A priori error analysis for transient problems using Enhanced Velocity approach in the discrete-time setting.

Transcription

1 arxiv: v1 [math.na] 1 Dec 018 A riori error analysis for transient roblems sing nhanced Velocity aroach in the discrete-time setting. Yerlan Amanbek 1, and Mary F. Wheeler 1 1 Center for Sbsrface Modeling, Institte for Comtational ngineering and Sciences, University of Texas at Astin School of Science and Technology, Nazarbayev University yerlan.amanbek@n.ed.kz, mfw@ices.texas.ed December 13, 018 Abstract Time discretization along with sace discretization is imortant in the nmerical simlation of sbsrface flow alications for long rn. In this aer, we derive theoretical convergence error estimates in discrete-time setting for transient roblems with the Dirichlet bondary condition. nhanced Velocity Mixed FM as domain decomosition method is sed in the sace discretization and the backward ler method and the Crank-Nicolson method are considered in the discrete-time setting. nhanced Velocity scheme was sed in the adative mesh refinement dealing with heterogeneos oros media [1, ] for single hase flow and transort and demonstrated as mass conservative and efficient method. Nmerical tests validating the backward ler theory are resented. This error estimates are sefl in the determining of time ste size and the sace discretization size. Keywords. a riori error analysis, enhanced velocity, mixed finite element method, error estimates, Darcy flow. 1 Introdction The most sbsrface flow eqations are dynamic and time-deendent roblems. In decision making rocess, nmerical simlation of flow lays vital role in many engineering 1

2 alications sch as oil and gas rodction evalation, CO seqestration and contaminate transort roblems. It is natral to deal with non-matching mltiblock grids in the reservoir simlation since sbsrface arameters sch as ermeability or orosity can vary over sbdomains sbstantially. The accracy of simlation can deend on discretization method of sace and time variables. For sace discretization, we are concerned with a wellestablished domain decomosition method, i.e. nhanced Velocity Mixed Finite lement Method VMFM, which rovides similar accracy as the Mltiscale Mortar Mixed FM [3]. VMFM is a mass conservative and an efficient domain decomosition method. By sing this method, several alications sch as single, two-hase flow, bio-remediation simlation and others were considered in [3, 4]. Recently, an adative mesh refinement strategy, which is based on nhanced Velocity scheme, has been roosed in the nmerical simlations of flow and transort throgh heterogeneos oros media [1,, 5, 6]. Sch a novel aroach demonstrates the efficiency and accracy of simlation in the heterogeneos oros media by allowing to catre an imortant featres of flow and transort roblems. A little attention has been given to discrete time setting analysis. Theoretical convergence analysis of VMFM has been shown in [4] for slightly comressible single hase flow for general continos in time aroximations. However, we cold not find the nmerical error tests that comare with analytical soltion. A few athors have begn to imlement time domain decomosition method to be flexible on selection of time ste size [7, 8, 9]. The key idea is to extend the VMFM in sace to time discretization by constrcting a monolithic system withot sbdomain iteration. In this aer, we are concerned with the soltion of time deendent roblem that is discretized by the backward ler method or the Crank-Nicolson method combined with VMFM, which ses the lowest order Raviart-Thomas saces on non-matching mltile sbdomains. In articlarly, we focs on deriving a riori error estimates for the transient sbsrface roblems in discrete-time setting. This gives asymtotive behavior of the nmerical error for a given mesh size, time ste size and others. This analsys allows s to conclde abot the convergence and the garantee of stability of the nmerical method. The reader is referred to [10, 1] for more different time and sace discretization in the transient roblems. We also rovide nmerical tests of the error estimate. This aer is organized as follows. In the next section, we describe the slightly comressible flow model formlation as well as give a strong and weak formlation sing nhanced Velocity sace. Section 3 is devoted to the error analysis with reliminary rojections, definitions and discrete formlation. This analysis carried ot for backward ler scheme and Crank-Nicolson schemes simltaneosly in time discretization settings. Nmerical reslts are resented in Section 4; Section 5 concldes the aer.

3 Model Formlation We describe in this section the slightly comressible flow formlate with initial and bondary conditions. Next, the transient roblem is resented in the strong and weak formlations..1 Slightly Comressible Flow Formlation Or focs is a single hase and slightly comressible flid in heterogeneos oros media. The classical mass conservation eqation is defined by φρ + ρ = q in Ω J 1 t where Ω R d d = 1, or 3, J = 0,T ], d is the nmber of satial dimensions, q is the sorce/sink term, φ is the orosity, ρ is the hase density, and is the hase velocity. We remark that a Peacemann correction is sed for modeling sorce/sink terms [11]. In slightly comressible flid, the hase density is given by ρ = ρ re f e C f re f, where where, C f is the flid comressibility, and ρ re f is the reference density at reference ressre re f. Using Taylor series exansion we obtain ρ ρ re f 1 +C f re f. Then it follows that φ ρ t = φ ρ t = φc 1 t for invariant-in-time φ and for C 1 = C f ρ re f. The hase velocity is defined by Darcy s law as, = K ρg, 3 µ where, µ is the viscosity, K is the ermeability absolte ermeability tensor, ρ is the density of the flid and g is the gravity vector. Althogh more general global bondary conditions can also be treated, we restrict orselves to the following, Additionally, the initial condition is given by, = g on Ω J. 4 x,0 = 0 x. 5 From now on or analysis focs on a transient arabolic roblems which might be involved to varios alications roblems.. Transient roblem with VMFM We start with the strong formlation of the transient slightly comressible flow roblems governing single hase flow model for ressre and the velocity, which is also case of 3

4 slightly comressible single hase flow model: t = K in Ω J, 6 + = f in Ω J, 7 = g on Ω J 8 = 0 at t = 0 9 where Ω R d d = or 3 is mltiblock domain, J = [0,T ] and K is a symmetric, niformly ositive definite tensor reresenting the ermeability divided by the viscosity with L Ω comonents, for some 0 < k min < k max < k min ξ T ξ ξ T Kxξ k max ξ T ξ x Ω ξ R d,d = 1,,3. 10 The Dirichlet bondary condition is considered for convenience. A weak soltion of arabolic qns. 6-9 is a air { h, h } : J V h W h, i.e. nhanced Velocity Mixed Finite lement aroximation K 1,v =, v g,v ν Ω v V 11 t,w +,w = f,w w W 1 In addition, there is an initial condition,w = 0,w t=0 w W 13 We formlate the variational roblem in semi-discrete sace as: Find { h, h } : J V h W h sch that K 1 h,v = h, v g,v ν Ω v V h 14 h t,w + h,w = f,w w W h 15 In addition, there is an initial condition h,w = 0,w t=0 w W h 16 Sbtracting qns from qns yields K 1 h,v h, v = 0 v V h 17 t h,w + h,w = 0 w W h 18 4

5 3 rror estimates In this section, we start with reliminaries inclding rojections oerators and some notations. Using them we resent discrete formlations for analysis. Next, we derive axiliary error estimates and a riori error estimate theorems. 3.1 Projections We shall write a rojection and define axiliary error of ressre and velocity as follows: I = ˆ, A = ˆ h, 19 I = Π, A = Π h. 0 Note that h = I + A, h = I + A. We sed ˆ the L -rojection of that is defined as ˆ,w = I,w = 0 w W h. 1 We know from original work [3] that the rojection oerator Π was introdced and was tilized for a riori error analysis of ellitic roblems. For convenience of the reader, we reeat the relevant and brief definition. Ths, we denote by Π the rojection oerator that mas H 1 Ω d onto V h that defined locally for any element T T h and any q H 1 T d sch that for all q H 1 T d Π q ν,1 e = q ν,1 e where e is either any edge in D or face in 3D of T not lying on Γ or an edge in D or face in 3D of a sb-element, T k. Sch rojection is develoed rior to condcting error analysis for a riori estimate. As can be seen in Figre 1, T k has a common edge with the interface grid T Γ. According to divergence theorem, we have For H 1 Ω Π q q,w = 0 w W h 3 Π,w = I,w = 0 w W h 4 Lemma. t I,w = 0 w W h 5 Proof. 0 = dt d I,w = t,w I + 0 I,w t, by 1, since wt W h. Usefl ineqalities of rojections, see [3]: C r h r 0 r 1, 6 I 5

6 e 1 e T 1 T Γ i, j Figre 1: Degrees of freedom for the nhanced Velocity sace. Recall Yong s Ineqality: for a, b 0 I C 1 h. 7 ab 1 ε a + ε b 8 The Inverse Ineqality can be given as h Ch 1 h 9 In this ineqality, we have been working nder the assmtion that T h,i qasi-niform rectanglar artition of Ω i. 3. Definitions In this section, we make analysis of discrete in time error estimates. Firstly, some definitions are made: for = T N, N is a ositive integer, t n = n and for given θ [0,1], f n = f x,t n, 0 n N, 30 f n,θ = θ f n θ f n, 0 n N Let s make also the following definitions: f l L = max 0 n N f n L f l L = 6 1 f n,θ. L

7 We note that the difference can be exressed t n,θ t n = 1 1+θ. Next, sing the Taylor series exansion abot t = t n,θ, for any sfficiently smooth fnction f t, we obtain: f n+1 = f f n = f t=t n,θ t=t n,θ + 1 f 1 θ t 1 f 1 + θ t t=t n,θ t=t n,θ θ f t θ f t + O 3 t=t n,θ + O 3 t=t n,θ After mltilying the first eqation by θ and the second eqation by 1 1 θ and then smming them, we obtain f n,θ = f + 1 t=t n,θ θ1 θ f t + O 3 t=t n,θ Note that if θ = 1 then f n,θ = f + O 3. In addition, we can get second order t=t n,θ aroximation of, details in [13] : x,t n,θ n,θ and x,t n,θ n,θ. According to Taylor series exansion [13], we obtain n+1 n = t x,t n,θ + ρ,n,θ, x Ω, 3 where ρ,n,θ deends on time-derivatives of and so ρ,n,θ = O θ. ρ,n,θ { C1 tt L t n,t n+1,h 1, i f θ = 1, C ttt L t n,t n+1,h 1, i f θ = 0, Discrete formlation We formlate variational form in semi-discrete sace as: Find { h, h } : J V h W h sch that h t,w + h,w = l 1 w w W h 34 K 1 h,v h, v = l v v V h 35 In addition, there is an initial condition h,w = 0,w t=0 w W h 36 7

8 where l 1 and l are bonded linear fnctionals, i.e. l 1 w = f,w, l v = g,v ν Ω. With this definitions, qns become as: Find { n,θ h 1,,...,N 1, sch that n+1 h n h,w, n,θ h } V h W h, n = + n,θ h,w = l n,θ 1 w w W h 37 K 1 n,θ h,v n,θ h, v = l n,θ v v V h 38 Note that if θ = 1 then the time discretization is the backward lerimlicit method, and if θ = 0 then the Crank-Nicolson scheme. We consider tre soltion L J,V and h H 1 J,W of qns. 11 and 1 at time t = t n,θ in the continos in time with satially discrete scheme. We sed qn. 3 and additional remark related to the Taylor series exansion in order to obtain the following eqations with at least order of O: n+1 n K 1 n,θ,v,w + n,θ,w n,θ, v = l 1 w + ρ,n,θ,w w W 39 = l v v V 40 We aroximate the vector integrals tye v,q T,M by traezoidal-midoint qadratre rles and K 1 q,v by traezoidal qadratre rles resectively. In [14], the eqivalence T between finite volme methods and the mixed finite element method was established for secial qadratre rle for K diagonal tensor and sing the lowest-order Raviart Thomas saces on rectangles. We emhasize that the VMFM with secial qadratre and velocity elimination in the discrete system can be redced to well-known a cell-centered finite difference method. For each time ste we se the Newton method to solve the system, in case of the slightly comressible flow: the flid comressibility C f term brings s to a nonlinear system. That is why consideration of it wold be beneficial for nonlinear roblems in the ftre. 3.4 Analysis We first derive the bonds of axiliary error terms. Theorem 1 Axiliary error estimate. For the velocity h and ressre h of the mixed method saces V h W h satisfying eqations 37-38, assme is sfficiently small 8

9 and ositive, K is niformly ositive definite and sfficient reglarity of tre soltion in eqations 6-9. Then, there exist a constant C sch that A where C = CT,K,, and + A l L r = l L C h + h + r 41 { 1, i f θ = 1, i f θ = 0 Proof. Sbtracting qns from qns resectively yields n+1 n+1 h n n h,w + K 1 n,θ n,θ h Take v = Π n,θ n,θ h n,θ n,θ h,w = ρ,n,θ,w w W h 4,v n,θ n,θ h, v = 0 v V h. 43 = I n+1 K and w = I n + in 43 and 4 resectively. +, =,, ρ,n,θ, +, = 0 9

10 After adding them, we can rewrite as I n I n + + K 1 + I n =, + + K 1,,, I n + + +, 0 +, +,, +, =, + 0 A n,θ, by 1 I n+1 0, =, + +1, by 1 I n 0, n,θ,a, + K 1 = + +1, = ρ,n,θ,, + 0, by 4, + K 1, +,, +1, = K 1 +, + K 1,, + ρ,n,θ, 44 10

11 +1, = 1 + θ = 1 + θ + 1 θ = 1 + θ +1 = +1, 1 + θ n+1,θ A + 1 θ n,θ A, θ +1 +1, θ, +1 = 1 = , θ θ,, +1 1 θ + θ +1 + θ +1 It follows immediately that +1,, }{{} , +1 By sing 45, mltily by and sm from 0 to N 1 in qation K 1 +, + K 1,, + ρ,n,θ, 11

12 A N 0, by 36 A 0 + K 1, K 1, } {{ } T 1 + ρ,n,θ, + } {{ } T, } {{ } T 3 T 1 = }{{} Holder s ineq. }{{} Yong s ineq. K 1, K 1 1 εkmin + ε We se the Holder ineqality and the Yong ineqality to get T = We shold note that ρ,n,θ, +, I A Ω =, I A ρ,n,θ. ρ,n,θ Ω + 0, I A Ω\Ω =, I A 46 Ω 1

13 since A W h and the roerty 1. Ω\Ω T 3 = = Ω, Ω, Ω Ω 1 + θ C n+1 1,Ω + 1 θ n 1,Ω h C 1 + θ n+1 1,Ω + 1 θ n 1,Ω + ε Remark 1. We sed the following roerties: = 1 + θ Ω 1 + θ 1 + θ 1 + θ n+1 I + 1 θ I n+1 I n Ω Ω + 1 θ I n Ω n+1 1,Ω h + 1 θ n 1,Ω h n+1 1,Ω + 1 θ n 1,Ω h 1 Ω h and C h 1 C Ω Ω h 1 Ω Next, we know that K 1, 1 k max 13

14 Therefore, 1 A N C A N [ k max ε +C + +C ] ρ,n,θ +C θ n+1 1,Ω + 1 θ n 1,Ω We can mltily by and make ε small enogh in order to have LHS with ositive coefficients. Later take minimm and divide both side of ineqality. C +C [ + A N + +C 1 + θ n+1 1,Ω + 1 θ ] n 1,Ω ρ,n,θ. We are ths aly the discrete Gronwall lemma, for sfficiently small, to obtain: + A N C C [ [ + + h 1 + θ n+1 1,Ω + 1 θ ] n 1,Ω +C 1 + θ Ch 1 + θ n θ n 1 + +Ch 1 + θ CT,,,K h + h + r. n+1 1,,Ω + 1 θ ] n 1,,Ω +C n+1 1,,Ω + 1 θ n 1,,Ω +C 14 ρ,n,θ ρ,n,θ ρ,n,θ

15 where r = { 1, i f θ = 1, i f θ = 0 We sed the fact that g n CT g n, Ω Ch and roerty that is given in eqn. 33. This finishes the roof of theorem. The axiliary error estimates theorem allows s to conclde the following theorem: Theorem rror estimate. Assme the same conditions as in the revios theorem. Then, where C = CT,K,, and h l L + h l L C h + h + r 47 r = { 1, i f θ = 1, i f θ = 0 Proof. By alying triangle ineqality,the Interolation rror Ineqalities and Theorem 1 reslts we obtain: h l L + h l L = I + A + I + A l L l L C I l L + I + l L A + A }{{} l L l L }{{} Interolation error Axiliary error CT,,,Kh + h + O r 15

16 4 Nmerical xamles In this section, we condct nmerical exeriment to verify the nmerical accracy of arabolic roblem soltion sing VMFM in sace and the backward ler in time. Based on or a riori error analysis estimates we assme a sfficiently smooth analytical soltion. In nmerical examles, we set Ω = 0,1 0,1, K i, j = δ i, j and the domain Ω is divided into for sbdomains Ω i ; Ω 1 and Ω 4 have fine grids, Ω and Ω 3 have coarse grids; sch mesh discretization is illstrated in Figre. 4.1 Nmerical examle 1 We se the known soltion x,y,t = tx1 xy1 y and se it to comte the forcing f, the Dirichlet bondary data g, and the initial data 0. We carry ot several levels of niform grid refinement in each sbdomains. The time ste and the element size are almost eqal to each other, see Table 1. The simlation time interval is 0;0.1, i.e. T = 0.1, and we se the Backward ler method to integrate with regard to time with niform time ste. We are interested in finding the exact error sing a given tre soltion, so the ressre is tre error and the velocity error is normalized error. On alying sfficient Newton iterations at each time ste rovided the residal is within the machine-recision tolerance, we obtain the nmerical soltion for evalating of the error in secified norm. We comte error, corresonds to h l L, which is maximm of vales among time stes that reslting for given time ste a discrete ressre L -norm that associates only the fnction vales at the cell-centers in sace. Also, error is defined as h l L where in sace a discrete L -norm that associates only the normal vector comonents at the midoint edges and then normalized by L and l -norm in time. The convergence rate is illstrated in Figre 3. Figre : xamle of non-matching grids for sbdomains. 16

17 Level h H error error 1 1/5 1/6 1/ e e-01 1/100 1/50 1/ e e /10 1/60 1/10.79e e-0 4 1/15 1/76 1/150.6e e-0 Table 1: Accracy reslts of ressre and velocity for varios levels. Figre 3: Convergence of the ressre and velocity error. 4. Nmerical examle We se manfactred the known soltion x,y,t = e t sinπxsinπy and se it to comte the forcing f, the Dirichlet bondary data g, and the initial data 0. The time ste is eqal to the root of the coarse mesh size. Ths first order convergence is exected from theoretical reslt. Sch mesh discretization are deicted in the Figre. The simlation time interval is 0,, i.e. T =, and we se the Backward ler method to integrate with regard to time with niform time ste, see Table. The convergence rate is illstrated in Figre 4. Level h H error error 1 1/100 1/50 1/7 7.8e e-01 1/18 1/64 1/8 6.38e e /164 1/8 1/9 5.69e e /00 1/100 1/ e e-01 Table : Accracy reslts of ressre and velocity for varios levels. 17

18 Figre 4: Convergence of ressre and velocity error. 5 Conclsion This research has rovided a riori error analysis for transient roblems or slightly comressible flow roblems throgh the heterogeneos oros media sing nhanced Velocity scheme as the domain decomosition method in sace that coled with backward ler or Crank-Nicolson method in the time setting. In these discretization settings, we obtained the first order convergence rate for the backward ler method and the second order convergence rate for the Crank-Nicolson method. Nmerical exeriments are rovided. The reslts sggest that this aroaches cold also be sefl for the engineering sbsrface alications inclding CO seqestration, etc. In or ftre research, we lan to concentrate on arareal algorithms to achieve efficiency in time discretization that allow to rn simlation efficiently for the long time range. Acknowledgements First athor thanks Drs. T. Arbogast and I. Yotov for some helfl discssions dring analysis of method. References [1] Yerlan Amanbek, Grreet Singh, Mary F Wheeler, and Hans van Dijn. Adative nmerical homogenization for scaling single hase flow and transort. ICS Reort, 1:17, 017. [] Grreet Singh, Yerlan Amanbek, and Mary F Wheeler. Adative homogenization for scaling heterogeneos oros medim. In SP Annal Technical Conference and xhibition. Society of Petrolem ngineers,

19 [3] John A Wheeler, Mary F Wheeler, and Ivan Yotov. nhanced velocity mixed finite element methods for flow in mltiblock domains. Comtational Geosciences, [4] Snil G Thomas and Mary F Wheeler. nhanced velocity mixed finite element methods for modeling coled flow and transort on non-matching mltiblock grids. Comtational Geosciences, 154:605 65, 011. [5] Benjamin Ganis, Gergina Pencheva, and Mary F Wheeler. Adative mesh refinement with an enhanced velocity mixed finite element method on semi-strctred grids sing a flly coled solver. Comtational Geosciences, ages 1 0, 018. [6] Yerlan Amanbek. ew adative modeling of flow and transort in oros media sing an nhanced Velocity scheme. PhD thesis, 018. [7] Grreet Singh and Mary F Wheeler. A sace time domain decomosition aroach sing enhanced velocity mixed finite element method. arxiv rerint arxiv: , 018. [8] Grreet Singh and Mary F Wheeler. A domain decomosition aroach for local mesh refinement in sace and time. arxiv rerint arxiv: , 018. [9] Yerlan Amanbek, Grreet Singh, and Mary F. Wheeler. Selective time-steing adativity for non-linear reactive transort roblems. SIAM CS 17. htts://doi. org/ /m9.figshare v4, 017. [10] Vidar Thomée. Galerkin finite element methods for arabolic roblems, volme Sringer, [11] Donald W Peaceman. Interretation of well-block ressres in nmerical reservoir simlation with nonsqare grid blocks and anisotroic ermeability. Society of Petrolem ngineers Jornal, 303: , [1] Mary F Wheeler. A riori l error estimates for galerkin aroximations to arabolic artial differential eqations. SIAM Jornal on Nmerical Analysis, 104:73 759, [13] Béatrice Rivière and Mary F Wheeler. A discontinos galerkin method alied to nonlinear arabolic eqations. In Discontinos Galerkin methods, ages Sringer, 000. [14] Thomas F Rssell and Mary Fanett Wheeler. Finite element and finite difference methods for continos flows in oros media, ages SIAM,