A linear domain decomposition method for partially saturated flow in porous media

Transcription

1 UHasset Computatona Mathematcs Preprnt Seres A near doman decomposton method for partay saturated fow n porous meda Davd Seus, Koondanbha Mtra, Iuu Sorn Pop, Forn Adran Radu and Chrstan Rohde UHasset Computatona Mathematcs Preprnt Nr. UP August 10th, 2017

2 A near doman decomposton method for partay saturated fow n porous meda Davd Seus a,, Koondanbha Mtra b,c, Iuu Sorn Pop c,d, Forn Adran Radu d, Chrstan Rohde a a Insttute of Apped Anayss and Numerca Smuaton, Char of Apped Mathematcs, Pfaffenwadrng 57, Stuttgart, Germany b Department of Mathematcs and Computer Scence, Technsche Unverstet Endhoven, PO Box 513, 5600 MB Endhoven, The Netherands c Facuty of Scences, Hasset Unversty, Campus Depenbeek, Agoraaan Budng D, BE3590 Depenbeek, Begum d Department of Mathematcs, Unversty of Bergen, P. O. Box 7800, N-5020 Bergen, Norway Abstract The Rchards equaton s a nonnear paraboc equaton that s commony used for modeng saturated/unsaturated fow n porous meda. We assume that the medum occupes a bounded Lpschtz doman parttoned nto two dsjont subdomans separated by a fxed nterface Γ. Ths eads to two probems defned on the subdomans whch are couped through condtons expressng fux and pressure contnuty at Γ. After an Euer mpct dscretsaton of the resutng nonnear subprobems a near teratve L-type) doman decomposton scheme s proposed. The convergence of the scheme s proved rgorousy. In the ast part we present numerca resuts that are n ne wth the theoretca fndng, n partcuar the uncondtona convergence of the scheme. We further compare the scheme to other approaches not makng use of a doman decomposton. Namey, we compare to a Newton and a Pcard scheme. We show that the proposed scheme s more stabe than the Newton scheme whe remanng comparabe n computatona tme, even f no paraesaton s beng adopted. Fnay we present a parametrc study that can be used to optmze the proposed scheme. Keywords: Doman decomposton, L-scheme Lnearsaton, Rchards Equaton 1. Introducton Unsaturated fow processes through porous meda appear n a varety of physca stuatons and appcatons. Notabe exampes are so remedaton, enhanced o recovery, CO 2 storage, harvestng of geotherma energy, or the desgn of fters and fue ces. Mathematca modeng and numerca smuaton are essenta for understandng such processes, snce measurements and experments are very dffcut Correspondng author: Ema address: davd.seus@ans.un-stuttgart.de Davd Seus) August 10, 2017

3 f not mpossbe, and hence ony mtedy avaabe. The assocated mathematca and computatona chaenges are manfod. The mathematca modes are usuay couped systems of nonnear parta dfferenta equatons and ordnary ones, nvovng argey varyng physca propertes and parameters, ke porosty, permeabty or so composton. Together wth the arge scae and possbe compexty of the doman, ths poses sgnfcant computatona chaenges, makng the desgn and anayss of robust dscretsaton methods a non-trva task. In ths work we focus on saturated/unsaturated fow of one fud water) n a porous medum e.g. the subsurface) occupyng the doman Ω R d d {1,2,3}). Besdes water, a second phase ar) s present, whch s assumed to be at a constant atmospherc) pressure. Ths stuaton s descrbed by the Rchards equaton, here n pressure formuaton Φ t Sp) [ K µ k ) r Sp) p + z )] = 0, 1) see e.g. [1], orgnay [2, 3]. In the above Φ denotes the porosty, S s the water saturaton, p s the water pressure, k r s the reatve permeabty, K the ntrnsc permeabty and z = ρ w gx 3 s the gravtatona term n drecton of the x 3 -axs. Fnay, g s the gravtatona acceeraton, ρ w the water densty and µ ts vscosty. Wth T > 0 beng a maxma tme, the equaton s defned for the tme t 0,T ) on the bounded Lpschtz doman Ω. Beow we propose a doman decomposton DD) scheme for the numerca souton of 1). To ths am we assume that Ω s parttoned nto two subdomans Ω {1,2}) separated by a Lpschtz-contnuous nterface Γ, see Fg. 1. In other words one has Ω = Ω 1 Ω 2 Γ. The restrcton to two subdomans s made for the ease of presentaton, but the scheme can be extended straghtforwardy to more subdomans. In each Ω {1,2}) we use the physca pressure p as prmary varabe. Furthermore, the permeabty and porosty n each of the subdomans may be dfferent and even dscontnuous, whch s the case of a heterogeneous medum consstng of bock-type heterogenety ke a fractured medum). In vew of ts reevance for manfod appcatons n rea fe, Rchards equaton has been studed extensvey, both anaytcay and numercay, and the dedcated terature s extremey rch. We restrct ourseves here by mentonng [4, 5] for the exstence of weak soutons and [6] for the unqueness. Numerca schemes for the Rchards equaton, or n genera for degenerate paraboc equatons, are anaysed n [7, 8, 9, 10, 11, 12, 13, 14, 15]. Most of the papers are consderng the backward Euer method for the tme dscretsaton n vew of the ow reguarty of the souton, see [4], and to avod restrctons on the tme step sze. Dfferent approaches wth regard to spata dscretsaton have been consdered. Gaerkn fnte eements were used n [8, 16, 17]. Dscontnuous Gaerkn fnte eement schemes for fows through 2

4 heterogeneous) porous meda have been studed n [18, 19]. Fnte voume schemes ncudng mutpont fux approxmaton ones for the Rchards equaton are anaysed n [20, 21, 13], and mxed fnte eements n [7, 22, 10, 11, 12, 15, 14]. Such schemes are ocay mass conservatve. Appyng the Krchhoff transformaton [4] brngs the mathematca mode to a form that smpfes mathematca and numerca anayss, see e.g. [8, 7, 10, 11]. However, the transformed unknown s not drecty reated to a physca quantty ke the pressure, and therefore a postprocessng step s requred after a numerca approxmaton of the souton has been obtaned. Aternatvey, one may deveop numerca schemes for the orgna formuaton and n terms of the physca quanttes. Nevertheess, when provng the convergence rgorousy, one often resorts to a Krchhoff transformed formuaton as ntermedate step. Aternatvey, suffcent reguarty of the souton, e.g. by avodng cases where the medum s competey saturated, or competey dry, has to be assumed. We pont out that n ths work we w not make use of the Krchhoff transformaton, keepng the equaton n a more reevant form for appcatons. If mpct methods are adopted for the tme dscretsaton, the eptc or fuy dscrete) probems obtaned at each tme step are nonnear. For sovng these, dfferent approaches have been proposed. Exampes are the Newton method [23, 24, 25], the Pcard/modfed Pcard method [26, 27], or the Jäger-Kacur method [28, 29]. We refer to [30] for the convergence anayss of such nonnear schemes. Assumng that the nta guess s the souton from the prevous tme step, the convergence of such schemes can ony be guaranteed under severe restrcton for the tme step n terms of the mesh sze. Addtonay, reguarzng the probem s requred, whch prevents the Jacoban from becomng snguar. Such dffcutes do not appear when the L-scheme s beng used, whch s a fxed pont scheme transformng the teraton nto a contracton, [31, 32, 16]. The convergence s merey near but n a better norm H 1 ) and requres no reguarzaton or severe constrant on the tme step. We aso refer to [33] for a combnaton of the Newton method and the L-scheme. Moreover, we menton [12] for the appcaton of the L-scheme to Höder nstead of Lpschtz contnuous nonneartes. Independent of the chosen dscretsaton method and of the nearsaton scheme, doman decomposton DD) methods offer an effcent way to reduce the computatona compexty of the probem, and to perform cacuatons n parae. Ths s n partcuar nterestng whenever domans wth bock type heterogenetes are consdered, as DD schemes aow decoupng the modes defned n dfferent homogeneous subdomans and sovng these numercay n parae. We refer to [34] for a detaed dscusson of near DD methods and to [35] for a genera ntroducton nto the subject. Comprehensve studes of nonnear DD schemes n the fed of fud dynamcs can be found n [36, 37, 38]. For artces strcty reated to porous meda fow modes, we refer to [39, 40] for an overvew of dfferent overappng doman decomposton strateges. Lnear and nonnear addtve Schwartz methods are compared, and the use 3

5 of such methods as near and nonnear precondtoners s dscussed. Regardess of the type of the DD scheme, choosng the optma parameters s a key ssue. Such aspects are anaysed e.g. n [41, 42]. We aso refer to [43] for a DD agorthm for porous meda fow modes, where a-posteror estmates are used to optmze the parameters and the number of teratons. Reca that the Rchards equaton s a nonnear evouton equaton. For sovng ths type of equaton, methods ke pararea [44] and wave-form reaxaton [45, 46] have been proposed. The man deas there are to decompose the probem nto separate probems defned n tme/space-tme domans. DD methods for the Rchards equaton are dscussed n [47, 48]. In these papers the doman s decomposed nto mutpe ayers and the Rchards modes restrcted to adjacent ayers are couped by Robn type boundary condtons. The approach uses nonoverappng doman-decomposton and generases the deas of the method ntroduced n [49] for near eptc probems see aso [50, 51]), eadng to decouped, nonnear probems n the subdomans. Here we consder a near DD scheme for the numerca approxmaton of the tme dscrete probems obtaned after substructurng nto subprobems and performng an Euer mpct tme steppng. A nonoverappng DD scheme referred to henceforth as LDD scheme) nspred by the DD method ntroduced n [49] s defned. The LDD teratons are near, based on an L-type scheme. Ths approach dffers from the one commony used when deang wth nonnear eptc probems n the context of DD. In most cases, the DD teratons ead to nonnear subprobems. For sovng these, teratve methods n each subdoman are apped. In our approach, the nearsaton step s part of the DD teratons, whch reduces the computatona tme. More precsey, the L-scheme dea s combned wth the nonoverappng DD scheme such that the equatons defned n each subdoman aong wth the Robn type coupng condtons on the nterface become near. For the resutng scheme we prove rgorousy the uncondtona convergence, and provde numerca exampes supportng the theoretca fndngs and demonstratng ts effectveness. The paper s structured as foows. In Sec. 2 we present the mathematca mode and ntroduce the DD scheme. Secton 3 contans the anayss of the scheme. Fnay, Sec. 4 provdes numerca experments n two spata dmensons, together wth an anayss of the practca performance of the scheme. Ths ncudes a comprehensve comparson ncudng robustness and effcency) between the proposed DD scheme and standard monothc schemes based on Newton, modfed Pcard as we as the L-scheme. 2. Probem formuaton and teratve scheme 2.1. Probem formuaton Reca that T > 0 and Ω R d s a bounded Lpschtz doman parttoned n two subdomans Ω 1,2, separated by the Lpschtz-contnuous nterface Γ. The boundary of Ω s denoted by Ω and the portons 4

6 Fgure 1: Iustraton of the doman Ω = Ω 1 Ω 2 R d wth fxed nterface Γ. Aso shown are the norma vectors aong the nterface. of Ω that are aso boundares of Ω are denoted by Ω see aso Fg. 1). To ease the presentaton, the two subdomans are assumed to be homogeneous and sotropc,.e. we can have two dfferent reatve permeabtes k r = k r, on each Ω, the ntrnsc permeabtes K = K are scaar and the two porostes Φ = 1,2) are constant. The product K k r, Φ µ n 1) s abbrevated by k henceforth. We sove equaton 1) n Ω together wth nta and homogeneous Drchet boundary condtons. We refer to [47, 52] for more genera condtons, ncudng outfow-type ones. On the two subdomans, the probem transforms nto two subprobems, couped through two condtons at the nterface Γ: the contnuty of the norma fuxes and the contnuty of the pressures. Wth the fuxes F := k S p ) ) p + z ), 1) becomes t S p ) + F = 0 n Ω 0,T ], 2) F 1 n 1 = F 2 n 2 on Γ [0,T ], 3) p 1 = p 2 on Γ 0,T ], 4) p = 0 on Ω 0,T ]. 5) Ths s cosed by the nta condtons p,0) := p,0 n Ω, where p s the water pressure on Ω, = 1,2, and k are gven) scaed reatve permeabty functons, that are assumed to be smooth enough. In the above, n stands for the outer unt norma vector at Ω. 5

7 Sem-dscrete formuaton dscretsaton n tme) For the tme dscretsaton we et N N be a gven and τ := N T be the correspondng tme step. Then p n s the approxmaton of the pressure p at tme t n = nτ. The Euer mpct dscretsaton of 2) 5) reads ) ) S p n S p n 1 + τ F n = 0 n Ω, 6) F n 1 n 1 = F n 2 n 2 on Γ, 7) p n 1 = pn 2 on Γ, 8) p n = 0 on Ω, 9) where F n := k S p n )) p n + z) s the fux at tme step t n. Observe that 7) and 8) are the coupng condtons at the nterface Γ The LDD teratve scheme If p n 1 1, p n 1 ) 2 s known, p n 1, p n ) 2 can be obtaned by sovng the nonnear system 6) 9). To ths end, we defne an teratve scheme that uses Robn type condtons at Γ to decoupe the subprobems n Ω, and nearses the terms due to the saturaton-pressure dependency by addng stabsaton terms that cance each other n the mt see e.g. [33, 31]). Specfcay, assumng that for some N the approxmatons p n, 1 and g 1 are known, we seek p n, 1, ) pn, 2 sovng the probems L p n, L p n, 1 + τ F n, = S p n, 1 Foowng the prevousy ntroduced notaton, F n, ) ) + S p n 1 n Ω, 10) F n, n = g + λ pn, on Γ [0,T ], 11) g := 2λ pn, 1 3 g ) := k S p n, 1 ) ) p n, + z ) denotes the nearsed fux at teraton. By λ 0, ), we denote a free to be chosen) parameter used to weght the nfuence of the pressure on the nterface condtons at Γ. The parameters L > 0 must adhere to some md constrants n order for the scheme to converge, whch w be dscussed ater, but other than that, are arbtrary. The teraton starts wth p n,0 := p n 1, and g 0 := F n 1 n λ p n 1, and ceary, the dfference L p n, L p n, 1 s vanshng n case of convergence. Remark 1. The usage of the terms g and of the parameter λ s motvated by the foowng. Wth the notaton f n := F n n, the transmsson condtons 7)-8) become f1 n = f 2 n and pn 1 = pn 2. For any λ 0, 6

8 these are equvaent to f1 n = f 2 n λ pn 2 ) + λ pn 1, 13) f2 n = f 1 n λ pn 1 ) + λ pn 2. Denotng the terms between brackets by g, one obtans f n 1 = g 1 + λ p n 1, f n 2 = g 2 + λ p n 2, and g 1 = 2λ p n 2 g 2, g 2 = 2λ p n 1 g 1. 14) The condtons n 11)-12) are the nearsed counterparts of 14). Remark 2 dfferent decoupng formuatons). as convex combnatons of the terms g and p, namey The decouped condtons n 7)-8) can be formuated F n, n = 1 λ)g + λ pn, 11 ) 1 λ)g := 2λ pn, λ)g ) The convergence anayss beow can be carred out for ths formuaton wthout any dffcuty. However, the DD scheme usng ths convex formuaton showed a sower convergence n the numerca experments than when 11)-12) was used. Moreover, t s easer to fnd cose to optma parameters for the atter. Such aspects are dscussed n Secton 4. In vew of ths, n what foows we restrct the anayss to the nta formuaton. Before formuatng the man resut we specfy the notaton that w be used beow. Notaton 1. L 2 Ω) s the space of Lebesgue measurabe, square ntegrabe functons over Ω. H 1 Ω) contans functons n L 2 Ω) havng aso weak dervatves n L 2 Ω). H 1 0 Ω) = C 0 Ω)H1, where the competon s wth respect to the standard H 1 norm and C0 Ω) s the space of smooth functons wth compact support n Ω. The defnton for H 1 Ω ) = 1,2) s smar. Wth Γ beng a d 1) dmensona manfod n Ω, H 1 2 Γ) contans the traces of H 1 functons on Γ see e.g. [53, 54, 34]. Gven u H 1 Ω), by ts trace on Γ s denoted by u Γ. Furthermore, the foowng spaces w be used V := { u H 1 Ω ) u Ω 0 }, 15) V := { u 1,u 2 ) V 1 V 2 u 1 Γ u 2 Γ }, 16) H 1/2 00 Γ) = { ν H 1/2 Γ) ν = w Γ for a w H 1 0 Ω) }. 17) Note, that V = H0 1 Ω). H1/2 00 Γ) denotes the dua space of H 1/2 00 Γ)., X w denote the L 2 X) scaar product, wth X beng one of the sets Ω, Ω = 1,2) or Γ. Whenever sef understood, the notaton of the 7

9 doman of ntegraton X w be dropped. Furthermore, stands aso for the duaty parng between, Γ H 1/2 00 Γ) and H 1/2 00 Γ). In what foows we make the foowng Assumptons 1. Wth = 1,2, we assume that a) k : R [0,1] are strcty monotoncay ncreasng and Lpschtz contnuous functons wth Lpschtz constants L k > 0, b) there exsts m R such that 0 < m k 1 S), k 2 S) for a S R, c) S : R R are monotoncay ncreasng and Lpschtz contnuous functons wth Lpschtz constants L S > 0. For ater use we defne L k := max{ L k1,l k2 } and L S := max{l S1,L S2 }. In a smpfed formuaton, the man resut n ths paper s Theorem 1. Assume there exsts a souton par p n 1, pn 2 ) to 6) 9) that addtonay fufs sup p n + z ) L M <. Let L obey L S < 2L for = 1,2 and assume that the tme step τ > 0 s chosen sma enough, so that for both one has τ < 2m 1 Lk 2 M 2 1 ). 18) L S 2L Then the sequence of souton pars { p n, 1, pn, 2 )} 1 of 10) 11) converges to pn 1, pn 2 ). Remark 3. The precse form of Theorem 1 w be formuated n Secton 3, after havng defned a weak souton. The theorem above s gven for the ease of presentaton. 3. Anayss of the scheme. Ths secton gves the convergence proof for the proposed scheme. The startng pont s the Euer mpct dscretsaton n Secton 2. Assumng p1 n 1, p n 1 ) 2 V to be known, a weak formuaton of 6) 9) s gven by Probem 1 Sem-dscrete weak formuaton). Fnd p n 1, pn 2 ) V such that Fn n H 1/2 00 Γ) for = 1,2 and for a ϕ 1,ϕ 2 ) V. S p n ),ϕ τ F n, ϕ + τ F n 3 n,ϕ Γ Γ = S 1 p n 1 1 ),ϕ 1, 19) 8

10 Remark 4. If p n 1, pn 2 ) V s a souton of Probem 1, we have pn 1 Γ = p n 2 Γ by defnton of V. Testng n 19) by an arbtrary ϕ C 0 Ω ) shows that the dstrbuton F n F n Hdv,Ω ) and s reguar and n L 2, yedng S p n ) S p n 1 ) = τ F n a. e. n Ω 20) by the varatona emma. By Lemma III. 1.1 n [53], F n n H 1/2 Ω ) and ntegratng by parts n 19) yeds for a ϕ 1,ϕ 2 ) V. Therefore 0 = F n n,ϕ Γ n H 1/2 00 Γ) snce the trace s a surjectve operator. Γ + F n 3 n,ϕ Γ Γ 21) F n n = F n 3 n 22) Note addtonay that Probem 1 s equvaent to the sem-dscrete Rchards equaton on the whoe doman, namey to fnd p n 1, pn 2 ) V such that S1 p n 1 ),ϕ 1 τ F n 1, ϕ 1 + S2 p n 2 ),ϕ 2 τ F n 2, ϕ 2 for a ϕ 1,ϕ 2 ) V. = S 1 p n 1 1 ),ϕ 1 + S2 p n 1 2 ),ϕ 2, 23) Remark 5. By appyng a Krchhoff transform n each subdoman Ω, Probem 1 can be reformuated as a nonnear transmsson probem. The exstence and unqueness of a souton for such probems has been studed n [55, 56] for the case when Ω 1 s surrounded by Ω 2, and the common boundary s smooth, however. p n 1 2 Now we can gve the weak form of the teratve scheme. Let n N and assume that the par p n 1 ) V s gven. Furthermore, et λ > 0 and L > 0 = 1,2) be fxed parameters and p n,0 := p n 1, as we as g 0 := F n 1 n λ p n 1 Γ. The teratve scheme s defned through Probem 2 L-scheme, weak form). Let N and assume that the approxmatons { p n,k } 1 k=0 and { g k } 1 k=0 are known for = 1,2. Fnd p n, 1, ) pn, 2 V such that L p n, + g,ϕ Γ,ϕ τ F n,, ϕ + τ λ p n, = L p n, 1,ϕ S p n, 1 ) ) S p n 1,ϕ g,ϕ Γ := 2λ p n, 1 3 g 1 3,ϕ 1, 24) Γ 25) 9

11 hods for a ϕ 1,ϕ 2 ) V Intutve justfcaton of the L-scheme We start the anayss by takng a coser ook at the forma mt of the L-scheme teratons n weak form and show that ths s actuay a reformuaton of Probem 1. Lemma 2 Lmt of the L-scheme). Let n N be fxed and assume that the functons p n V and g H 1/2 00 Γ) = 1,2) exst such that S p n ),ϕ ) S p n 1,ϕ τ F n, ϕ + τ λ p n,+g,ϕ g,ϕ Γ = 2λ p n 3 g 3,ϕ hod for a ϕ 1,ϕ 2 ) V. Then the nterface condtons Γ = 0, 26) Γ, 27) p n 1 Γ = p n 2 Γ n H 1/2 00 Γ), 28) F n 1 n 1 = F n 2 n 1 n H 1/2 00 Γ) 29) are satsfed and p n 1, pn 2 ) soves Probem 1. Moreover, g = λ p n Γ + F n n 30) n H 1/2 00 Γ). Conversey, f p n 1, pn 2 ) V s a souton of Probem 1 and g := λ p n Γ + F n n, then p n and g sove the system 26), 27). Remark 6. Lemma 2 states that sovng Probem 1 s equvaent to fndng a souton to 26), 27). Ths reformuaton w be used to show, that the L-scheme converges to a souton of Probem 1 Proof. Wrtng out 27) for = 1,2 and subtractng the resutng equatons yeds p n 1 Γ = p n 2 Γ n the sense of traces. On the other hand, addng up these equatons eads to g 1 + g 2 ) = λp n 1 Γ + p n 2 Γ). Insertng ths nto the sum of the equatons 26) eads to 23), and by equvaence to the sem-dscrete formuaton 19). Moreover, by 20) one has S p n ) S p n 1 ) = τ F n a.e. and therefore ntegratng by parts n 26) gves g = λ p n Γ + F n n n H 1/2 00 Γ). Conversey, f p n 1, pn 2 ) soves Probem 1, then pn 1 Γ = p n 2 Γ and g = λ p n Γ + F n n = λ p n 3 Γ + F n 3 n 3 = 2λ p n 3 Γ g 3 31) s deduced by the fux contnuty 22). Fnay, 26) now foows by ntegratng 20) by parts and usng the defnton of g. 10

12 3.2. Convergence of the scheme The convergence of the L-scheme nvoves two steps: frst, we prove the exstence and unqueness of a souton to Probem 2 defnng the near teratons, and then we prove the convergence of the sequence of such soutons to the expected mt. Lemma 3. Probem 2 has a unque souton. Proof. Ths s a drect consequence of the Lax-Mgram emma. We now prove the convergence resut, whch was announced n Theorem 1. We assume that the souton p n 1 1, p n 1 ) 2 of Probem 1 at tme step n 1) s known and et p n,0 V be arbtrary startng pressures however, a natura choce s p n,0 := p n 1 ). Lemma 3 enabes us to construct a sequence { p n, } V N N 0 of soutons to Probem 2 and prove ts convergence to the souton p n 1, pn 2) of Probem 1 at the subsequent tme step. Theorem 4 Convergence of the DD scheme). Assume there exsts a souton p n 1, pn 2 ) V to Probem 1 s.t. sup p n + z) L M < and et g be as n 30). Let Assumptons 1 hod, λ > 0 and L R be gven wth L S /2 < L for = 1,2. For arbtrary startng pressures p n,0 := v,0 V = 1,2) et { p n, 1, pn, 2 )} be the sequence of soutons of Probem 2 and et { g } N 0 be defned by 25). Assume N 0 further that the tme step τ satsfes τ < Then p n, p n n V and g g n V as for = 1,2. 2m 1 Lk 2 M 2 1 ). 32) L S 2L Remark 7. The essenta boundedness of the pressure gradents can be proven under the addtona assumpton that the functons S are strcty ncreasng and the doman s of cass C 1,α, see e.g. [57, Lemma 2.1]. Proof. We ntroduce the teraton errors e p, := pn pn, L p n,ϕ to 26) and subtract 24) to arrve at [ L e p,,ϕ + τλ e p,,ϕ Γ + τ e g,,ϕ Γ + τ F n k S p n, 1 +k S p n, 1 ) ) ] p n + z) + F n,, ϕ = L e 1 p,,ϕ as we as e g, := gn g, add L p n,ϕ Insertng ϕ := e p, n 33) and notng that L e p, e 1 p,,e p, = L [ e 2 2 p, e 1 2 p, + e p, e 1 2] p,, ) ) p n + z) S p n ) S p n, 1 ),ϕ. 33) 11

13 yeds L 2 [ e 2 p, e e p, e 1 2] + S p n ) S p n, 1 p, = S p n ) S p n, 1 ),e 1 p, e p, τ }{{} =:I 2 p, ),e 1 p, } {{ } =:I 1 e g,,e p, τ k S p n )) k S p n, 1 ) )) p n + z), e p, }{{} τ =:I 3. k S p n, 1 ) ) e p,, e p, }{{} =:I 4 Γ +τλ e p,,e p, Γ 34) We estmate now the terms I 1 I 4 n 34) one by one. By Assumpton 1c), for I 1 we have 1 S p n L ) S p n, 1 ) 2 S p n ) S p n, 1 S ),e 1 p,. 35) I 2 s estmated by I 2 = S p n ) S p n, 1 L e 1 2 p, e p, ),e 1 p, e p, L S p n ) S p n, 1 ) 2. 36) For an ε > 0 to be chosen beow we use Young s nequaty to dea wth I 3, whch can be estmated by I3 = τ k S p n )) k S p n, 1 ) )) p n + z), e p, τ k S p n )) k S p n, 1 τl k M S p n ) S p n, 1 τl k Mε S p n ) S p n, 1 ) )) p n + z) e p, ) e p, ) 2 + τ L k M 4ε e 2 p,, 37) where we used the Lpschtz contnuty of k and the assumpton sup p n + z) L < M. Fnay, by Assumpton 1b) one has τ k S p n, 1 ) ) e p,, e p, for I 4. Usng the estmates 35) 38), 34) becomes [ e 2 p, e 1 2] p, + 1 S p n L ) S p n, 1 S ) 1 S + τl k Mε p n 2L ) S p n, 1 ) 2 + τ L 2 τm e 2 p, 38) ) 2 + τλ e p,,e p, Lk M 4ε ) e m Γ + τ e g,,e p, p, Γ ) 12

14 In order to dea wth the nterface term τ e g,,e p, reca, that denotes the dua parng of H1/2 Γ, Γ 00 Γ) and H 1/2 00 Γ) and the H1/2 00 Γ)-norm smutaneousy. Subtractng 25) from 27),.e. e g, = 2λe 1 p,3 e 1 g,3, we get e p, 2 Γ = 1 4λ 2 e +1 g,3 2 Γ e 2 g, Γ 4λ e ) p,,e g,. 39) Γ Wth b {p,g} we et e b := e b,1,e b,2 ) V 1 V 2 and e b 2 := =1 e b, 2. Smary, on Γ we et e b,e b Γ := 2 =1 e b,,e b, Γ and correspondngy e b 2 Γ = 2 =1 e b, 2 Γ. Summng n 39) over = 1,2 gets e p 2 Γ = 1 4λ 2 e +1 g Dong the same for 34 ) and nsertng 40), eaves us wth L 2 [ e p 2 e 1 2] + + τ 4λ 2 =1 p 2 Γ e g 2 Γ 4λ e p,e ) g. 40) 2 =1 e +1 g 2 Γ e g 2 Γ 1 L S S p n ) S p n, 1 ) 2 ) 2 + τ =1 m L k M 4ε Γ ) e 2 p, ) 1 S + τl k Mε p n 2L ) S p n, 1 ) 2. 41) Now, summng for the teraton ndex = 1,...,r and notcng teescopc sums one gets r 2 =1 =1 + τ L 2 1 L S 1 2L τl k Mε ) S p n ) S p n, 1 ) 2 r 2 =1 =1 m L k M 4ε ) e [ e 0 p 2 e r p 2 ] + τ 4λ 2 p, e 1 g 2 Γ e r+1 g 2 Γ ). 42) Now we choose ε = L k M 2m, hence m L k M 4ε = m 2 > 0 for both. Recang the restrcton on L, 1 LS 1 2L > 0, as we as that by the tme step restrcton L 1 S 2L 1 τ L2 k M 2 2m > 0 for = 1,2, the estmates r 2 =1 =1 τ 1 1 τ L2 k M 2 L S 2L 2m r m e 2 p L 2 2 =1 ) S p n ) S p n, 1 ) 2 L 2 e 0 p 2 + τ 4λ e1 g 2 Γ, 43) e 0 p 2 + τ 4λ e1 g 2 Γ 44) foow for for any r N. Snce the rght hand sdes are ndependent of r, we thereby concude that the seres on the eft are absoutey convergent and therefore S p n ) S p n, 1 ), e p, 0 as. Moreover, 44) mpes e p, 0, ) as we, by the Poncaré nequaty. 13

15 to get To show that e g, 0 n V we subtract agan 24) from 26) and consder test functons ϕ C 0 Ω ) τ F n Fn,, ϕ = L e p,,ϕ + L e 1 p,,ϕ S p n ) S p n, 1 ),ϕ. 45) ) Thus, F n Fn, exsts n L 2 and τ F n Fn, ) = L e p, e 1 p, amost everywhere. Therefore, for any ϕ V one has ) + S p n ) S p n, 1 ) 46) F n ) L Fn,,ϕ e τ p, e 1 p, ϕ 1 + S p n τ p n, 1 ) ϕ. 47) Abbrevatng the eft hand sde of 47) as Ψ n, ) ϕ, 47) means Ψ n, ) ϕ sup L e ϕ V ϕ V τ p, e 1 p, ϕ S p n τ p n, 1 ) 0 ) 48) as a consequence of 44). In other words Ψ n, V 0 as. Startng agan from 33) wthout the added zero term), ths tme however nsertng ϕ V, ntegratng by parts and keepng n mnd 46) one gets e g,,ϕ Γ = λ e p,,ϕ [F Γ + n ] Fn, n,ϕ. 49) Γ We aready know that e p, V 0 as 0 so by the contnuty of the trace operator the frst term on the rght vanshes n the mt. For the ast summand n 49) we use the ntegraton by parts formua to obtan [F n ] Fn, n,ϕ Γ Γ = Ψn, ϕ ) + F n Fn,, ϕ. 50) Whe the frst term on the rght approaches 0, the second can be estmated by k S p n )) p n + z) k S p n, 1 ) ) p n, + z ), ϕ L k M ) S p n S p n, 1) ϕ V + p n, ϕ V, 51) where we used the same reasonng as n 37). Wth ths we et n 50) to obtan [F sup n ] Fn, n,ϕ ϕ V ϕ V =1 Ψ n, Γ V + L k M S p n ) S p n, 1 ) + p n, 0. 52) 14

16 Fnay, usng the above and ettng n 49) gves e g, sup,ϕ Γ 0. ϕ V ϕ V ϕ 0 Ths shows e g, 0 n V for both and concudes the proof. Remark 8. Note that Theorem 4 states that f a souton to the sem-dscrete couped probem exsts, then t s the mt of the teraton scheme. Snce n the convergence proof we use the exstence of a souton to Probem 1, the argument cannot be used to prove exstence. The dffcuty es n the fact that the nonneartes encountered n the dffuson terms are space dependent and may be dscontnuous w.r.t. x over the nterface. 4. Numerca Experments Ths secton s devoted to numerca experments and the mpementaton of the proposed doman decomposton L-scheme. As our formuaton and anayss dd not specase to a partcuar spaca dscretsaton, the numerca mpementaton of the LDD scheme can n prncpa be done wth fnte dfference, fnte eements as we as fnte voume schemes. Snce mass conservaton s an essenta feature of porous meda fow modes, we adopted a ce-centred two pont fux approxmaton varant of a fnte voume scheme to refect ths on the numerca eve. The doman Ω s assumed to be rectanguar and a rectanguar unform mesh was used. Remark 9 dfferent decoupng formuatons revsted). We saw n Remark 2 that another decoupng formuaton s possbe. In fact, ths can be taken a step further. Equatons 11), 12) as we as 11 ), 12 ) can be embedded nto a combned formuaton. For some 0 < η < 1 and M > 0, consder the generased decoupng [ ] F n, n = M 1 η)g + η pn,, 11 ) 1 η)g = 2η pn, η)g ) Observe that the λ-formuaton 11), 12), as we as the convex-combnaton formuaton 11 ), 12 ), are speca cases of ths genera formuaton: In partcuar, M = 1 η) 1 and λ = η1 η) 1 recovers the λ-formuaton, M = 1 and η = λ yeds the convex-combnaton formuaton. Athough 11 ) and 12 ) mght gve even greater parametrc contro over the numercs, n ths paper we adhere to the λ-formuaton because of ts smpcty. Fg. 10 and Fg. 11 show the nfuence of λ and η n both formuatons. 15

17 We start by consderng an anaytcay sovabe exampe. The LDD scheme s tested aganst other frequenty used schemes that do not use a doman decomposton. A of them are defned on the entre doman and the contnuty of norma fux and pressure over Γ s mantaned mpcty. The frst scheme to be compared s a fnte voume mpementaton of the orgna L-scheme on the whoe doman see [16, 31, 33]), henceforth referred to as LFV scheme. Comparson s aso drawn to the modfed Pcard scheme, whch performs better than the Pcard method, see [26]), whch s gven by S p n, 1 ) p n, ) p n, 1 + τ F n, = τ f S p n, 1 ) ) ) S p n 1 on Ω, 53) F n, n 1 = 0 on Γ. 54) Here, the brackets denote the jump over the nterface. Fnay, a comparson wth the quadratcay convergent Newton scheme s made. Wrtng δ p = pn, p n, 1, t reads as foows: S p n, 1) δ p τ [k S p n, 1 ) ) δ p + k S p n, 1 ) ) S p n, 1) δ p p n, + z )] = τ f S p n, 1) ) ) S p n 1 τ k S p n, 1 ) ) p n, 1 + z )) on Ω 55) k S p n, 1 ) ) δ p n 1 + k S p n, 1 ) ) δ p p n, 1 + z ) n 1 = k S p n, 1 ) ) p n, 1 + z ) n 1 on Γ. 56) We refer to [33] for a recent study on nearsatons for Rchards equaton Resuts for a case wth known exact souton To demonstrate the robustness of the proposed scheme, we sove 2) 5) wth both Drchet and Neumann type boundary condtons. In the frst case we dsregard gravty. Specfcay, we consder Ω 1 = 1,0) 0,1), Ω 2 = 0,1) 0,1), and Γ = {0} [0,1]. 57) The reatve permeabtes are k 1 S 1 ) = S1 2 on Ω 1, k 2 S 2 ) = S2 3 on Ω 2 and the saturatons 1 for p < 0, S p) = 1 p) +1 1, = 1,2. 58) 1 for p 0 The boundares and rght hand sdes are chosen to make the exact souton p 1 x,y,t) = 1 1 +t 2 )1 + x 2 + y 2 ), t > 0, x,y) Ω 1, p 2 x,y,t) = 1 1 +t 2 )1 + y 2 ), t > 0, x,y) Ω 2, 16

18 Fgure 2: The doman used n the numerca exampes. The boundary condtons are gven n Tabe 1. The exact souton s aso gven n each subdoman. Ω 1 Ω 2 t = 0 p 1 x,y,0) = x 2 + y 2 ) p 2 x,y,0) = y 2 BCy = 0 y p 1 = 0 y p 2 = 0 y = 1 k 1 S1 p 1 ) ) y p 1 = 2 2+x 2 k 2 S2 p 2 ) ) y p 2 = 1 x = 1 p 1 1,y,t) = 1 1 +t 2 )2 + y 2 ) x = 1 p 2 1,y,t) = 1 1 +t 2 )1 + y 2 ) Tabe 1: Inta and boundary condtons for the exampe wth exact souton. and ths corresponds to the rght hand sdes f 1 x,y,t) = x 2 + y 2 ) 2 t 1 +t 2 ) x 2 + y 2 ), f 2 x,y,t) = 21 y2 ) 1 + y 2 ) 2 2t t 2 ) y 2 ), for t > 0, and x,y) Ω respectvey. The boundary and nta condtons are summed up n Tabe 1. A near systems were soved usng a restarted generased mnmum resdua method gmres) [58]. To boost up speed, sparse trpet format was used n the matrx computaton. The programs are mpemented n ANSI C. For the mpementaton we took the same L n both sub-domans,.e. L := L 1 = L 2. The resuts are shown n Fgures 3 and 4a. Fg. 3 shows the pressure dstrbuton of the exact souton p := χ Ω1 p 1 + χ Ω2 p 2 wth the numerca souton p n, := χ Ω1 p n, 1 + χ Ω 2 p n, 2 potted on top of t. For 17

19 Fgure 3: Comparson between the exact pressure and the numerca pressure provded by the LDD scheme. x = 10 2, t = as we as parameters L = 0.25 and λ = 4, the maxmum reatve error was ess than 0.03%,.e. pn p n, p n L Ω) < The reatve errors of the LDD, LFV and Newton schemes at the md-ne y = 0.5 are potted n Fg. 4a. The LDD scheme preserves the fux contnuty and pressure contnuty at the nterface at every tme step wthout havng to sove for the entre doman. We test ths theory numercay. Fg. 4b shows how dfferent knds of errors behave wthn one tme step. The errors p n, p n, 1 L 2 Ω), pn, p n, 1 L Ω) defned on the doman Ω, as we as p n, L 2 Γ) and F n, n L 2 defned on the nterface Γ, are shown. We observe that the fux and pressure jump tend Γ) to zero whch mpes that fux and pressure contnuty s acheved. Note that the fux at x = 0 from the exact souton s 0. Next, we compare the LDD scheme wth other schemes and study ther dependence on dscretsaton parameters. We compare the Newton scheme, the modfed) Pcard teraton, the aready mentoned LFV scheme and the LDD scheme, nvestgatng the dependence of tme step refnement and space grd refnement separatey. The frst study, shown n Fg. 5, pots og 10 p n, p n, 1 L 2 Ω)) for a schemes, at the fxed tme step correspondng to t = 0.2. As expected, Newton s the fastest and shows a quadratc convergence rate. But at the same tme, t s most susceptbe to change n mesh sze as observed from the sopes of the eft-most curves. The convergence rate of the Pcard teraton s near, faster than both the L-schemes and s stabe wth respect to varaton n mesh sze. The L-schemes aso exhbt near convergence, abet sower than Pcard, and the convergence speed does not vary much wth mesh sze. LFV and LDD schemes have practcay the same convergence rate. Tabe 2 compements the pot n Fg. 5 and sts expermenta average convergence rates, defned as e n,+1 p / e n, p, for a schemes Newton data s not shown for 18

20 a) Comparson between the numerca soutons pro- b) Dfferent errors vs nner teratons for the case wth vded by the LDD, LFV and the Newton schemes. Pot- exact souton. Here t = 0.2, L = 0.25 and λ = 4. ted are the reatve errors p exact p num p exact as functons of x, for y = 0.5 and t = 1. x = 0.1, 0.05, 0.02 as t reaches an error ower than n 3 teratons). Secondy, we study the dependence of the convergence rates on tme step sze for a fxed mesh x Newton Pcard LFV LDD sze x = 0.02). The error characterstcs of a four schemes n Fg. 6 are shown for t = 0.5. In Fg. 6a both, Newton and Pcard, dverge, whereas both L-schemes converge for L = The LFV scheme exhbts some oscatons, the reason beng the dependence of the choce of L on the tme Type Quadratc Lnear Lnear Lnear Tabe 2: The average convergence rate, e n,+1 / e n,, for the step τ. Hgher vaues of τ mght requre hgher dfferent schemes and wth respect to the mesh-sze. vaues of L. Indeed, f we substtute L = 0.5 n the LFV scheme marked as LFV* n the dagram), we see a more robust behavour. Note, that the LDD scheme converges for a τ and s at east as fast as the LFV scheme n a the cases. For smaer vaues of τ the Newton and Pcard teraton converge faster than both L-schemes, as shown n Fgures 6b and 6c. Accordng to the theory, the convergence of the Newton and Pcard schemes s ony guaranteed f the nta guess s cose enough to the exact souton. Therefore, startng the teraton wth the numerca souton at the prevous tme step ths suggests that the tme step shoud be taken sma enough to have a 19

21 Fgure 5: Performance comparson and mesh study for the convergence of the LDD, LFV, Pcard and Newton schemes. Here L = 0.25 and λ = 4. guaranteed convergence see [24, 30, 33]. Contrarwse, L-schemes are free of ths constrant. To ustrate ths behavour, we have nvestgated the convergence of the schemes for a constant nta guess. Specfcay, p n,0 = 5 has been used nstead of p n,0 = p n 1. In ths case, the Newton and Pcard schemes are dvergent whereas both L-schemes st produce a good approxmaton after severa teratons. Ths s dspayed n Fg. 7. A smar behavour w be observed agan whe dscussng a numerca exampe wth reastc parameters. Remark 10. The convergence behavour of the Fgure 7: Error decay for the dfferent schemes for a constant LDD scheme can be optmzed by choosng λ nta guess, p n,0 = 5. Here L = 0.25, λ = 4. propery. In the above comparson λ was chosen dfferenty for every choce of mesh sze. The optmaty of λ s dependent on the mesh and the tme step 20

22 a) t = 0.1 b) t = 0.01 c) t = Fgure 6: Convergence study for the tme-steps τ = 0.1, 0.01, Here, L = 0.25 for the LFV scheme and L = 0.5 for the LFV* scheme. For the LDD scheme one has L = 0.25, λ = 2 n case 6a, L = 0.25, λ = 4 n case 6b, and L = 0.25, λ = 10 n case 6c. 21

23 sze. Wth a good choce of λ, one can make the LDD scheme at east as fast as the LFV scheme. Ths s dscussed n more deta n Secton Resuts for a reastc case wth van Genuchten parameters We demonstrate the appcabty of the LDD scheme for a case wth reastc parameters, ncorporatng aso gravty effects. We consder a van-genuchten-muaem parametrsaton [59] wth the curves k and S S p) = S,r + S,s S,r )Φ p), Φ p) = α p) ˆn ) m, m = 1 1ˆn, k S) = Φ p) 1 m 1 Φ p) 1 ) ) m 2. 59) The specfc parameter vaues are sted n Tabe 3 and are characterstc for partcuar types of materas, st oam G.E. 3 Ω 1 ) and sandstone Ω 2 ). These materas have very dfferent absoute permeabtes κ 1,κ 2, whch makes the numerca cacuatons more chaengng. The dmensona governng equatons and boundary condtons become = 1,2) In ths case F n, = κ µ k r, L p n, + τ F n, = L p n, 1 φ S p n, 1 ) S p n 1 ) ), on Ω, 60) F n, n = g + 2λ pn,, on Γ, 61) p n,, = 0 on Ω. 62) S p n, 1 ) ) p n, ρ g ). Here g = ge x s the gravtatona acceeraton agned wth the postve x-drecton, ρ, µ are the densty and the vscosty of the fud and κ, φ are the absoute permeabty as we as the porosty of the medum. Note that Fg. 2 s rotated by 90 degrees. The probem s nondmensonased by usng the characterstc pressure p := Pa, ength 1.48m and tme s. Ths eads to the nondmensona quanttes p, x, y) and t. After nondmensonasaton, the doman used s agan taken to be Ω 1 = 1,0) 0,1), Ω 2 = 0,1) 0,1). The nta condton used s px,y,0) = 1 63) and boundary condtons are 1 +ty f y < 1 ε)t 1 p 1,y,t) =, ε f y 1 ε)t 1 p1,y,t) = 1, 22

24 Parameter Unt St Loam G.E. 3 Ω 1 ) Sandstone Ω 2 ) Porosty φ ) Water Densty ρ) kg m Water Vscosty µ) Pa s Absoute permeabty κ ) m 2 s Retenton exponent ˆn ) Retenton parameter α ) Pa Irreducbe water saturaton S,r ) Irreducbe ar saturaton 1 S,s ) Tabe 3: The van Genuchten-Muaem parameters n the reastc test case. together wth a no-fow condton at y = 0,1. We take ε > 0 to avod degeneracy. Fg. 8a shows the dfferent errors for ths case and t can be seen that a the errors are decreasng for the LDD scheme. Errors at the nterface and nsde the doman tend to 0, the convergence s sower compared to the case wth exact souton, however. Ths s due to the arge varance of the parameters as we as the hghy nonnear nature of the assocated functons. Because of ths, both Newton and Pcard schemes dverge. The behavour of dfferent schemes for the same set of parameters s shown n Fg. 8b. Observe that for the Newton scheme the startng error as we as the number of teratons requred ncreases steady wth t unt t = 0.94, at whch pont the errors start dvergng. The Pcard scheme becomes dvergent even before t = 0.2. In contrast, both L-schemes reman stabe n ths case Tme Performance Ths secton s devoted to the comparson of tme performance of the schemes. We have seen that L-schemes are more stabe than Newton and Pcard. But f they are convergng, then Newton and Pcard schemes converge faster than the L-schemes. Beow we nvestgate how the schemes compare to one another wth respect to actua computatona tme. We set an error toerance for the schemes that stops the teratons wthn one tme step, after reachng an error ower than 10 6,.e. p n, p n, 1 L 2 Ω) < Ths s to ensure that we get comparatve CPU-cock-tme for dfferent schemes for the same degree of accuracy. We computed the exacty sovabe case on a LINUX server mammoth.wn.tue.n) for a four schemes usng the same set of parameters x = 0.02, τ = 0.001, L = 0.25 and λ = 10). Fgure 9 ustrates the tme-performances of these schemes over the whoe computatona tme doman. Tabe 5 shows how many 23

25 a) Dfferent errors vs nner teratons for the reastc b) Error vs nner teratons for the reastc case. LDD, case at t = 0.2. The parameters are τ = 0.01, x = 0.02, LFV and Newton errors are potted at t = 0.2. Newton L = 0.25 and λ = 10. Ony the LDD scheme s shown n denotes the errors of Newton scheme at t = 0.9. Pcard s ths pot. potted at t = Here, L = 0.5, λ = 10. Fgure 8: Error pots and scheme comparson for the reastc case. nner-teratons are requred on average for dfferent schemes to reach the error crteron at dfferent ponts n tme. Iteraton requrement per tme step ncreases for a schemes as the boundary condtons change more rapdy wth tme. Tabe 5 shows the average tme taken and how many gmres teratons outer and nner) were requred by each scheme to execute one nner teraton. Unsurprsngy, the Newton scheme s st fastest, foowed by Pcard and the LDD scheme. But LDD competes cosey wth Newton and Pcard. Even more surprsng s the fact that the LFV scheme takes consderaby more tme to reach the Fgure 9: Tme performance of the L-DD, L-FV and the Newtondesred accuracy compared to the LDD scheme, FV schemes. despte both havng amost the same convergence rate. The reason becomes apparent from Tabe 5: The LDD scheme requres much ess tme per nner 24

26 Average nner teratons requred Tme-step/Scheme LDD LFV Pcard Newton Avg. tme per ter Avg. GMRES teratons Tabe 5: The average number of nner teratons per tme step requred by the dfferent schemes to reach the stoppng crteron p n, p n, 1 L 2 Ω) < The ast two rows gve the average tme and gmres-teratons per nner teraton. teraton than a other schemes. The LFV scheme has the second fastest average tme per teraton. For the Pcard teraton, the dervatve of the saturaton functon needs to be evauated whch n turn costs more tme than an teraton n the LFV scheme. The Newton scheme s computatonay most expensve per teraton because t cacuates the Jacoban at every teraton. The schemes that do not decoupe the doman requre much more tme and many more gmresteratons Condton number per nner teraton. The reason s that x the doman decomposton schemes nvove smaer L-DD Ω 1 ) matrces and and they have smaer condton numbers. L-DD Ω 2 ) Ths s ustrated by the ast row of Tabe 5. The LDD scheme requres on average 119 gmresteratons L-FV Ω) Tabe 4: The condton number vs mesh sze for the LDD and on Ω 1 and 123 gmres-teratons on Ω 2 and both domans have eements. Compare ths wth Newton, whch takes amost 400 gmresteratons LFV schemes. Here, τ = 0.001, t = 0.2, L = 0.25, λ = 10. The condton numbers are cacuated for the 200 th tme step for the matrces of the frst nner teraton. and deas wth varabes on each gmres-teraton. Ths expans why the LDD scheme takes so much ess tme per nner teraton. Tabe 4 compares the condton numbers of the LDD and the LFV scheme. It shows that the matrces for the LFV scheme are worse condtoned than the ones of the LDD scheme. The atter has two condton numbers, one for each doman. The 2-norm condton numbers were cacuated wth MATLAB s bud n cond) functon. 25

27 Remark 11. The fact that the LDD scheme performance competes cosey wth Newton and Pcard, means that, LDD can potentay be made much faster than even Newton as t s paraesabe. Ths s the key advantage of the LDD scheme aong wth ts goba convergence property Parameter dependence and key features Havng outned the robustness and speed of the proposed LDD scheme we turn to nvestgate some of ts propertes. Two mportant parameters have been ntroduced n the L-DD scheme,.e. L and λ, and apart from a ower bound on L nothng has been specfed about these parameters. Ths means that they can freey be adjusted to gve optma convergence rate. In fact, n ths secton we w see that the convergence rate depends strongy on these parameters. The nfuence of λ a) The decay of the pressure error n terms of λ. b) The decay of the g-error n terms of λ. Fgure 10: The nfuence of λ on the convergence rate. The parameters for the LDD scheme are τ =.01, x = 0.02, L = 0.25 at t = 0.2. Fgure 10 shows the nfuence of the parameter λ on error characterstcs. A the resuts shown are for the case wth exact souton. Fgure 10a focuses on the errors p n, p n, 1 L 2 Ω) on the doman Ω, whe Fg. 10b depcts the L 2 -errors g g 1 L 2 Γ) on the nterface for the same tme step. Ceary, λ has tremendous mpact on the convergence rate. The convergence rate rapdy ncreases wth λ at frst but after a certan pont the convergence rate starts decreasng. Ths trend s notceabe n both pots of Fgure 10. Ths ndcates that there s an optma ambda λ opt for whch the whoe scheme has a fastest convergence rate. The optmaty of λ s actuay a we studed behavour n the doman decomposton 26

28 terature. In [60, 50] t has been shown that λ opt depends at east on mesh sze and sub- doman sze. Later we w show that t aso depends on L and τ n our case. Ths contro over the convergence rate s the reason why the λ-formuaton was chosen over the convex-combnaton formuaton gven n Remark 2. To ustrate ths, Fg. 11 shows the same pots as Fgure 10 but for the convex-combnaton formuaton. In order to dfferentate between pots more easy, we use the combned formuaton 11 ), 12 ) and set M = 1. For η = 0.01 the convex-combnaton formuaton even fas to converge. In a other cases the convergence s consderaby sower. a) The decay of the pressure error n terms of η. b) The decay of the g-error n terms of η. Fgure 11: The nfuence of η on the convergence rate n the convex-combnaton formuaton M = 1 n Remark 2). The parameters for the LDD scheme are τ = 0.01, x = 0.02, L = 0.25 at t = 0.2. The nfuence of L We brefy gve an overvew over the nfuence of L on the convergence rate. Fgure 12 depcts ths for L := L 1 = L 2. For L-schemes t s common to dverge f L s too sma, whch seems to be the case for L = 0.1. On the other hand, the convergence rate decreases sgnfcanty for very arge L, a behavour that s a common trat of L-schemes as we, cf. [10]. It s best to choose L as sma as possbe, yet great enough to ensure convergence of the scheme. Note that L = 0 represents the orgna nonmodfed) Pcard teraton case and Fgure 12 suggests that the orgna Pcard scheme fas for these probems. The dependence of λ opt on L, τ and x In ths ast secton we nvestgate numercay how λ opt depends on the choce of L, τ and x. For a fxed grd n tme and space Tabe 6 sts convergence rates for dfferent λ and L. 27

29 Wth ths tabe we can guess the nterva n whch λ opt es. Wthn ths estmated nterva, Fg. 13 shows how the convergence rate vares wth λ for fxed L. For L = 0.25, x = 0.02 and τ = 0.01 the fastest convergence s acheved for λ = 4 ths s why λ = 4 was chosen for the above comparsons, wherever the specfed L, x, τ set was used). The λ dependence for hgher vaues of L s ess pronounced. For a fxed L, Tabes 7a, 7b show the varance of λ opt wth respect to tme-step and mesh sze respectvey. The shown tabes are of course ony a rough estmate of λ opt. Due to computaton tme, t s a tedous process to fnd a cose to exact vaue of λ opt, Fgure 12: The nfuence of L on the convergence rate, as obtaned for the nner teratons for the 50 th tme step. especay for very sma tme-step szes. In practce the vaues are numercay guessed. The resuts ndcate qute a strong correaton of λ opt wth the tme-step sze, contrasted by a rather mnor correaton wth the mesh sze. L λ = 0.1 λ = 1 λ = 10 λ = 100 λ opt 0.1 dverged dverged dverged dverged ,10) 1 dverged ,10) 5 dverged ,100) Tabe 6: The dependence of the convergence rates on λ and L: the geometrc average of the contracton rates over the frst 20 teratons and for dfferent L,λ) pars s gven n the frst coumns, whereas the ast gves the nterva for λ opt. Here, x = 0.02, t = 0.01, t = 0.2. t Nr ter.? Avg. CR a) λ opt for x = 0.02, L = 0.25 x Nr ter.? Avg. CR b) λ opt for t = 0.01, L = 0.25 Tabe 7: The dependence of λ opt on t and on x. 28

30 5. Concuson We consdered a nonnear paraboc probem appearng as mathematca mode for varaby saturated fow n porous meda. For the numerca souton of the nonnear, tme dscrete probems we proposed a combned scheme LDD) that s based on a fxed pont teraton the L-scheme), and on a doman decomposton scheme nvovng Robn type coupng condtons at the nterface separatng dfferent subdomans. The resut s a scheme featurng the advantages of both approaches: an uncondtona convergence, regardess of tme step and startng pont, as we as a decoupng Fgure 13: Convergence rate vs λ for L = 0.25 and L = 1. of the tme dscrete probems nto subprobems that For L = 0.25, λ opt 4. can be soved n parae. The stabty, robustness and effcency of the method s tested for varous cases and aso compared to Newton and Pcard schemes. The tests ncude stuatons where the atter dverge whereas the proposed scheme s convergng. In summary, the key advantages of the method are: The LDD scheme converges uncondtonay. It can provde accurate resuts even n stuatons where the Pcard or Newton teratons fa. In conjuncton wth a sutabe space dscretsaton, t provdes a decouped, mass conservatve approach. Ths s very usefu n partcuar when deang wth modes defned n meda wth bocktype heterogenetes, where the matera propertes n dfferent bocks may vary sgnfcanty. Though neary convergent, the computatona tme requred by the LDD scheme for achevng a certan accuracy of the approxmaton s comparabe to the tme needed by Newton and Pcard schemes, and much faster than a standard L-scheme apped to the mode n the entre doman. Ths effcency s due to the fact that the scheme needs ess tme per nner teraton than a scheme defned n the entre doman. Moreover LDD s paraesabe, whch gves the possbty of ncreasng ts effcency even further. The convergence rate of LDD schemes depends on the choce of L and λ. Wth the optma choce of parameters, the convergence order can be reduced sgnfcanty. 29

31 Acknowedgements Ths work was supported by Netherands Organsaton for Scentfc Research NWO Vstors Grant , Odysseus programme of the Research Foundaton - Fanders FWO G0G1316N), by UHasset Speca Research Fund project BOF17BL04, the Norwegan Research Counc through the projects NRC CHI) and NRC IMMENS) as we as She-NWO/FOM CSER programme project 14CSER016) and the German Research Foundaton through IRTG 1398 NUPUS project B17). References [1] R. Hemg, Mutphase fow and transport processes n the subsurface: a contrbuton to the modeng of hydrosystems, Sprnger, Bern, [2] L. F. Rchardson, Weather predcton by numerca process, Camebrdge Unversty Press, [3] L. A. Rchards, Capard conducton of quds through porous medums, Physcs 1 5) 1931) [4] H. W. At, S. Luckhaus, Quasnear eptc-paraboc dfferenta equatons, Math. Z ) 1983) [5] H. W. At, S. Luckhaus, A. Vsntn, On nonstatonary fow through porous meda, Anna d Matematca Pura ed Appcata 136 1) 1984) [6] F. Otto, L1-contracton and unqueness for quasnear eptc paraboc equatons, Journa of Dfferenta Equatons 131 1) 1996) [7] T. Arbogast, M. F. Wheeer, A Nonnear Mxed Fnte Eement Method for a Degenerate Paraboc Equaton Arsng n Fow n Porous Meda, SIAM J. Numer. Ana. 33 4) 1996) [8] R. H. Nochetto, C. Verd, Approxmaton of degenerate paraboc probems usng numerca ntegraton, SIAM J. Numer. Ana. 25 4) 1988) [9] I. S. Pop, Error estmates for a tme dscretzaton method for the Rchards equaton, Comput. Geosc. 6 2) 2002) [10] F. A. Radu, I. S. Pop, P. Knabner, Order of Convergence Estmates for an Euer Impct, Mxed Fnte Eement Dscretzaton of Rchards Equaton, SIAM J. Numer. Ana. 42 4) 2004)