg,z DG,v n y [w ], (7.5 zdg, w 1 DG (w x Γ vr 1 DG (wv y Γ or { z DG, n }[w ] z DG, w, (7.6 ( 1 DG n x [w ] ( 1 DG v n y[w v ] Agan, w s (3.8 to approxmat t onvtv trms n t sm abov, wt t on-sdd loal spds gvn by: ( ( a ot, := max (χ DG 1, (χ DG, 0, a n, := mn (χ DG 1, (χ DG, 0, ( ( b ot,v := max (χvdg 1, (χvdg, 0, b n,v := mn (χvdg 1, (χvdg, 0, ( ( a ot,z := max (χzdg, 1, (χz DG,, 0, a n,z := mn (χzdg, 1, (χz DG,, 0 (7.9, ( ( b ot,z v := max (χzdg,v 1, (χz DG,v, 0, b n,z v := mn (χzdg,v 1, (χz DG,v, 0, Dnot by m 1 :=, z, m := v, z v and not tat w mak t sam sttngs r as n t Rmark n ston 3. Also not tat t nqalts smlar to (3.10, a ot,m 1 a ot,m 1 a n,m 1 1, a n,m 1 a ot,m 1 a n,m 1 1, b ot,m b ot,m b n,m 1, and b n,m b ot,m b n,m 1, (7.10 w ar ndd n or onvrgn proof, ar satsfd for t loal spds dfnd n (7.9 as wll (for smplty, w assm tat a ot,m 1 a n,m 1 0 and b ot,m b n,m 0 trogot t omptatonal doman and t ntal ondtons: 0 DGw = 0 w, 0 DGw = 0 w, 0 DG w = 0 w, vdg 0 wv = v 0 w v. (7.11 (7.7 (7.8

24 4 Y. pstyn and A. Izmrlogl Now, lt s s a smlar da to [41] and dfn nw varabls z, z, z, z v : z (x, y, t = (x, y, t t (x, y, t t (7.1 z (x, y, t = (x, y, t t (x, y, t t (7.13 z (x, y, t = (x, y, t t (x, y, t t (7.14 z v (x, y, t = v(x, y, t v t (x, y, t t (7.15 Nxt, sng t rglarty of t dnsty, t onntraton, and Taylor xpanson, w obtan for and : (x, y, t t (x, y, t t (x, y, t t t(x, y, t t t = O( t3 (x, y, t t (x, y, t t (x, y, t t t(x, y, t t t = O( t3 Hn, w obtan: t (x, y, t t = (χ(x, y, t t(x, y, t t x (χ(x, y, t tv(x, y, t t y (x, y, t t = ( χ((x, y, t t (x, y, t t O( t ((x, y, t t (x, y, t t O( t x ( χ((x, y, t t (x, y, t t O( t (v(x, y, t v t (x, y, t t O( t y ((x, y, t t (x, y, t t O( t = (χz z x (χz z v y z O( t In t sam way, t an b sown tat t (x, y, t t = z z z O( t From (7.1 and t Taylor xpanson abov for, t follows tat: Smlarly, for w gt: Not tat for and v w wll av: z = ((χ x (χ v y t 1 = 1 1 z ((χz z x (χz z v y z t O( t3, (7.16 z = ( t 1 = 1 1 z ( z z z t O( t3, (7.17 z = t t = x ( x t t = ( t t x = (z x Hn, t follows tat: z v = (z y z = (z x 1 = ( 1 x, (7.18

25 Analyss of a DG Mtod for t Cmotaxs Modl 5 z v = (z y v 1 = ( 1 y, (7.19 Dnotng by rr(x, y, = O( t 3 and mltplyng t abov qaton (7.16- (7.19 by t tst fntons w, w, w and w v, ntgratng by parts, and sng onsstny of t DG sm w obtan t sm for z, 1, z, 1, z, 1 and zv, v 1 : z w = w A (,, v, w t, 1 w = 1 w 1 z w A (z, z, z v, w t z w = w A (,, w t, 1 w = 1 w 1 z w A (z, z, w t z w σ 1 w σ z v wv σ v v 1 w v σ v r Γ vr r Γ vr r v Γ or r v Γ or [z ][w ] = A (z, w, rr(x, y, w (7.0 rr(x, y, w (7.1 [ 1 ][w ] = A ( 1, w (7. [zv ][wv ] = A v (z, wv, [v 1 ][w v ] = A v ( 1, w v, (7.3 wr w dnotd by ( A (x 1, x, x 3, w := x 1 w Γ { x 1 n }[w ] ε Γ { w n }[x 1 ] σ r [x 1 ][w ] χx 1 x 3 (w y ( A (x 1, x, w := Γ or χx 1 x (w x Γ vr (χx 1 x n x [w ] (χx 1 x 3 n y [w ], (7.4 x 1 w Γ { x 1 n }[w ] ε Γ { w n }[x 1 ] σ r [x 1 ][w ] x 1 w x w, (7.5

26 6 Y. pstyn and A. Izmrlogl ( A (x 1, w := ( A v (x 1, w v := x 1 (w x Γ vr x 1 (w v y Γ vr ( x 1 n x[w ] ( x 1 v n y[w v ] vr vr x 1 n x w, (7.6 x 1 n y w v, (7.7 and Lt s agan ntrod smlar notatons to (6.14 τ := DG, ξ :=, τ := DG, ξ :=, τ := DG ũ, ξ = ũ, τv := v DG ṽ, ξv := v ṽ, τz := zdg, z, ξ z := z z, τ z := zdg, z, ξ z := z z, τz := zdg, z, ξz = z z, τz v := zdg,v z v, ξz v := zv z v. (7.8 (7.9 and sbtrat (7.0 from (7.1 and (7.5 rsptvly. W obtan t followng rror qatons: τz w = τw M(w, (7.30 τ 1 w = ( 1 τ 1 τ z w 1 N (w = τ w 1 M (w 1 N (w (7.31 wr M (w = N (w = (ξ z ξ w (A ( DG, DG, v DG, w A (,, v, w t, (7.3 (ξ 1 ξ ξ z (x, y, w (A (z DG,, z DG,, z DG,v, w A (z, z, z v, w t, (7.33 In t sam way w obtan t rror qatons for onntraton, w obtan t followng rror qatons: τz w = τw M(w, (7.34 τ 1 w = ( 1 τ 1 τ z w 1 N (w = τ w 1 M (w 1 N (w (7.35 wr M (w = N(w = (ξ z ξ w (A ( DG, DG, w A (,, w t, (7.36 (ξ 1 ξ ξz (x, y, w (A (zdg,, zdg,, w A (z, z, w t, (7.37

27 Analyss of a DG Mtod for t Cmotaxs Modl 7 T rror qatons for and v ar : τz w σ r Γ vr r Γ vr τ 1 w σ [τz ][w ] = M (w, (7.38 [τ 1 ][w ] = N (w (7.39 wr M (w = N (w = ξ z w σ r Γ vr r Γ vr ξ 1 w σ [ξz ][w ] (A (zdg,, w A (z, w, (7.40 [ξ 1 ][w ] (A ( 1 DG, w A ( 1, w (7.41 τ z v w v σ v r v Γ or r v Γ or τv 1 w v σ v [τz v ][w v ] = Mv(w v, (7.4 [τ 1 v ][w v ] = N v (wv (7.43 wr Mv (wv = N(w = ξ z v w v σ v r v Γ or r v Γ or ξv 1 w v σ v [ξz v ][w v ] (A v (zdg,, wv A v (z, wv, (7.44 [ξ 1 v ][w v ] (A v ( 1 DG, wv A v ( 1, w v (7.45 Nxt, st w = τ n t qaton of (7.30 and w = τ z n (7.31, add t two qatons and aftr asy allatons, obtan t mportant qalty tat wll b sd to drv t rror stmat for t dnsty : τ 1 0, τ τ = 1 0, τz 0, M (τ N (τ z (7.46 Smlarly, w obtan for t onntraton : τ 1 0, τ = 0, τ 1 τz 0, M (τ N(τ z (7.47 As for Forward lr Sm, lt s mak t followng ndton ypotss: { SR = ( DG, v DG W r, Wv r : v, n t DG ũ 0, =0

28 8 Y. pstyn and A. Izmrlogl ( mn(r 1,s C r r s 3 Cr t t 4, n mn(r1,s r s 3 mn(r1,s r s 3 mn(rv1,sv rv sv 3 (7.48 =0 t v DG ṽ 0, C rv ( mn(r 1,s C t rv t4, sp 0 n sp 0 n r s 3 DG ũ 0, C r DG 0, C r mn(r1,s r s 3 rmn rmn 4, sp 0 n, sp 0 n T ndton ypotss SR mpls t followng lmma. mn(r1,s r s 3 v DG ṽ 0, C rv DG 0, C r rmn rmn 4 mn(rv1,sv rv sv 3 (7.49, (7.50 }, (7.51 Lmma 7.1 For ( DG, DG, DG, v DG SR, tr xst postv onstants M, M, M,, M v, N, N, N, and N v ndpndnt of, r, r, r, and r v, s tat sp DG, M, 0 n sp zdg, N,, 0 n sp DG, M, 0 n sp zdg, N,, 0 n sp DG, M, 0 n sp zdg, N,, 0 n sp vdg, M v, 0 n (7.5 sp 0 n z DG,v, N v. (7.53 Proof: For t dtals of t proof s Appndx 9. Torm 7. (l (H 1 and l (L Rng-Ktta rror stmats. Lt t solton,, and v of t Kllr-Sgl systm (1. b sffntly rglar. Frtrmor, w assm tat pnalty paramtrs σ, σ, σ, σ v ar sffntly larg. Tn t ndton ypotss olds tr for n1. Frtrmor, tr xsts onstants C r and C r, ndpndnt of and r, s tat ( DG l ([0,T];L ( N t1 (DG l ([0,T];L ( t r [ DG ] 1 0, Γ C r ( mn(r1,s 1 r s 3 mn(r1,s 1 r s 3 mn(r1,s 1 r s 3 =0 mn(rv1,sv 1 r sv 3 v =0 t ( DG l ([0,T];L ( N t1 (DG l ([0,T];L ( t r [ DG ] 1 0, Γ C r ( mn(r1,s 1 r s 3 wr (r, r, r, r v. mn(r1,s 1 r s 3 mn(r1,s 1 r s 3 mn(rv1,sv 1 r sv 3 v t,

29 Analyss of a DG Mtod for t Cmotaxs Modl 9 T proof of t abov torm s postpond to Appndx 9. Rmark. T rror stmats obtand n ts papr ar -optmal and r-sboptmal (by 1/, and t nmral tsts rportd n [18] onfrm t tortal rror stmats. 8 Nmral xampl In ts ston, w onsdr t ntal-bondary val problm for t Kllr-Sgl systm and ompar t solton obtand by t proposd Forward lr DG mtod and by t Rng- Ktta DG mtod. W tak t motat snstvty χ = 1 and t bll-sapd ntal data (x, y, 0 = (x y, (x, y, 0 = (x y. W onsdr t abov problm n t sqar doman [ 1, 1] [ 1, 1 ]. Aordng to t rslts n [3], bot omponnts and of t solton ar xptd to blow p at t orgn n fnt tm. In Fgrs , w plot t ontors of t dnsty along t ln L = [ 1, 1] 0, omptd at dffrnt tms bfor blow-p on nform grd wt = 1/101 sng qadrat polynomal approxmaton (Fgrs 8.1 and 8. and b polynomal approxmaton (Fgrs 8.3 and 8.4. T dasd ln rprsnts t solton obtand by t Forward lr DG mtod and t sold ln rprsnts t solton obtand by t Rng-Ktta DG mtod. As on an s, t gr ordr tm sms ar mportant for s typ of problms wt rapdly angng solton n tm. 9 Appndx: Proof of Svral stmats W ollt n t Appndx dtals of t proofs of t svral stmats. 9.1 Drvaton of t stmat (6.37 To obtan t stmat (6.37, w agan sbtrat (6.16 from (6.3 and oos w = τ 1 τ. τ 1 τ = t τ 0, (τ1 τ t Γ r Γ { τ n }[τ 1 τ] ε t { (τ 1 τ n }[τ] tσ Γ t τ x t t t χ ξ (τ 1 χτ v DG (τ1 χτ DG (τ1 τ x t Γ vr τ y t χ τ (τ1 τ x ((χ DG DG (χ n x [τ 1 τ ] χ τ v (τ1 τ y t χ ξ v (τ1 [τ ][τ 1 τ ] τ y

30 30 Y. pstyn and A. Izmrlogl t Γ or ((χ DG vdg (χ v n y [τ 1 τ ] t ξ t (t (τ 1 τ t ε t Γ t t ξ (τ1 { (τ 1 χξ (τ 1 ( t (t 1 t τ t Γ (τ 1 τ τ n }[ξ ] tσ τ x t =: TT 1 TT... TT 0. { ξ n }[τ 1 τ ] r χξ v (τ 1 [ξ ][τ 1 τ ] (9.1 τ y Now lt s bond a trm on t RHS of (9.1. Usng smlar tnqs as for t stmaton of (6.17, xpt now w wll s nvrs nqalty for t stmaton of t (τ 1 τ, and nqalty (.6 to stmat [τ 1 τ ]. Hn w obtan t followng stmats: 0, TT 1 1 τ 1 1 τ 0, R1 t r 4 τ 0, (9. TT 1 τ 1 τ 1 r 4 0, R t τ 0, (9.3 TT 3 1 τ 1 τ 1 r 4 0, R3 t [τ ] 0, (9.4 TT TT 4 1 τ 1 1 TT TT τ τ 1 τ 1 TT 7 1 τ 1 1 τ 1 R 11 t r 4 0, R4 t r 4 Γ r r Γ [τ ] 0, (9.5 τ r 4 0, R5 t τ 0, (9.6 τ r 4 0, R6 t τ 0, (9.7 τ 0, R7 t r 4 mn(r1,s r s τ r 4 0, R9 t τ r 4 0, R10 t τ ( mn(r 1,s mn(r1,s TT 9 1 τ 1 1 r s r s τ r 4 0, R1 t τ 0, (9.8. (9.9 0, (9.10

31 Analyss of a DG Mtod for t Cmotaxs Modl 31 TT TT TT τ 1 R 18 t r 4 TT TT TT TT TT TT TT TT τ 1 τ 1 τ r 4 0, R13 t v τ v 0, (9.11 τ r 4 0, R14 t mn(rv1,sv (9.1 r s r sv v τ r 4 0, R16 t τ r 4 0, R17 t τ v ( mn(r 1,s mn(rv1,sv rv sv τ 1 τ 1 τ 1 τ 1 τ 1 τ 1 τ 0, R19 t mn(r1,s r s 0,. (9.13 (9.14 t 1 τ 0, R0 t 3 tt (s 0, ds. (9.15 t τ r 4 0, R1 t mn(r1,s (9.16 τ 1 τ 1 τ τ τ τ τ 0, R t r 4 0, R3 t r 4 0, R4 t r 4 0, R5 t r 4 0, R6 t r 4 r s mn(r1,s r s mn(r1,s r s 3 mn(r1,s r s 3 mn(r1,s r s mn(r1,s r s (9.17 (9.18 (9.19 (9.0 (9.1 Now, ombnng all t bonds (9.-(9.1 and sng t assmpton tat < 1, r > 1, t stmat (6.37 follows. 9. Drvaton of t stmat for t Conntraton (6.44 Frst, from t onsstny Lmma 4.1 w obtan tat t xat solton of (1. satsfyng t wak formlaton of t form of t qaton (3.4, w may b rwrttn as t (t w w { n }[w ] ε { w n }[ ] σ r [ ][w ] w w = ξ t (t w ξ w { ξ n }[w ] Γ ε Γ { w n }[ξ ] σ r [ξ ][w ] ξ w ξ w. (9.

32 3 Y. pstyn and A. Izmrlogl W tn sbtrat qaton (9. from qaton (6.4 and st w = τ to obtan τ 1 0, τ 0, t τ 0, tσ r Γ t ξ τ t ξ τ t t(1 ε Γ { τ n }[τ] t [τ ] 0, = τ 1 τ τ t τ 0, t ξ t (t τ t τ 0, ( t (t 1 t τ ξ τ t Γ { ξ n }[τ ] tε Γ { τ n }[ξ ] tσ r [ξ ][τ ] = T 1 T... T 1. (9.3 Lt s not agan tat t trm on t LHS of (9.3 s rwrttn smlar to t trm on t LHS of (6.17: τ 1 τ t τ = τ1 0, τ 0, t t τ1 τ 0, t (9.4 T trms on t RHS of (9.3 ar bondd sng t sam tnqs as n (6.17. As bfor, w start wt trm T: T ε t τ tmn(r1,s 0, C (9.5 r s T 3 ε 3 t τ 0, C 3 t mn(r1,s r s T 4 ε 4 t τ 0, C 4 t τ 0, (9.6 (9.7 T 5 = t τ 0, (9.8 T6 ε 6 t t 1 τ 0, C 6 t tt (s 0, ds. (9.9 t T 7 ε 7 t τ 0, C 7 t r Γ [τ ] 0, (9.30 T 8 ε 8 t τ 0, C 8 t mn(r1,s r s T9 ε 9 t τ 0, C 9 tmn(r1,s r s T10 ε 10 t r [τ ] mn(r1,s 0, C 10 t r Γ s T11 ε 11 t τ 0, C11 t mn(r1,s r s 3 (9.31 (9.3 (9.33 (9.34

33 Analyss of a DG Mtod for t Cmotaxs Modl 33 T1 ε 1 t r [τ ] mn(r1,s 0, C 1 t r Γ s 3 (9.35 Nxt, w nd to bond t trm T1 as t was don for t T 1. W sbtrat (9. from qaton (6.4 and st w = τ 1 τ to obtan: τ 1 τ = t ξ 0, (τ 1 τ t ξ(τ 1 τ t τ(τ 1 τ t τ(τ 1 τ t ε t Γ t ( t (t 1 tσ r { (τ 1 t ξ (τ 1 [ξ ][τ1 (τ 1 τ t τ (τ1 τ t τ ] t τ n }[τ ] tσ τ t Γ τ ] = TT 1 TT r Γ { ξ n }[τ 1 [τ ][τ1 τ ] tε Γ Usng smlar tnqs as n (9.1, w obtan t followng stmats: TT TT 1 15 TT τ 1 τ 1 TT TT τ 1 τ 1 τ 1 TT6 1 τ 1 15 TT7 1 TT τ 1 15 τ 1 TT9 1 τ 1 15 TT10 1 τ 1 15 { τ n }[τ 1 τ ] ξ t(t (τ 1 τ { (τ 1 τ n }[ξ ]... TT14. (9.36 τ 0, t KC τ 0, t KC 1 mn(r1,s r s mn(r1,s r s τ 0, t KC 3 τ 0, (9.37 (9.38 (9.39 τ 0, t KC 4 τ (9.40 0, t 1 τ 0, KC5 t 3 tt (s 0, ds. (9.41 t τ r 4 0, KC6 t τ 0, (9.4 τ r 4 0, KC7 t τ 0, (9.43 τ r 4 0, KC8 t [τ ] (9.44 0, τ 0, KC9 t r 4 τ 0, t KC Γ r r Γ 10 mn(r1,s r s [τ ] 0, (9.45 (9.46

34 34 Y. pstyn and A. Izmrlogl TT11 1 τ 1 15 TT1 1 τ 1 15 TT13 1 τ 1 15 TT14 1 τ 1 15 τ τ τ τ 0, KC11 t r 4 0, KC1 t r 4 0, KC13 t r 4 0, KC14 t r 4 mn(r1,s r s mn(r1,s r s mn(r1,s r s 3 mn(r1,s r s 3 (9.47 (9.48 (9.49 (9.50 W ombn all t bonds (9.5-(9.35 and s bonds (9.37-(9.50 to stmat τ 1 τ 0, (smlar to (6.37. Tn w plg t stmats n (9.3. Coosng t C, smmng r 4 for = 0,.., n, applyng Lmma.4 for τ, sng stmat (6.43 and applyng dsrt 0, Gronwall s lmma w obtan t fnal stmat for t τ 1 : n τ n1 0, =0 t( τ 0, r Γ mn(r1,s r s 3 [τ ] 0, C ( mn(r 1,s r s 3 mn(r1,s r s 3 mn(rv1,sv r C t v sv 3 t ( Proof of t Indton Hypotss (6.1 for n1 DG Lt s onsdr (6.46 and bond a trm on t RHS of (6.46. Consdr t trm T1. Usng nvrs nqalty, Cay-Swarz, assmpton (3. and Yong s nqalty, w obtan: T1 ε 1 τ 1 UU r 4 0, 1 τ 1 (9.5 0, A bond for T an b obtand n a smlar way to t bond on T 8 : T Γ vr ( a ot a n aot a ot a n a ot a n ( ( 1 DG 1 [ 1 DG 1 ]n x [τ 1 ] ( 1 1 n x [τ 1 ] a ot a n a n ( ( 1 DG ( 1 n x [τ 1 ] := I II III. (9.53 From (6.8 and (6.14, t frst trm on t RHS of (6.46 an b stmatd by I ( (τ 1 1 n x[τ 1 ] (ξ 1 1 n x[τ 1 ] := Ĩ. Γ Tn, sng t Cay-Swarz nqalty, t tra nqalty (.4, t nqalty (.6, and t assmpton (3., w stmat Ĩ as follows: Ĩ KU 1 τ 1 0, KU r [τ1 Γ ] 0, KU 3 mn(r1,s r s.

35 Analyss of a DG Mtod for t Cmotaxs Modl 35 A smlar bond an b drvd for t sond trm on t RHS of (6.46. T trd trm on t RHS of (6.46 s smlar to t trd trm on t RHS of (6.3, n t an b bondd by ( KU1 III 1 [τ1 ] 0, KU 3 mn(r1,s 1. r s By oosng small nog, w obtan: r III Γ r Γ r [τ1 ] 0, mn(r1,s 1. r s Combnng t abov bonds on I, II, and III, and sng lmma.4 w arrv at T UU τ 1 UU ( 0, 3 ] mn(r 1,s 1 0, UU 4 r [τ1 Γ vr r s mn(r1,s r s (9.54 To bond t trm T3, w s t Cay-Swarz nqalty, Yong s nqalty and t nqalty (.6 w yld T3 UU 5 τ 1 UU 0, 6 r [τ1 vr. ] 0,. (9.55 T trm T4 s bondd wt t lp of Cay-Swarz nqalty, Yong s nqalty, and t approxmaton nqalty (.: T 4 ε τ1 0, UU 7 Usng smlar tnqs as for (6.5 to obtan T 5 ε 3 τ 1 0, ŨU 8 r Γ vr mn(r1,s r s. (9.56 [τ1 ] mn(r1,s 0, UU 8. (9.57 r s 3 T trm T 6 s bondd sng t Cay-Swarz nqalty, t tra nqalty (.4, and t approxmaton nqalty (.: T last trm T 7 T 6 UU 9 r [τ1 vr ] 0, UU 10 mn(r1,s r s s bondd sng t approxmaton rslt (.3. Hn, T 7 UU 11 r [τ1 Γ vr ] 0, UU 1 mn(r1,s r s 3. (9.58. (9.59 Aftr obtanng t stmats (9.5 (9.59, w plg tm nto (6.46 and s t assmpton < 1, r > 1, assmpton (3., oos t pnalty paramtrs larg nog, oos an approprat salng and s t stmat (6.44 for t τ n1 to obtan τ n1 C ( mn(r 1,s 4 0, mn(r1,s 4 mn(r1,s 4 mn(rv1,sv 4 r C r s 7 r s 7 r t 4 v sv 7. t r s 7 (9.60

36 36 Y. pstyn and A. Izmrlogl T stmat (9.60 provs t ndton ypotss on DG (6.1 for n 1 by makng t approprat o of t = O( : r Proof of Lmma 7.1 τ n1 0, C r mn (9.61 Frstly, t proof of t statmnt (7.5 of Lmma 7.1 s t drt onsqn of t ndton ypotss SR, and s t sam as for t Lmma 6.1. Sondly, to prov t nxt statmnt (7.53 of Lmma 7.1, lt s frst onsdr (7.30, and st w = τz, to obtan: τz = τ τ z M (τ z, (9.6 wr M (w = ε Γ Γ vr 0, ( (ξz ξ w t { w n }[τ ] σ χτ DG(w x r τ w Γ [τ ][w ] χ τ (w x ((χ DG DG (χ n x [w ] χ ξ v(w y Γ or ξ w Γ { τ n }[w ] χ ξ (w x χτ v DG (w y ((χ DGv DG (χ v n y [w ] { ξ n }[w ] ε Γ { w n }[ξ ] χ τ v (w y σ r = T 1 T... T 19. [ξ ][w ] χξ (w x χξ v (w y (9.63 Rallng t dfntons (7.8, (7.9 of ξ and ξ z, w an tn asly obtan t followng bond, w wll b sd svral tms trogot t rror analyss: ξ 1 ξ ξ 0, z ξ 0, mn(r1,s C ξ t (9.64 wr postv onstant C dpnds on t and s ndpndnt of, and r (smlar bond s vald for t onntraton w wll b sd n t drvaton of t rror stmat for. Usng tnqs smlar to t stmaton of t trms TT 1 TT 0, (9.-(9.1 and applyng t alrady vrfd rslt (7.5, w obtan: r s

37 Analyss of a DG Mtod for t Cmotaxs Modl 37 T 1 ε 1 w 0, t R 1 mn(r1,s r s T ε w 0, R t r 8 τ 4 0, T 3 ε 3 w 0, R t r 8 3 τ 4 0, T 4 ε 4 w 0, R t r 8 4 τ 4 0, T 5 ε 5 w 0, R t r 8 5 τ 4 0, (9.65 (9.66 (9.67 (9.68 (9.69 T 6 ε 6 w 0, R t r 4 6 τ 0, (9.70 T 7 ε 7 w 0, R t r 8 7 τ 4 r 4 0, (9.71 T 8 ε 8 w 0, R t r 4 mn(r1,s 8 r s T 9 ε 9 w 0, R t r 4 9 τ R t r 8 0, 10 τ 4 r 4 0, ( mn(r 1,s R 11 t r 4 r s mn(r1,s r s (9.7. (9.73 T10 ε 10 w 0, R t r 4 1 τ 0, (9.74 T11 ε 11 w 0, R t r 8 13 τ 4 rv 4 v 0, (9.75 T1 ε 1 w 0, R t r 4 mn(rv1,sv 14 rv sv T 13 ε 13 w 0, R t r 4 15 R 17 t r 4 ( mn(r 1,s r s τ R 0, 16 t r 8 4 mn(rv1,sv r sv v r 4 v τ v 0, (9.76. (9.77 T14 ε 14 w 0, R t r 4 mn(r1,s 18 r s T 15 ε 15 w 0, R t r 4 mn(r1,s 19 r s (9.78 (9.79

38 38 Y. pstyn and A. Izmrlogl T 16 ε 16 w 0, R t r 5 mn(r1,s 19 r s T 17 ε 17 w 0, R t r 5 mn(r1,s 0 r s T18 ε 18 w 0, R 1 t r mn(r1,s r s (9.80 (9.81 (9.8 T19 ε 19 w 0, R t r mn(r1,s r s (9.83 Now ombnng t stmats (9.65-(9.83, sng t assmpton tat < 1, r > 1, t < 1, and plggng tm nto (9.63, w obtan t followng stmat for M(w : M (w ε w 0, M t r 8 1 τ M t r 8 4 0, 4 r 4 t r 5 ( mn(r 1,s M 4 r s τ M t r 8 0, 3 τ 4 rv 4 v 0, mn(r1,s r s mn(rv1,sv r sv v (9.84 Nxt, oosng w = τz n (9.84, makng ε small nog, takng t C, and sng t r 4 ndton ypotss SR (7.50 and (7.51, w onld tat w obtan from (9.6 τ z 0, As wt stmat (9.85, t an b sown tat C 4 r 8 mn τ z 0, C 4 r 8 mn (9.85 (9.86 Now, lt s onsdr t qaton of (7.38, t an b sown sng t sam tnqs as n t drvaton of (9.84 and (9.60: M (w ε τ z r 0, M1 4 τ z 0, M ( mn(r 1,s M 3 r s 3 r Γ vr [τ z ] 0, mn(r1,s r s 3 (9.87 Consdrng w = τ z, makng ε small nog, takng pnalty paramtr σ larg nog, and sng rslt (9.86, w obtan from (7.38 tat: In t sam way, t an b sown tat: τ z 0, C r 4 mn τ zv 0, C v r 4 mn (9.88 (9.89 Applyng ts stmats (9.88 and (9.89, t nvrs nqalty, and t s of Lmma.1, onlds t proof of Lmma 7.1.

39 Analyss of a DG Mtod for t Cmotaxs Modl Proof of Torm 7. Frst lt s onsdr, (7.46, w an b rwrttn n t followng way: τ 1 0, τ 0, t( τ 0, σ r Γ = τ 1 τz wr M (τ = Ñ (τ z = 0, [τ ] 0, t( τ z 0, σ z r Γ [τz ] 0, M (τ Ñ (τ z, (9.90 (ξ z ξ τ (A ( DG, DG, v DG, τ A (,, v, τ ( τ 0, σ (ξ 1 ( τ z 0, σ z r Γ ξ ξ z (x, y, τ z (A (z DG,, z DG,, z DG,v, τ z A (z, z, z v, τ z r Γ [τz ] 0, t, Lt s stmat, t trms on t RHS of (9.90, startng wt trms M and Ñ. To stmat ts trms, nstad of tnqs sd to obtan stmats n (9.84, w wll s tnqs smlar to t ons sd n stmatng t trms T T 10 and T 13 T 18 n (6.17 (nldng Lmma 6.1. Ts, w wll obtan t followng bonds for t M (τ : r Γ M (τ M 1 t τ 0, ε M t( τ 0, C M ( mn(r 1,s M 4 t r s 3 mn(r1,s r s mn(rv1,sv Smlarly, w an drv t followng stmat Ñ (τ z (applyng now Lmma 7.1: Ñ (τ z Ñ1 t τz 0, ( mn(r 1,s Ñ5 t r s 3 r sv v ε N t( τ z 0, C N mn(r1,s r s r Γ mn(rv1,sv r sv v [τ ] 0, ( [τ ] 0, M t τ 0, M 3 t τ v 0, [τz ] 0, (9.93 Ñ3 t τ z 0, Ñ4 t τ zv 0, O( t 5 (9.94 Nxt, w nd to bond t frst trm of t RHS (9.90. Frst, lt s not tat by sbtratng t qaton (7.30 from t qaton (7.31, and sttng w = τ 1 τz, w obtan tat t frst trm on t RHS of (9.84 w an b xprssd as : τ 1 τ z Hn, n ordr to stmat 0, = 1 (N (τ 1 τ z M (τ 1 τ z (9.95 τ 1 τ z 0, w nd to stmat t RHS of (9.95. It an b sown, sng das smlar to t das sd n t stmaton of t trms TT 1 TT 1 (9.-(9.13, (

40 40 Y. pstyn and A. Izmrlogl and TT 15 TT 0 (9.16-(9.1, tat t followng stmat olds: τ 1 τ z 0, C MN 1 t r 4 ( τ 0, r [τ ] 0, Γ C MN 3 C MN C MN 6 t r 4 t r 4 C MN 5 τ t r 0, CMN 4 3 t r 4 τ z 0, ( mn(r 1,s r s 3 t r 4 ( τ z 0, r Γ C MN 7 τ t r 0, CMN 4 4 [τz ] 0, τv 0, t r 4 τ t r z 0, CMN 4 τ 8 zv 0, mn(r1,s r s mn(rv1,sv r sv v O( t 5 (9.96 At ts pont, w nd to gt stmats for τ z and τ z v. Consdr qaton n (7.38. To obtan an stmat for τ z n trms of τ z, w apply smlar tnqs as n (6.46 to t trms T 1 T 7 (6.47-(6.55. Ts, w obtan t followng stmat: τ z 0, r Γ vr [τ z ] 0, Cz 1 ( τz 0, r Γ Smlarly, for τ z v w av: Cz 1 v ( τz 0, r Γ [τ z ] 0, C z ( mn(r 1,s τ z v 0, r v Γ or r s 3 [τ z v ] 0, [τ z ] 0, C z v ( mn(r 1,s r s 3 mn(r1,s r s 3 mn(rv1,sv r sv 3 v. (9.97. (9.98 Nxt, lt s plg t stmat (9.97 and (9.98 nto (9.96, and aftr smpl modfatons w obtan: τ 1 τz t MN C r 4 1 ( τ 0, 0, r [τ ] 0, Γ t MN C r 4 τ t MN C r 4 0, 3 τ t MN C r 4 0, 4 τ v 0, MN C 6 C MN 5 t r 4 ( τ z 0, r Γ [τz ] 0, t r 4 (( τ z 0, r [τ z ] t C r 4 MN τ 0, 7 z 0, Γ

41 Analyss of a DG Mtod for t Cmotaxs Modl 41 t MN C r 4 ( mn(r 1,s 3 r s 3 mn(r1,s r s 3 mn(r1,s r s 3 mn(rv1,sv r sv 3 v O( t 5 (9.99 Nxt, w ombn stmat (9.94, (9.93 and (9.99, plg tm nto (9.90 and s t assmpton tat t 1, < 1, r > 1 to obtan (aftr som smplfatons t followng: τ 1 0, τ 0, t( τ 0, σ C r 1 r Γ [τ ] 0, t( τ z 0, σ z t r 4 ( τ 0, r [τ ] t r 0, Cr 4 τ t r 0, Cr 4 3 τ 0, Γ t r 4 C4 r τ v 0, Cr 5 C r 6 t r 4 ( τ z 0, r Γ ( mn(r 1,s t r 4 C8 r t r 4 ( τ z 0, r Γ [τz ] 0, [τ z ] 0, Cr 7 t r 4 τ z 0, mn(r1,s mn(rv1,sv rv sv 3 r Γ [τz ] r s 3 mn(r1,s r s 3 r s 3 O( t 5 (9.100 Nxt, onsdr t rror qatons (7.47. Usng smlar tnqs as for t stmats n t qaton (7.46, w obtan t followng stmat for τ : τ 1 0, τ 0, t( τ 0, σ [τ ] 0, t( τ z 0, σ z Γ r C1 t r 4 ( τ 0, r [τ ] 0, C t τ 0, Γ C 3 t τ 0, C 4 t r 4 ( τ z 0, r Γ C 6 t τ t r 4 ( z 0, C mn(r 1,s 7 r s Now, oos t C, apply lmma.4 to stmat r 4 dsrt Gronwall s lmma to (9.101 to obtan: n τ n1 0, K 1 K 3 =0 t( τ 0, r Γ [τ z ] 0, C 5 t τ z mn(r1,s r s 3 τ z [τ ] 0, n n t n τ 0, K t( τz 0, r [τ z ] =0 =0 Γ n ( mn(r 1,s t mn(r1,s r s 3 =0 r s =0 0, 0, r Γ 0, [τ z ] 0, O( t 5 (9.101, sm for = 0,..., n, and apply t( τz 0, r [τ z ] 0, Γ 0,

42 4 Y. pstyn and A. Izmrlogl O( t 4 (9.10 Nxt, smmng t qaton (9.100 for = 0,...n, sng t abov stmat (9.10, onsdrng t C, oosng t pnalty paramtrs larg nog, and sng t assmpton r 4 t < 1, < 1, r > 1, w obtan: n τ n1 t( τ 0, 0, r [τ ] n t( τ 0, z 0, r [τ z ] 0, =0 Γ =0 Γ n R 1 t n τ R 0, t n τ R 0, 3 t τ v 0, =0 =0 n ( mn(r 1,s R 4 t =0 r s 3 mn(r1,s r s 3 =0 mn(r1,s r s 3 mn(rv1,sv r sv 3 v O( t 4 (9.103 Now, applyng ndton ypotss SR (7.48, (7.49 and sng dsrt Gronwall s lmma w obtan: n τ n1 t( τ 0, 0, r [τ ] n t( τ 0, z 0, r [τ z ] 0, =0 Γ =0 Γ ( mn(r 1,s C r s 3 mn(r1,s mn(r1,s mn(rv1,sv r s 3 r s 3 rv sv 3 O( t 4 (9.104 T stmat (9.104 also onfrms t ndton ypotss (7.51 for for 1 = n 1. Usng t fnal stmat for τ n1 (9.104 w drv t followng bond for τ n1 : n τ n1 0, =0 ( mn(r 1,s C t( τ 0, r Γ [τ ] n t( τ 0, z 0, r =0 Γ mn(rv1,sv [τ z ] 0, r s 3 mn(r1,s mn(r1,s r s 3 r s 3 rv sv 3 O( t 4 (9.105 Agan, t abov stmat (9.105 onfrms t ndton ypotss (7.51 for, for 1 = n. Nxt, to obtan t stmat for τ n1 w onsdr sond rror qaton n (7.39. mployng smlar tnqs as n t as of t rror qaton (6.46 and mltplyng by t, t an b sown tat : n n t τ 1 0, C 1 t C 3 n =0 = 1 t( τ 0, Γ r = 1 r [τ1 ] 0, Γ vr C ( tr4 τ n1 tr4 τ n1 0, 0, [τ ] n ( mn(r 1,s 0, C 4 t =0 r s 3 mn(r1,s r s 3. (9.106

43 Analyss of a DG Mtod for t Cmotaxs Modl 43 Fnal stmat s obtand by oosng t C r 4 and by sng t stmat for τ n1 (9.105: ( mn(r 1,s U r s 3 n = 1 n t τ 1 0, t = 1 mn(r1,s r s 3 r [τ1 ] 0, Γ vr mn(r1,s r s 3 mn(rv1,sv U rv sv 3 t t 4. (9.107 T stmat (9.107 provs t ndton ypotss (7.48 for 1 = n 1: n1 ( t τ mn(r 1,s 0, U =0 mn(r1,s r s 3 r s 3 mn(r1,s r s 3 mn(rv1,sv U rv sv 3 t t 4 Indton ypotss SR (7.50 an b sown sng smlar tnqs as n t proof of t ndton ypotss S (6.1. Smlarly, w prov t ndton ypotss SR for τ v and sow tat: ( mn(r 1,s V r s 3 n =0 t τ 1 v mn(r1,s r s 3 n 0, t =0 r v [τ1 v ] 0, Γ or mn(r1,s r s 3 mn(rv1,sv V rv sv 3 t t 4. (9.108 Aknowldgmnt: T rsar of Y.pstyn s basd pon work spportd by t Cntr for Nonlnar Analyss (CNA ndr t Natonal Sn Fondaton Grant # DMS Rfrns [1] J. Adlr, Cmotaxs n batra, Ann. Rv. Bom., 44 (1975, pp [] S. Agmon, Ltrs on llpt Bondary Val Problms, Van Nostrand, Prnton, NJ, [3] D.N. Arnold, An ntror pnalty fnt lmnt mtod wt dsontnos lmnts, SIAM J. Nmr. Anal., 19 (198, pp [4] I. Babška and M. Sr, T -p vrson of t fnt lmnt mtod wt qasnform mss, RAIRO Modél. Mat. Anal. Nmér., 1 (1987, pp [5] I. Babška and M. Sr, T optmal onvrgn rats of t p-vrson of t fnt lmnt mtod, SIAM J. Nmr. Anal., 4 (1987, pp [6] J.T. Bonnr, T lllar slm molds, Prnton Unvrsty Prss, Prnton, Nw Jrsy, nd d., 1967.