Journal of Computational and Applid Matmatics 235 (2011) 5422 5431 Contnts lists availabl at ScincDirct Journal of Computational and Applid Matmatics journal ompag: www.lsvir.com/locat/cam An adaptiv discontinuous finit volum mtod for lliptic problms Jiangguo Liu a,, Lin Mu b, Xiu Y c a Dpartmnt of Matmatics, Colorado Stat Univrsity, Fort Collins, CO 80523-1874, USA b Dpartmnt of Applid Scinc, Univrsity of Arkansas at Littl Rock, Littl Rock, AR 72204, USA c Dpartmnt of Matmatics, Univrsity of Arkansas at Littl Rock, Littl Rock, AR 72204, USA articl info abstract Articl istory: Rcivd 13 Dcmbr 2010 Rcivd in rvisd form 16 May 2011 MSC: 65N15 65N30 65N50 An adaptiv discontinuous finit volum mtod is dvlopd and analyzd in tis papr. W prov tat t adaptiv procdur acivs guarantd rror rduction in a msdpndnt nrgy norm and as a linar convrgnc rat. Numrical rsults ar also prsntd to illustrat t tortical analysis. 2011 lsvir B.V. All rigts rsrvd. Kywords: Adaptiv ms rfinmnts a postriori rror stimats lliptic boundary valu problms Finit volum mtods 1. Introduction Adaptiv procdurs combind wit finit lmnt mtods or finit volum mtods av bcom important tools for scintific computing and nginring applications; s [1 4] and t rfrncs trin. Ts adaptiv procdurs usually rly on a postriori typ rror stimats of rsiduals [3 6] or quantitis of intrst [7]. Convrgnc of adaptiv finit lmnt mtods for lliptic problms as bn invstigatd for continuous finit lmnts in [3] and for discontinuous finit lmnts in [2,8]. For adaptiv finit volum mtods, t rsults in [9] by Lazarov and Tomov rprsnt noticabl arly work on diffusion and convction diffusion raction quations in tr dimnsions, in wic continuous trial functions ar usd. A rcnt work on convrgnc of an adaptiv continuous finit volum mtod for lliptic problms can b found in [10]. T discontinuous finit volum mtod dvlopd in [11] for scond ordr lliptic boundary valu problms incorporats t idas of t discontinuous Galrkin finit lmnt mtods and t control or dual volums. T discontinuous finit volum mtod can b applid to lliptic intrfac problms and Darcy s flows [12]. It as bn obsrvd tat t discontinuous finit volum mtod as asir implmntation tan t traditional nod-orintd or cll-orintd (continuous) finit volum mtods and offrs local consrvation on sub-triangls [12]. As a continuation of our work on a postrior rror stimation for t discontinuous finit volum mtod, tis papr stabliss convrgnc of an adaptiv procdur for t discontinuous finit volum mtod for scond ordr lliptic problms. T rsidual typ a postriori stimator in [6] will b usd as an indicator for adaptiv ms rfinmnts. Our analysis of t adaptiv discontinuous finit volum mtod in tis papr is similar to tos for adaptiv discontinuous finit lmnt mtods in [2,8]. Corrsponding autor. -mail addrsss: liu@mat.colostat.du (J. Liu), lxmu@ualr.du (L. Mu), xxy@ualr.du (X. Y). 0377-0427/$ s front mattr 2011 lsvir B.V. All rigts rsrvd. doi:10.1016/j.cam.2011.05.051
J. Liu t al. / Journal of Computational and Applid Matmatics 235 (2011) 5422 5431 5423 Fig. 2.1. A triangular lmnt along wit its dual volums. Fig. 2.2. A triangular lmnt wit dg. For as of prsntation, w considr t following modl omognous Diriclt boundary valu problm Lu := u = f in Ω, u = 0 on Ω, (1.1) wr Ω R 2 is a boundd polygonal domain. Howvr, our adaptiv discontinuous finit volum mtod and convrgnc analysis apply to mor gnral boundary valu problms, as sown in t numrical rsults in Sction 5. W will us t standard dfinitions [13,11] for t Sobolv spacs H s (D) and tir associatd innr products (, ) s,d, norms s,d, and sminorms s,d, s 0. T spac H 0 (D) coincids wit L 2 (D), in wic cas t norm and innr product ar dnotd by D and (, ) D, rspctivly. Wn D = Ω, w drop t subscript D. T rst of tis papr is organizd as follows. In Sction 2, a discontinuous finit volum mtod is introducd. In Sction 3, ana postriori rror stimator is prsntd. Convrgnc of t adaptiv procdur is drivd in Sction 4. Numrical rsults ar prsntd in Sction 5 to illustrat t rror analysis. T papr is concludd wit som rmarks in Sction 6. 2. A discontinuous finit volum mtod Lt T b a quasi-uniform triangulation of Ω. ac triangular lmnt T T is dividd into tr sub-triangls by conncting t barycntr to t tr vrtics of t triangl, as sown in Fig. 2.1. All ts sub-triangls form a dual partition of T, wic is dnotd as T. W dfin a finit dimnsional spac of picwis linar trial functions on T as V ={v L 2 (Ω) : v T P 1 (T), T T }, and a finit dimnsional spac W ={q L 2 (Ω) : q T P 0 (T ), T T } for picwis constant tst functions on t dual partition T. Lt V() = V + H 2 (Ω) H 1 0 (Ω). Dfin a mapping γ : V() W as γv T = 1 v T ds, T T. S Fig. 2.2. Lt b an intrior dg common to lmnts T 1 and T 2 in T, and n 1 and n 2 b t unit normal vctors on xtrior to T 1 and T 2, rspctivly. For a scalar q or a vctor w, w dfin rspctivly its avrag { } on and jump [ ] across as {q} = 1 2 (q T 1 + q T2 ), [q] =q T1 n 1 + q T2 n 2, {w} = 1 2 (w T 1 + w T2 ), [w] =w T1 n 1 + w T2 n 2.
5424 J. Liu t al. / Journal of Computational and Applid Matmatics 235 (2011) 5422 5431 Not tat t jump of a vctor is a scalar, wras t jump of a scalar is a vctor. If is an dg on t boundary of Ω, w dfin {q} =q, [w] =w n. T quantitis [q] and {w} on boundary dgs ar dfind analogously. For D Ω, lt (D) b t st of dg in D and = (Ω). Dnot 0 := \ Ω, t collction of all intrior dgs of T. For convninc, w also dfin (v, w) T = v w dx, (v,w) = v w ds. T T T T picwis gradint oprator on T is dfind as ( v) T = (v T ), T T. Tsting q. (1.1) by γvfor v V givs (Lu,γv) T = (f,γv). Intgrating by parts and using t fact tat γvis a constant on ac T T, w obtain (Lu,γv) T = uγv dx = u nγv ds T T T T T T = u nγv ds + u nγv ds u nγv ds, T T T T T T T T T wr w av addd and subtractd t last trm to bring in t ffct of t primal triangulation T. Nxt w dfin a bilinar form on V() V() as a(u,v)= u nγv ds + u nγv ds. T T T T T T Utilizing t facts tat [ u] =0 and u nγv ds = T T T [γv] { u}ds + 0 {γv}[ u n]ds, (2.1) w av (Lu,γv) T = a(u,v) ({ u}, [γv]) = (f,γv). (2.2) Sinc [u] =0, w can add a pnalty trm to t abov quation and still maintain consistncy of t mtod: a(u,v) ({ u}, [γv]) + α( 1 [u], [v]) = (f,γv). Tn w dfin A (u,v)= a(u,v) ({ u}, [γv]) + α( 1 [u], [v]). (2.3) Now our discontinuous finit volum mtod can b formulatd as Sk u V suc tat A (u,v)= (f,γv) v V. (2.4) T formulation (2.4) is consistnt, i.., t tru solution u satisfis A (u,v)= (f,γv) v V. (2.5) Subtracting (2.4) from (2.5), w obtain t Galrkin ortogonality A (u u,v)= 0, v V. (2.6) For v, w V(), it as bn provd in [11] tat a(v, w) = ( v, w) + v n(γ w w)ds + ( v, w γw) T. (2.7) T T T T Furtrmor, w dfin a ms-dpndnt norm v 2 = v 2 + 1, { v} 2 + 1 [v] 2. T following a priori rror stimat as bn stablisd in [11]. T Torm 2.1. Lt u and u b rspctivly t solutions of (1.1) and (2.4). Tn u u C inf v V u v, wr C is a constant indpndnt of t ms siz.
J. Liu t al. / Journal of Computational and Applid Matmatics 235 (2011) 5422 5431 5425 3. An a postriori rror stimator First, w mak an assumption as in [8] tat f is a picwis constant, sinc t data oscillation is ssntially a igr ordr trm. Tcniqus for andling data oscillations can also b found in [2]. It is clar tat (f,v γv) T = 0. (3.1) W dfin η 2 = T T η 2 T + η 2 + η 2,1, (3.2) wr and η T = T f T, T T, η = 1/2 [ u ], η,1 = 1/2 [u ],. Lt T b an lmnt wit dg. It is wll known [14] tat tr xists a constant C suc tat for any function g H 1 (T), g 2 C( 1 T g 2 + T T g 2 T ). (3.3) W dnot T = T 1 T 2 wit T 1, T 2 in T and T 1 T 2 =. T following two torms stablisd in [6] provid an uppr bound and a lowr bound for t rror. Torm 3.1. Lt u and u b rspctivly t solutions of (1.1) and (2.4). Tn tr xists a positiv constant C suc tat (u u ) 2 Cη 2. (3.4) Torm 3.2. Tr xists a constant C > 0 suc tat and 2 T f 2 T C (u u ) 2 T, T T, (3.5) [ u ] 2 C(2 T f 2 T + (u u ) 2 T ),. (3.6) Now w cit a rsult in [8] about approximating a discontinuous picwis polynomial in V by a continuous picwis polynomial. Lmma 3.3. Lt T b a conforming triangular ms. Tn for any v V, tr xists v I V H 1 0 (Ω) satisfying (v v I ) 2 T C 1 [v] 2, (3.7) T T v v I 2 T C [v] 2, (3.8) T T wr C is indpndnt of t ms siz. Sinc [ u] =0 and [ u ] is a constant, t dfinition of A (, ) along wit (2.7) and (3.1) imply tat for any v V(), A (,v) = (, v) T + ([γv v], { }) + ({γv v}, [ ]) (f,v γv) ([γv], { }) α( 1 [u ], [v]) = (, v) T ([v], { }) α( 1 [u ], [v]). (3.9) For v V, w av (, v) T ([v], { }) α( 1 [u ], [v]) = 0. (3.10) Lmma 3.4. Lt u b t solution of (2.4). Tn w av [u ] 2 C α 2 1 T T 2 T. (3.11)
5426 J. Liu t al. / Journal of Computational and Applid Matmatics 235 (2011) 5422 5431 Proof. Substituting v by u u I in (3.10) givs α 1 [u ] 2 = (, (u u I )) T ({ }, [u u I ]). Applying Lmma 3.3 and intgration by parts, w rwrit t abov quation as α 1 [u ] 2 = (, u u I ) T + ({ }, [u u I ]) + ([ ], {u u I }) ({ }, [u u I ]) = (, u u I ) T ([ u ], {u u I }) ( ( )) 1 C η 2 T α + ( 2 η 2 α 1/2 1 [u ] ) 2. T T Combining t abov inquality wit Torm 3.2 givs (3.11). Lmma 3.5. Tr xists a constant C indpndnt of suc tat A (, ) C 2, (3.12) A (, ) C( 2 + α 1 [u ] 2 ). (3.13) Proof. It follows from (3.10) tat ([u ], { }) = (, (u u I )) T α( 1 [u ], [u ]) 1 2 2 + (α + C) 1 [u ] 2. Using t abov inquality and (3.11), w obtain A (, ) = 2 + α 1 [u ] 2 ([u ], { }) 1 2 2 C 1 1 2 2 C α 2 2 C 2. [u ] 2 Similarly, w can prov (3.13). 4. Convrgnc of t adaptiv procdur W adopt t marking stratgy in [2]: for a givn paramtr θ (0, 1), w mark substs M T tat η 2 θ T η 2, T T M T T T H η 2 θ η 2. M H T H and M H suc (4.1) (4.2) T rfinmnt stratgy in [2] tat dos not rquir t intrior nod proprty is usd to rfin M T and M. Any T M T will b rfind by biscting t longst dg, wras t two triangls saring any M will b rfind by bisction. Lt T b a rfind ms obtaind in suc a way from T H. As follows, Lmmas 4.1 and 4.2 rspctivly stat ow t rrors rlatd to t to-b-rfind lmnts and dgs can b controlld. Lmma 4.1. T following olds T M T η 2 T C (u u H ) 2 + Cα 1 [u ] 2. (4.3)
J. Liu t al. / Journal of Computational and Applid Matmatics 235 (2011) 5422 5431 5427 Proof. Lt T T H b a triangl rfind as T = T 1 T 2, T 1, T 2 T.Asin[2], lt φ V b a Crouzix Raviart typ linar sap function suc tat φ T = 0 for T T \{T} and 2 2 T f 2 T i = 2 (f,φ ) Ti, φ 2 T i C 4 T f 2 T i, i = 1, 2, {φ }ds = 0, T. Applying t fact φ V and (4.4) (4.6), w obtain A (u,φ ) = = 2 ( u, φ ) Ti + α 2 (f,φ ) Ti. (T) 1 ([u ], [φ ]) Using (4.4), (4.6), (4.7), and intgration by parts, w av 2 2 2 2 f 2 T i = (f,φ ) Ti + ( u H,φ ) Ti = = C 2 (f,φ ) Ti 2 ( u H, φ ) Ti 2 ( (u u H ), φ ) Ti + α ( 2 (u u H ) 2 T i ) 1 2 + (T) ( α 1 ([u ], [φ ]) (T) Summing t abov inquality ovr all T M T givs (4.3). ) 1/2 ( ) 1/2 2 1 [u ] 2 2 T f 2 T i. (4.4) (4.5) (4.6) (4.7) Lmma 4.2. T following olds wit C bing a constant indpndnt of t mss η 2 C (u u H ) 2 + Cα 1 [u H] 2 + Cα 1 [u ] 2. (4.8) M H Proof. Lt M b an dg wit = T 1 T 2, T 1, T 2 T H. Lt φ H V H b a Crouzix Raviart typ sap function suc tat φ H T = 0 for any T T \{T } and [ u H ] 2 = ([ u H], {φ H }), (4.9) φ H 2 3 2 T i C [ u H], i = 1, 2, (4.10) {φ H }ds = 0. (4.11) Intgration by parts yilds 0 = ( u H,φ H ) T = ( u H, φ H ) T ([ u H ], {φ H }). (4.12) Sinc φ H V H, w av ( u H, φ H ) T + α H (T ) It follows from (4.9), (4.12) and (4.13) tat 1 ([u H], [φ H ]) = (f,φ H ) T. (4.13) [ u H ] 2 = ([ u H], {φ H }) = ( u H, φ H ) T = (f,φ H ) T α 1 ([u H], [φ H ]). H (T )
5428 J. Liu t al. / Journal of Computational and Applid Matmatics 235 (2011) 5422 5431 Applying (4.3), (4.10) and (4.11), w obtain [ u H ] 2 C ( T T 2 T f 2 T + H (T ) ) 1 [u H] 2. Summing t abov inquality ovr all M and applying Lmma 4.1 giv (4.8) as dsird. Sinc T is a rfinmnt obtaind from T H, it is asy to s [8] tat A ( H, H ) A H ( H, H ) + α H 1 [u H] 2. (4.14) It is also clar from t fact V H V tat A (, ) = A ( H, H ) + A (u u H, u u H ). (4.15) A combination of (4.14) and (4.15) lads to t main tortical rsult on rror rduction statd in t torm blow. Torm 4.3. Tr xists ρ (0, 1) suc tat A (, ) ρa H ( H, H ). Proof. Applying (3.4) and (3.13), w obtain A H ( H, H ) Cη 2 H. (4.16) Combining t corcivity of A (, ), Lmmas 4.1 and 4.2 givs A (u u H, u u H ) C (u u H ) 2 Cθη 2 Cα H 1 [u H] 2 Cα 1 [u ] 2. (4.17) H Using (3.4), (3.11), (3.12) and (4.14) (4.17), w av A (, ) = A ( H, H ) A (u u H, u u H ) A H ( H, H ) + α 1 [u H] 2 C (u u H ) 2 H ( A H ( H, H ) Cθ C ) η 2 H α + Cα 1 [u ] 2 Trfor, (1 Cθ)A H ( H, H ) + C α A (, ). A (, ) ρa H ( H, H ). for som constant ρ (0, 1). Rmark. As on can cck, t abov proof rquirs t pnalty factor α to b larg noug. But t torm implis guarantd rror rduction and linar convrgnc of t adaptiv procdur. 5. Numrical rsults In tis sction, w validat t adaptiv discontinuous finit volum mtod troug numrical rsults. T algoritms and Matlab implmntation in [1] av bn adoptd for our numrical xprimnts. xampl 1. W considr an lliptic boundary valu problm on an L-sapd domain Ω = ( 1, 1) 2 \([0, 1] [ 1, 0]) wit a known xact solution u(r,θ)= r 2/3 sin(2θ/3), wr (r,θ)ar t polar coordinats. Nonomognous Diriclt boundary conditions ar spcifid using t valus of t xact solution.
J. Liu t al. / Journal of Computational and Applid Matmatics 235 (2011) 5422 5431 5429 Tabl 5.1 Numrical rsults for xampl 1. Lvl #lts #Nwlts rror η ffindx Rlrr η r 1 12 0.5714 0.2255 0.3946 0.4217 0.1664 2 16 3 0.4975 0.2172 0.4366 0.3671 0.1603 3 20 2 0.4337 0.1975 0.4554 0.3201 0.1457 4 23 3 0.3910 0.1848 0.4726 0.2885 0.1364 5 32 6 0.2982 0.1720 0.5768 0.2201 0.1269 6 43 9 0.2691 0.1580 0.5871 0.1986 0.1166 7 70 13 0.2005 0.1273 0.6349 0.1480 0.0939 8 95 14 0.1646 0.1117 0.6786 0.1215 0.0824 9 138 26 0.1396 0.0983 0.7042 0.1030 0.0725 10 188 34 0.1145 0.0864 0.7546 0.0845 0.0638 11 251 48 0.0973 0.0751 0.7718 0.0718 0.0554 12 351 67 0.0787 0.0625 0.7942 0.0581 0.0461 13 478 93 0.0670 0.0543 0.8104 0.0494 0.0401 14 653 134 0.0554 0.0464 0.8375 0.0409 0.0342 15 888 187 0.0460 0.0398 0.8652 0.0339 0.0294 16 1 211 254 0.0393 0.0344 0.8753 0.0290 0.0254 17 1 660 347 0.0327 0.0293 0.8960 0.0241 0.0216 18 2 265 488 0.0277 0.0250 0.9025 0.0204 0.0184 19 3 068 674 0.0234 0.0215 0.9188 0.0173 0.0159 20 4 178 927 0.0199 0.0184 0.9246 0.0147 0.0136 21 5 667 1275 0.0168 0.0158 0.9405 0.0124 0.0117 22 7 617 1708 0.0143 0.0136 0.9510 0.0106 0.0100 23 10,258 2351 0.0121 0.0117 0.9669 0.0089 0.0086 Tis is a widly usd tst problm on a nonconvx domain for wic t xact solution dos not av full lliptic rgularity and nc adaptiv ms rfinmnts ar ndd to rsolv t cornr singularity. To calibrat t adaptiv discontinuous finit volum mtod dvlopd in tis papr, w comput rrors and rlativ rrors as follows rror = u u, Rlrr = u u u, (5.1) wr u, u ar rspctivly t xact and numrical solutions. T rror indicator η = η dfind in (3.2) will b computd for all mss. For convninc, w also comput a rlativ rror indicator η r = η u. Mor importantly t ffctivnss indx is calculatd as η ffindx: = u u. (5.2) A stopping critrion Rlrr tol basd on t rlativ rror is adoptd. Tabulatd in Tabl 5.1 ar our numrical rsults. W coos tol = 0.01, tat is, 1% as a trsold for t rlativ rror. W start from a rgular triangular ms tat as only 12 lmnts. Sown in Fig. 5.1 is t adaptivly rfind ms at lvl 19. Wit about 3000 triangular lmnts, t rlativ rror is smallr tan 2%. Aftr 22 adaptiv ms rfinmnts, w nd up wit 10,258 triangular lmnts and a rlativ rror 0.89%. rror rductions in Columns 4 & 5 can b clarly obsrvd. As mss ar rfind, t ffctivnss indx (Column 6) clarly approacs 1. Our adaptiv discontinuous finit volum mtod is wll validatd. Sown in Fig. 5.1 is t adaptivly rfind ms at Lvl 19. Prsntd in Fig. 5.2 is a log log plot of t rror of t adaptiv numrical solution vrsus t numbr of nods. T slop 1 indicats tat t adaptiv discontinuous finit 2 volum mtod xibits asymptotical optimality in nonlinar approximation [15,16,8,3], tat is, t rror is proportional to N 1/d dof, d = 2, wr N dof is t dgr of frdom. 6. Concluding rmarks In tis papr, w av dvlopd and analyzd an adaptiv discontinuous finit volum mtod for solving scond ordr lliptic boundary valu problms. A prviously stablisd a postriori rror stimator [6] as bn usd for adaptiv ms rfinmnts. T fficincy of t rror indicator as bn vrifid by numrical rsults on a widly tstd problm. rror rduction as bn clarly dmonstratd by numrical rsults. T adaptiv discontinuous finit volum mtod is asymptotically optimal. A rsidual typ a postriori rror stimator as bn usd to stablis an adaptiv procdur for t discontinuous finit volum mtod. It sould b intrsting to xplor combination of t adjoint-basd a postriori rror stimation in [7] and t discontinuous finit volum mtod in tis papr.
5430 J. Liu t al. / Journal of Computational and Applid Matmatics 235 (2011) 5422 5431 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Fig. 5.1. T adaptivly rfind ms at Lvl 19. 10 0 10 1 10 2 η/ u u u / u slop= 1/2 10 3 10 1 10 2 10 3 10 4 Numbr of nods Fig. 5.2. Convrgnc of t rror of t adaptiv solution. It sould also b noticd tat t discontinuous finit volum mtod w av analyzd in tis papr is nonsymmtric, i.., nonsymmtric intrior pnalty Galrkin (NIPG), and nc stabl for any pnalty factor α>0. It is wll known tat tr ar otr formulations suc as symmtric intrior pnalty Galrkin (SIPG) and incomplt intrior pnalty Galrkin (IIPG). Along tis lin, anotr intrsting formulation is to drop bot trms for t avrags/jumps of trial and tst functions, but to apply wak pnalization in t pnalty trm. Tis approac as bn invstigatd for finit lmnts [17,18]. A wakly ovr-pnalizd discontinuous finit volum mtod for lliptic problms as bn dvlopd in [19]. It is sown tat t wakly ovr-pnalizd discontinuous finit volum mtod offrs vn asir implmntations, spcially in t construction of prconditionrs. stablising a postrior rror stimators and adaptiv procdurs for t wakly ovrpnalizd finit volum mtod is currntly undr our invstigation and will b rportd in our futur work. Acknowldgmnts J. Liu was supportd partially by US National Scinc Foundation undr Grant No. DMS-0915253. X. Y was supportd in part by US National Scinc Foundation undr Grant No. DMS-0813571. Rfrncs [1] L. Cn, C.-S. Zang, AFM@Matlab: A MATLAB packag of adaptiv finit lmnt mtods, Tcnical Rport, Univrsity of Maryland, 2006. [2] R. Hopp, G. Kanscat, T. Warburton, Convrgnc analysis of an adaptiv intrior pnalty discontinuous Galrkin mtod, SIAM J. Numr. Anal. 47 (2008) 534 550. [3] P. Morin, R. Noctto, K. Sibrt, Convrgnc of adaptiv finit lmnt mtods, SIAM Rv. 44 (2002) 631 658. [4] R. Vrfürt, A postriori rror stimation and adaptiv ms rfinmnt tcniqus, J. Comput. Appl. Mat 50 (1994) 67 83. [5] R. Vrfürt, A not on constant-fr a postriori rror stimats, SIAM J. Numr. Anal. 47 (2009) 3180 3194. [6] X. Y, A postrior rror stimat for finit volum mtods of t scond ordr lliptic problm, Numr. Mt. PDs 27 (2011) (in prss). [7] D. stp, M. Prnic, D. Pam, S. Tavnr, H. Wang, A postriori rror analysis of a cll-cntrd finit volum mtod for smilinar lliptic problms, J. Comput. Appl. Mat. 233 (2009) 459 472.
J. Liu t al. / Journal of Computational and Applid Matmatics 235 (2011) 5422 5431 5431 [8] O. Karakasian, F. Pascal, Convrgnc of adaptiv discontinuous Galrkin approximations of scond ordr lliptic problms, SIAM J. Numr. Anal. 45 (2007) 641 665. [9] R.D. Lazarov, S.Z. Tomov, Adaptiv finit volum lmnt mtod for convction diffusion raction problms in 3-D, in: P. Minv, Y. Lin (ds.), Scintific Computing and Application, in: Advancs in Computation: Tory and Practic, vol. 7, Nova Scinc Publising Hous, 2001, pp. 91 106. [10] J. Xu, Y. Zu, Q. Zou, Nw adaptiv finit volum mtods and convrgnc analysis, Prprint, 2010. [11] X. Y, A nw discontinuous finit volum mtod for lliptic problms, SIAM J. Numr. Anal. 42 (2004) 1062 1072. [12] J. Liu, L. Mu, X. Y, A comparativ study of locally consrvativ numrical mtods for Darcy s flows, Procdia Comput. Sci. 4 (2011) 974 983. [13] R.A. Adams, J.J.F. Fournir, Sobolv Spacs, 2nd d., Acadmic Prss, 2003. [14] S.C. Brnnr, L.R. Scott, T Matmatical Tory of Finit lmnt Mtods, 3rd d., Springr, 2008. [15] P. Binv, W. Damn, R.A. DVor, Adaptiv finit lmnt mtods wit convrgnc rats, Numr. Mat. 97 (2004) 219 268. [16] R.A. DVor, Nonlinar approximation, Acta Numr. 7 (1998) 51 150. [17] S.C. Brnnr, T. Gudi, L.-Y. Sung, A postriori rror control for wakly ovr-pnalizd symmtric intrior pnalty mtod, J. Sci. Comput. 40 (2009) 37 50. [18] S.C. Brnnr, L. Owns, L.-Y. Sung, A wakly ovr-pnalizd symmtric intrior pnalty mtod, l. Trans. Numr. Anal. 30 (2008) 107 127. [19] J. Liu, M. Yang, A wakly ovr-pnalizd finit volum lmnt mtod for lliptic problms, Prprint, Colorado Stat Univrsity, 2010.