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RESEARCH Open Access A stabilized mixed discontinuous Galerkin method for the incompressible miscible displacement problem Yan Luo 1 , Minfu Feng 2 and Youcai Xu 2* * Correspondence: xyc@scu.edu.cn 2 School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China Full list of author information is available at the end of the article Abstract A new fully discrete stabilized discontinuous Galerkin method is proposed to solve the incompressible miscible displacement problem. For the pressure equation, we develop a mixed, stabilized, discontinuous Galerkin formulation. We can obtain the optimal priori estimates for both concentration and pressure. Keywords: Discontinuous Galerkin methods, a priori error estimates, incompressible miscible displacement 1 Introduction We consider the problem of miscible displacement which has co nsiderable and practi- cal importance in petroleum engineering. This problem can be considered as the result of advective-diffusive equation for concentrations and the Darcy flow equation. The more popular approach in application so far has been based on the mixed formulation. In a previous work, Douglas and Roberts [1] presented a mixed finite element (MFE) method for the compressible miscible displacement problem. For the Darcy flow, Masud and Hughes [2] introduced a stabilized finite element formulation in which an appropriately weighted residual of the Darcy law is added to the standard mixed for- mulation. Recently, discontinuous Galerkin for mi scible displacement has been investi- gated by numerical experiments and was reported t o exhibit good numerical performance [3,4]. In Hughes-Masud-Wan [5], the method of [2] was extended to the discontinuous Galerkin framework for the Darcy flow. A family of mixed finite element discretizations of the Darcy flow equations using totally discontinuous elements was introduced in [6]. In [7] primal semi-disc rete discontinuous Galerkin methods with interior penalty are proposed to so lve the coupled system of flow and reactive trans- port in porous media, which arises from many applications including miscible displace- ment and acid-stimulated flow. In [8], stable Crank-Nicolson discretization was given for incompressible miscible displacement problem. The discontinuous Galerkin (DG) method was introduced by Reed and Hill [9], and extended by Cockburn and Shu [10-12] to conservation law and system of conserva- tion laws,respectively. Due to localizability of the discontinuous Galerkin method, it is easy to construct higher order element to obtain higher order accuracy and to derive highly parallel algorithms. Because o f these advantages, the discontinuous Galerkin Luo et al. Boundary Value Problems 2011, 2011:48 http://www.boundaryvalueproblems.com/content/2011/1/48 © 2011 Luo et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Common s Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited . method has become a very active area of research [4-7,13-18]. Most of the literature concerning discontinuous Galerkin methods can be found in [13]. In this paper, we analyze a fully discrete finite element method with the stabilized mixed discontinuous Galerkin methods for the incompressible miscible displacement problem in porous media. For the pressure equation, w e develop a mix ed, stabilized, discontinu ous Galerkin formulation. To some extent, we develop a more general stabi- lized formulation and because of the proper choose of the parameters g and b,this paper includes the methods of [2,6] and [5]. All the schemes are stable for any combi- nation of discontinuous discrete concentration, velocity and pressure spaces. Based on our results, we can assert that the mixed stabilized discontinuous Galerkin formulation of the incompressible miscible displacement problem is mathematically viable, and we also believe it may be practically useful. It generalizes and encompasses all the success- ful elements described in [2,6] and [5]. Optimal error estimate are obtained for the concentration, velocity and pressure. An outline of the remainder of the paper follows: In Section 2, we describe the mod- eling equations. The DG schemes for the concentration and some of their properties are introduced in Section 3. Stabilized mixed DG methods are introduced for the velo- city and pressure in Section 4. In Section 5, we propose the numerical approximation scheme of incompressible miscible displacement problems with a fully discrete in time, comb ined with a mixed, stabilized and discontinuous Galerkin method. The bounded- ness and stab ility of the finite element formulation are studied in Section 6. Error esti- mates for the incompressible miscible displacement problem are obtained in Section 7. Throughout the paper, we denote by C a generic positive constant that is indepen- dent of h and Δt, but might depend on the partial differential equation solution; we denote by ε a fixed positive constant that can be chosen arbitrarily small. 2 Governing equations Miscible displacement of one incompressible fluid by another in a porous medium Ω Î R d ( d =2,3)overtimeintervalJ =(0,T] is modeled by the system concentration equation: φ ∂c ∂t + u ·∇c −∇·(D(u)∇c)=qc ∗ ,(x, t) ∈  × J . (2:1) Pressure equation: u = −a ( c ) ∇p, ( x, t ) ∈  × J , (2:2) ∇ · u = q, ( x, t ) ∈  × J . (2:3) The initial conditions c ( x,0 ) = c 0 ( x ) , x ∈  . (2:4) The no-flow boundary conditions  u ·n =0, x ∈ ∂ , (D(u)∇c − cu) · n =0, x ∈ ∂ . (2:5) Luo et al. Boundary Value Problems 2011, 2011:48 http://www.boundaryvalueproblems.com/content/2011/1/48 Page 2 of 17 Dispersion/diffusion tensor D ( u ) = φ d m I + |u| (d l E ( u ) + d t ( I − E ( u ))), (2:6) where the unknowns are p (the pressure i n the fluid mixture), u (the Darcy velocity ofthemixture,i.e.,thevolumeoffluidflowing cross a unit across-section per unit time) and c (the concentration of the interested species, i.e., the amount of the species per unit volume of the fluid mixture). j = j(x) is the porosity of the medium, uni- formly bounded above and below by positive numbers. The E(u) is the tensor that pro- jects onto the u direction, whose (i,j) component is (E(u)) ij = u i u j | u | 2 ; d m is the molecular diffusivity and assumed to be strictly positive; d l and d t are the longitudinal and the transverse dispersivities, respectively, and are assumed to be nonnegative . The impos ed external total flow rate q is sum of sources (injection) and sinks (extraction) and is assumed to be bounded. Concentration c* in the source term is the injected concentra- tion c w if q ≥ 0 and is the resident concentration c if q < 0. Here, we assume that the a (c) is a globally Lipschitz continuous function of c, and is uniformly symmetric positive definite and bounded. 3 Discontinuous Galerkin method for the concentration 3.1 Notation Let T h =(K)beasequenceoffiniteelementpartitionsofΩ.LetΓ I denote the set of all interior edges, Γ B the set of the edges e on ∂Ω,andΓ h = Γ B + Γ I . K + , K - be two adjacent elements of T h ;letx be an arbitrary point of the set e = ∂K + ∩ ∂K - ,whichis assumed to have a nonzero (d - 1) dimensional measure; and let n + , n - be the corre- sponding outward unit normals at that point. Let (u, p) be a function smooth insid e each element K ± and let us denot e by (u ± , p ± ) the traces of ( u, p)one from the inter- ior of K ± . Then we define the mean values {{·}} and jumps [[·]] at x Î {e}as  [u]  = u + ·n + + u − ·n − , {{u}} = 1 2 (u + + u − ), {{p}} = 1 2 (p + + p − ), [[p]] = p + n + + p − n − . For e Î Γ B , the obvious definitions is {{p}} = p,[[u]] = u·n,withn denoting the out- ward unit normal vector on ∂Ω. we define the set 〈K, K’〉 as K, K   :=   if meas d−1 (∂K ∩∂K  )=0 , interior of ∂K ∩ ∂K  otherwise. For s ≥ 0, we define H s ( T h ) = {v ∈ L 2 (  ) : v| K ∈ H s ( K ) .K ∈ T h } . (3:1) The usual Sobolev norm on Ω is denoted by ||·|| m, Ω [19]. The broken norms are defined, for a positive number m,as |v| 2 m =  K∈T h v 2 m,K . (3:2) The discontinuous finite element space is taken to be D r ( T h ) = {v ∈ L 2 (  ) : v| K ∈ P r ( K ) , K ∈ T h } , (3:3) Luo et al. Boundary Value Problems 2011, 2011:48 http://www.boundaryvalueproblems.com/content/2011/1/48 Page 3 of 17 where P r (K) denotes the space of polynomials of (total) degree less than or equal to r (r ≥ 0) on K. Note that we present error estimators in this paper for the local space P r ,but the results also apply to the local space Q r (the tensor product of the polynomial spaces of degree less than or equal to r in each spatial dimension) because P r (K) ⊂ Q r (K). The cut-off operator M is defined as M (c)(x) = min(c(x), M), M(u)(x)=  u(x)if|u(x)|≤M, M u(x)/|u(x)| if |u(x)| > M , (3:4) where M is a large positive constant. By a straightforward argument, we can show that the cut-off operator M is uniformly Lipschitz continuous in the following sense. Lemma 3.1 [7] (Property of operator M ) The cut-off operator M defined as in Equa- tion 3.4 is uniformly Lipschitz continuous with a Lipschitz constant one, that is M(c) −M(w) L ∞ () ≤c − w L ∞ () , ∀c ∈ L ∞ (), w ∈ L ∞ (),  M(u) − M(v) ( L ∞ (  )) d ≤u −v ( L ∞ (  )) d , ∀u ∈ (L ∞ ()) d , v ∈ (L ∞ ()) d . We shall also use the following inverse inequalities, which can be derived using the method in [20]. Let K Î T h , v Î P r (K) and h K is the diameter of K. Then there exists a constant C independent of v and h K , such that  D q v 0,∂K ≤ Ch −1/2 K D q v K , q ≥ 0 . D q+1 v 0,K ≤ Ch −1 K D q v 0,K , q ≥ 0 . (3:5) 3.2 Discontinuous Galerkin schemes Let ∇ h · v and ∇ h v be the functions whose restriction to each element K ∈ are equal to ∇ · v, ∇v, respectively. We introduce the bilinear form B(c, w; u) and the linear func- tional L(w; u, c) B(c, w; u)=(D(u)∇ h c, ∇ h w)+   h {{D(u)∇ h w}}[[c]]ds −   h {{D(u)∇ h c}}[[w]]d s +   h C 11 [[c]][[w]]ds +(u ·∇ h c, w) −   cq − wdx, L(w; u, c)=   c w q + wdx, with C 11 =  c 11 max{h −1 K + , h −1 K − } x ∈K + , K − , c 11 h −1 K + x ∈ ∂K + ∩ ∂ , (3:6) here c 11 > 0 is a constant independent of the meshsize. We now defi ne the weak form ulatio n on which our mixed discontinuous method is based ( φc t , w ) + B ( c, w; u ) = L ( w; u, c ) , ∀w ∈ H k ( T h ). (3:7) Let N be a positive integer,  t = T N and t m = mΔt for m = 0, 1, , N.Theapproxi- mation of c t at t = t n+1 can be discreted by the forward difference. The DG schemes for approximating concentration are as follows. We seek c h Î W 1,∞ (0, T; D k-1 ( T h )) Luo et al. Boundary Value Problems 2011, 2011:48 http://www.boundaryvalueproblems.com/content/2011/1/48 Page 4 of 17 satisfying (φ c n+ 1 h − c n h t , w h )+B(c n+1 h , w h ; u n M )=L(w h ; u n M , c n+1 h ), ∀w h ∈ W 1,∞ ( 0, T; D k−1 ( T h )), (3:8) where u n M = M(u n h ) with the DG velocity u h defined below u n h = −a(M(c n h ))∇p n h , x ∈ K, K ∈ T h . 4 A stabilized mixed DG method for the velocity and pressure 4.1 Elimination for the flux variable u Letting a(c)=a(c) -1 . For the velocity and pressure, we define the following forms a ( u, v; c ) = ( α ( c ) u, v ) , (4:1) b (p, v )=(p, ∇ h · v) −   I {{p}}[[v]]ds −   B {{v}}[[p]]ds . (4:2) The discrete problem for the velocity and pressure can be written as: find u h Î (D l-2 (T h )) d ,(l ≥ 2), p h Î D l-1 (T h ) such as  a(u h , v; c) −b(p h , v)=0, ∀v ∈ (D l−2 (T h )) d , b(ψ, u h )=(ψ, q), ∀ψ ∈ D l−1 (T h ). (4:3) In order to eliminate the flux variable, we first recall a useful identity, that holds for vectors u and scalars ψ piecewise smooth on T h :  K∈T h  ∂K v · nψds =   h {{v}} · [[ψ]]ds +   I [[v]]{{ψ}}ds . (4:4) Using (4.4) we have  K  K (∇·u h ψ + u h ·∇ψ)dx =   h {{u h }} · [[ψ]]ds +   I [[u h ]]{{ψ}}ds . (4:5) Substituting (4.5) in the first equation of (4.3) we obtain (α(c)u h + ∇ h p h , v) −   I [[p h ]] ·{{v}}ds =0 . (4:6) We introduce the lift operator R:L 1 (∪∂K) ® (D l-2 (T h )) d defined by   R[[ψ]] · vdx = −   I [[ψ]] ·{{v}}ds, ∀v ∈ (D l−2 (T h )) d . (4:7) From (4.6) and (4.7) we have ( α ( c ) u h + ∇ h p h + R[[p h ]], v ) =0 . (4:8) We also introduce the L 2 -projection π onto (D l-2 (T h )) d ( πw, v ) = ( w, v ) , ∀v ∈ ( D l−2 ( T h )) d . (4:9) Luo et al. Boundary Value Problems 2011, 2011:48 http://www.boundaryvalueproblems.com/content/2011/1/48 Page 5 of 17 Equation 4.8 gives now α( c ) u h = − ( π∇ h p h + R [[ p h ]]). (4:10) Noting that ∇ h D l-1 (T h ) ⊂ (D l-2 (T h )) d , we have π∇ h p h ≡ ∇ h p h for all p h Î D l-1 (T h ). The Equation 4.10 gives α( c ) u h = − ( ∇ h p h + R[[p h ]] ). (4:11) Using (4.5) and the lifting operator R defined in (4.7) we have b (ψ, u h )=−(u h , ∇ h ψ)+   I [[ψ]] ·{{u}}ds , = − ( u h , ∇ h ψ + R[[ψ]] ) . (4:12) Substituting (4.12) in the second equation of (4.3) and using (4.11) we have ( a ( c )( ∇ h p h + R[[p h ]] ) , ∇ h ψ + R[[ψ]] ) = ( q, ψ ). (4:13) For future reference, it is convenient to rewrite (4.13) as follows A BR ( p h , ψ ) = ( q, ψ ) , ∀ψ ∈ D l−1 ( T h ), (4:14) where A BR (p h , ψ)=(a(c)(∇ h p h + R[[p h ]]), ∇ h ψ + R[[ψ]]). (4:15) 4.2 Stabilization of formulation (4.3) We write first (4.3) in the equivalent form: find (u h , p h ) Î (D l-2 (T h )) d × D l-1 (T h )such that A ( u h , v; p h , ψ;c ) = l ( ψ ) , ∀ ( v, ψ ) ∈ ( D l−2 ( T h )) d × D l−1 ( T h ), (4:16) where A ( u h , v; p h , ψ;c ) = a ( u h , v; c ) − b ( p h , v ) + b ( ψ, u h ) , l ( ψ ) = ( q, ψ ). (4:17) In a sense, (4.16) can be seen as a Darcy problem. The usual way to stabilized it is to introduce penalty terms on the jumps of p and/or on the j umps of u.In[2],Masud and Hughes introduced a stabilized finite element formulation in which an appropri- ately weighted residual of the Darcy law is added to the standard mixed formulation. In Hughes-Masu d-Wan [5], the method was extend within the discontinuous Galerkin framework. A family of mixed finite element discretizations of the Darcy flow equa- tions using totally discontinuous elements was introduced in [6]. In this paper, we con- sider the following stabilized formulation which includes the methods of [2,6] and [5]. The stabilized formulation of (4.16) is A stab ( u h , v; p h , ψ;c ) = l stab ( ψ ) , ∀ ( v, ψ ) ∈ ( D l−2 ( T h )) d × D l−1 ( T h ), (4:18) where A stab (u, v; p, ψ; c)=A(u, v; p, ψ; c)+γ e(p, ψ) +βθ  u + a(c)∇ h p, −α(c)v + δ∇ h ψ  , l stab (ψ)=l(ψ), e(p, ψ)=a(c)   h C 11 [[p]][[ψ]]ds, (4:19) Luo et al. Boundary Value Problems 2011, 2011:48 http://www.boundaryvalueproblems.com/content/2011/1/48 Page 6 of 17 where g and b are chosen as the following (i) g =1,b = 1. (ii) g =0,b =1,δ could assume either the value +1 or the value -1. The definition of θ will be given in the fol- lowing content. 5 A mixed stabilized DG method for the incompressible miscible displacement problem By combining (3.8) with (4.18), we have the stabil ized DG for the approximating (2.1)- (2.5): seek c h Î W 1,∞ (0, T; D k-1 (T h )) =: W h , p h Î W 1,∞ (0, T; D l-1 (T h )) =: Q h and u h Î (W 1,∞ (0, T; D l-2 (T h ))) d =: V h satisfying ⎧ ⎨ ⎩ (φ c n+1 h − c n h t , w)+B(c n+1 h , w; u n M )=L(w; u n M , c n+1 h ), ∀w ∈ W h , A stab (u n h , v; p n h , ψ;M(c n h )) = l stab (ψ), ∀(v × ψ) ∈ (V h × Q h ) . (5:1) We define the “stability norm” by (u, p) stab =  1 2 |α 1/2 (c)u| 2 0 + p 2 1,h  1/2 , (5:2) where p 2 1,h = 1 2 a 1/2 (c)∇ h p 2 0 + a 1/2 (c)[[p]] 2 0, h , a 1/2 (c)[[p]] 2 0, h =   h a(c)C 11 [[p]] · [[p]]ds, ∇ h p 2 0 =  K ∇p 2 0,K . (5:3) 6 Stability and consistency From [6], we can state the following results. Lemma 6.1 [6]There exist two positive constants C 1 and C 2 , depending only on the minimum angle of the decomposition and on the polynomial degree C 1 R[[ψ]] 2 0, ≤  e∈ I h −1 e [[ψ]] 2 0,e ≤ C 2 R[[ψ]] 0, . (6:1) Lemma 6.2 [6]There exists two positive constants C 1 and C 2 , depending only on the minimum angle of the decomposition such that C 1 R[[ψ]] 2 0, ≤  e∈ I h −1 e [[ψ]] 2 0,e ≤ C 2 (R[[ψ]] 2 0, + ∇ h ψ 2 0 ), ψ ∈ H 2 (T h ) . (6:2) Lemma 6.3 [6]Let H be a Hilb ert spaces, and l and μ positive constants. Then, for every ξ and h in H we have λξ + η 2 H + μη 2 H ≥ λμ 2 ( λ + μ ) (ξ 2 H + η 2 H ) . (6:3) Theorem 6.1 (Stability) For δ =1,problem (4.18) is stable for all θ Î (0,1). Proof Consider first the case g =1,b = 1. From the definition of A stab (·,·;·,·;·), we have A stab ( u h , u h ; p h , p h ; c ) = a ( u h , u h ; c ) + e ( p h , p h ) + θ ( u h + a ( c ) ∇ h p h , −α ( c ) u h + ∇ h p h ). (6:4) We remark that (6.4) can be rewritten as A stab (u h , u h ; p h , p h ; c)=(1−θ )|α 1/2 (c)u| 2 0 + θa 1/2 (c)∇ h p 2 0 + a 1/2 (c)[[p]] 2 0, h , (6:5) Luo et al. Boundary Value Problems 2011, 2011:48 http://www.boundaryvalueproblems.com/content/2011/1/48 Page 7 of 17 and the stability in the norm (5.2) follows from θ = 1 2 . Consider now the case g =0,b =1. Using the equivalent expressions (4.11) and (4.12) for the first and second equation of (4.3), respectively, the problem (4.18) for g = 0 can be rewritten as: find u h Î (D l-2 (T h )) d , p h Î D l-1 (T h ) such that  (α(c)u h + ∇ h p h + R[[p h ]], v) − θ (α(c)u h + ∇ h p h , v)=0, −( u h , ∇ h ψ + R[[ψ]]) + δθ(u h + a(c)∇ h ψ, ∇ h ψ)=(q, ψ) . (6:6) From the first equation in (6.6) and (4.9) we have α (c)u h = −(∇ h p h + 1 1 − θ R[[p h ]]) . (6:7) Substituting the expression (6.7) in the second equation of (6.6) for δ = 1, we have A BR (p h , ψ)+ θ 1 − θ   a(c)R[[p h ]] ·R[[ψ]]dx =(q, ψ), ∀ψ ∈ D l−1 (T h ) . (6:8) Denote by B 1h (·,·) the bilinear form (6.8), we have B 1h (ψ, ψ)=a(c)(∇ h ψ + R[[ψ]]) 0, + θ 1 − θ a(c) 1/2 R[[ψ]] 2 0, , (6:9) and the stability in the norm (5.3) follows from Lemma 6.1. This completes the proof. □ Theorem 6.2 For δ = -1, problem (4.18) is stable for all θ <0. Proof Consider first the case g =1,b = 1. The problem (4.18) for δ = -1 reads A stab (u h , u h ; p h , p h ; c)=a(u h , u h ; c)+θ(u h + a(c)∇ h p h , −α(c)u h −∇ h p h ) +e ( p h , p h ) . (6:10) Using the arithmetic-geometric mean inequality, we have A stab (u h , u h ; p h , p h ; c) ≥ (1 − 2θ)|α 1/2 (c)u| 2 0 − 2θ a 1/2 (c)∇ h p 2 0 +a 1/2 (c)[[p]] 2 0, h , (6:11) and since θ < 0 the result follows. Consider now the case g =0,b = 1. From (6.7) the second equation of (6.6) for δ = -1 can be written as A BR (p h , ψ)+ 2 θ 1 − θ (R[[p h ]], a(c)∇ h ψ)+ θ 1 − θ   a(c)R[[p h ]] · R[[ψ]]dx =(q, ψ ) . (6:12) We remark that formulation (6.12) can be rewritten as 1 1 − θ A BR (p h , ψ) − θ 1 − θ A BO (p h , ψ)=(q, ψ) , (6:13) where A BO (p h , ψ) is introduced by Baumann and Oden [14], and given by A BO (p h , ψ):=   a(c)(∇ h p h −R[[p h ]]) ·(∇ h ψ + R[[ψ]])dx +   a(c)R[[p h ]] ·R[[ψ]]dx . (6:14) Denote by B 2h (·,·) the bilinear form (6.13), we have B 2h (ψ, ψ)= 1 1 − θ a 1/2 (c)(∇ h ψ + R[[ψ]]) 0, − θ 1 − θ a 1/2 (c)∇ h ψ 2 0, , (6:15) Luo et al. Boundary Value Problems 2011, 2011:48 http://www.boundaryvalueproblems.com/content/2011/1/48 Page 8 of 17 and since θ < 0 the result follows again from Lemma 6.3 and 6.1. □ Theorem 6.3 (Consistency) If p,c and u are the solut ion of (2.1)-(2.5) and are essen- tially bounded, then  (φc t , w)+B(c, w; u)=L(w; u, c), ∀w ∈ L 2 (0, T; H k (T h )) A stab (u, v; p, ψ; c)=l stab (ψ), ∀(v × ψ) ∈ ((L 2 (0, T; H l−1 (T h ))) d × L 2 (0, T; H l (T h )) ) (6:16) provided that the constant M for the cut-off operator is sufficiently large. To summarize, for all the bilinear forms in (6.4), (6.10), (6.8) or (6.13) we have: ∃C > 0 such that B 1h (ψ, ψ) ≥ Cψ 2 1 , h , B 2h (ψ, ψ) ≥ Cψ 2 1 , h , ∀ψ ∈ D l−1 (T h ) , (6:17) and ∃C > 0 such that A(v, v; ψ, ψ; c) stab ≥ C(v, ψ) 2 stab , ∀(v, ψ) ∈ (D l−2 (T h )) d × D l−1 (T h ) , (6:18) where (6.17) clearly holds for every θ Î (0,1) for the case ((6.4 ), (6.8)), and for every θ < 0 for the case ((6.10), (6.13)). On the other hand, since ∇ h D l-1 (T h ) ⊂ (D l-2 (T h )) d holds, boundedness of the bilinear form in (6.8) and (6.13) follows directly from the boundedness of the bilinear forms A BR and A BO , as proved in [13], thanks to the equivalence of the norms (6.1) and (6.2). Thus, we have: ∃C > 0 such that B 1h ( p h , ψ ) ≤ Cp h  1,h ψ 1,h , B 2h ( p h , ψ ) ≤ Cp h  1,h ψ 1,h , ∀p h , ψ ∈ D l−1 ( T h ). (6:19) 7 Error estimates Let ( ˜u, ˜ p, ˜ c ) be an interpolation of the exact solution (u, p, c) such that ⎧ ⎨ ⎩ a( ˜u, v; c) − b( ˜ p, v)=0, ∀v ∈ (D l−2 (T h )) d , b(ψ, ˜ u)+e( ˜ p, ψ)=(q, ψ), ∀ψ ∈ D l−1 (T h ), ( ˜ c − c, w)=0, ∀w ∈ D k−1 (T h ). (7:1) Let us define interpolation errors, finite element solution errors and auxiliary errors ξ 1 = ˜u − u h , ξ 2 = ˜u − u, e u = u − u h = ξ 1 − ξ 2 ; η 1 = ˜ p − p h , η 2 = ˜ p − p, e p = p −p h = η 1 − η 2 ; τ 1 = ˜ c − c h , τ 2 = ˜ c − c, e c = c −c h = τ 1 − τ 2 . It was proven in [18] that |α 1/2 (c)ξ 2 | 2 0 + a 1/2 (c)[[η 2 ]] 2 0, h ≤ Ch 2l−2 (u 2 l−1 + p 2 l ) . (7:2) hold for all t Î J with the constant C independent only on bounds for the coefficient a(c), but not on c itself. Theorem 7.1 (Error est imate for the velocity and pressure) Let (u, p, c) be the solu- tion to (2.1)-(2.5), and assume p Î L 2 (0, T; H l (T h )), u Î (L 2 (0, T; H l-1 (T h ))) d and c Î L 2 (0, T; H k (T h )). We further assume that p, ∇p, cand∇c are essentially bounded. If the constant M for the cut-off operator is sufficiently large, then there exists a constant C independent of h such that (u −u h , p − p h ) 2 stab (t ) ≤ C(c −c h  2 0 (t )+h 2l−2 ) . (7:3) Luo et al. Boundary Value Problems 2011, 2011:48 http://www.boundaryvalueproblems.com/content/2011/1/48 Page 9 of 17 Proof Forthesakeofbrevitywewillassume θ = 1 2 , δ = 1 in the following content. Consider the case g =1,b = 1. From the second equation of (5.1) and (6.16) we have (α(c)u −α(M(c h ))u h , v) − b(p − p h , ψ)+b(ψ, u − u h )+e(p − p h , ψ ) − 1 2 ( u h + a(M( c h ))∇ h p h ), −α(M(c h ))v + ∇ h ψ) + 1 2 (u + a(c)∇ h p, −α(c)v + ∇ h ψ)=0. (7:4) That is (α(c)( u −˜ u ), v )+(α( M (c h ))( ˜ u − u h ), v )+((α(c) − α( M (c h ))) ˜ u , v ) − b(p − p h , v ) +b(ψ, u − u h )+e(p − p h , ψ)+ 1 2 (α( M (c h )) u h − α(c) u , v )+ 1 2 ( u − u h , ∇ h ψ) − 1 2 (∇ h p −∇ h p h , v )+ 1 2 (a(c)∇ h p − a( M (c h ))∇ h p h , ∇ h ψ)=0. Choosing v = ξ 1 , ψ = h 1 and splitting e p according e p = h 1 - h 2 ,from(7.1)andwe obtain 1 2 (α( M (c h ))ξ 1 , ξ 1 )+e(η 1 , η 1 )+ 1 2 (a( M (c h ))∇ h η 1 , ∇ h η 1 )= 1 2 ((α( M (c h )) −α(c)) ˜ u , ξ 1 ) − 1 2 (α(c)ξ 2 , ξ 1 )+ 1 2 (a(c)∇ h η 2 , ∇ h η 1 ) − 1 2 ((a(c) − a( M (c h )))∇ h ˜ p, ∇ h η 1 ) + 1 2 (ξ 2 , ∇ h η 1 ) − 1 2 (∇ h η 2 , ξ 1 ). (7:5) Let us first consider the left side of error equation (7.5) 1 2 (α( M(c h ))ξ 1 , ξ 1 )+e(η 1 , η 1 )+ 1 2 (a( M(c h ))∇ h η 1 , ∇ h η 1 ) = 1 2 (|α 1/2 (M(c h ))ξ 1 | 2 0 + a 1/2 (M(c h ))∇ h η 1  2 0 )+[[η 1 ]] 2 0, h . We know that (7.2) and quasi-regularity that ∇ h ˜ p , ˜ u are bounded in L ∞ (Ω). So the right side of the error equation (7.5) can be bounded from below. Noting that |α ( M ( c h )) − α ( c ) |≤C|c h − c | , we have | (α(M(c h )) − α(c)) ˜u, ξ 1 )|≤Cc −c h  2 0 + ε|ξ 1 | 2 0 . (7:6) The second and the third terms of t he right side of the error equation (7.5) can be bounded using Cauchy-Schwartz inequality and approximation results, | (α(c)ξ 2 , ξ 1 )|≤α(c) 0,∞ ξ 2  0 ξ 1  0 ≤ ε|ξ 1 | 2 0 + Ch 2l−2 , (7:7) | (a(c)∇ h η 2 , ∇ h η 1 )|≤ε∇ h η 1  2 0 + Ch 2l−2 . (7:8) The fourth term can be bounded in a similar way as that for the first term |(a(c) − a(M(c h ))∇ h ˜ p, ∇ h η 1 )|≤Cc − c h  2 0 + ε∇ h η 1  2 0 . (7:9) The last two terms can be bounded as follows (ξ 2 , ∇ h η 1 ) ≤ ε∇ h η 1  2 0 + Ch 2l−2 ,(∇ h η 2 , ξ 1 ) ≤ ε|ξ 1 | 2 0 + Ch 2l−2 . (7:10) Luo et al. Boundary Value Problems 2011, 2011:48 http://www.boundaryvalueproblems.com/content/2011/1/48 Page 10 of 17 [...]... for incompressible miscible displacement problem of low regularity SIAM J Numer Anal 47, 3720–3743 (2009) doi:10.1137/070712079 4 Rivière, B: Discontinuous Galerkin finite element methods for solving the miscible displacement problem in porous media Ph.D Thesis, The University of Texas at Austin (2000) 5 Hughes, TJR, Franca, LP, Balestra, M: A stabilized mixed discontinuous Galerkin method for Darcy... http://www.boundaryvalueproblems.com/content/2011/1/48 Page 17 of 17 20 Schwab, Ch: p-and hp-Finite Element Methods, Theory and Applications in Solid and Fluid Mechanics Oxford Science Publications, Oxford (1998) doi:10.1186/1687-2770-2011-48 Cite this article as: Luo et al.: A stabilized mixed discontinuous Galerkin method for the incompressible miscible displacement problem Boundary Value Problems 2011 2011:48 Submit your manuscript to a journal and... Rivière, B, Wheeler, MF: A combined mixed element and discontinuous Galerkin method for miscible displacement problem in porous media Proceedings of the International Conference on Recent Progress in Computational and Applied PDEs, Zhangjiaje 321–348 (2001) 19 Adams, RA: Sobolev Spaces Academic Press, San Diego, CA (1975) Luo et al Boundary Value Problems 2011, 2011:48 http://www.boundaryvalueproblems.com/content/2011/1/48... R: Stable Crank-Nicolson discretisation for incompressible miscible displacement problem of low regularity Numer Math Adv Appl 469–471 (2009) 9 Reed, WH, Hill, TR: Triangular mesh methods for the neutron transport equation Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory (1973) 10 Cockburn, B, Shu, CW: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation... the study, drafted the manuscript MF conceived the study, and participated in its design and coordination YX participated in the design and the revision of the study All authors read and approved the final manuscript Competing interests The authors declare that they have no competing interests Received: 8 May 2011 Accepted: 25 November 2011 Published: 25 November 2011 References 1 Douglas, J Jr, Roberts,... Nos.11101069, Grant Nos.11126105)and the Youth Research Foundation of Sichuan University (no 2009SCU11113) Author details 1 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China 2School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China Authors’ contributions YL participated in the design and theoretical analysis of the study,... element methods for the Stokes equations Chin J Numer Math and Appl 28, 163–174 (2006) 16 Luo, Y, Feng, M: A discontinuous element pressure gradient stabilization for the Stokes equations based on local projection Chin J Numer Math and Appl 30, 25–36 (2008) 17 Luo, Y, Feng, M: Discontinuous element pressure gradient stabilizations for compressible Navier-Stokes equations based on local projections Appl Math... )d3/2is a fixed number (d = 2 or 3 is the dimension of domain t l Ω) Theorem 7.2 (Error estimate for concentration) Let (u, p, c) be the solution to (2.1)(2.5), and assume p Î L2(0, T; Hl(Th)), u Î (L2(0, T; Hl-1(Th)))d and c Î L2(0, T; Hk (Th)) We further assume that p, ∇p, c and ∇c are essentially bounded If the constant M for the cut-off operator is sufficiently large, then there exists a constant C... Numerical methods for a model for compressible miscible displacement in porous media Math Comp 41, 441–459 (1983) doi:10.1090/S0025-5718-1983-0717695-3 2 Masud, A, Hughes, TJR: A stabilized mixed finite element method for Darcy flow Comput Methods Appl Mech Eng 191, 4341–4370 (2002) doi:10.1016/S0045-7825(02)00371-7 3 Bartels, S, Jensen, M, Müller, R: Discontinuous Galerkin finite element convergence for. .. estimate for flow in coupled system) Let (u, p, c) be the solution to (2.1)-(2.5), and assume p Î L2(0, T; Hl(Th)), u Î (L2(0, T; Hl-1(Th)))d and c Î L2(0, T; Hk(Th)) We further assume that p, ∇p, c and ∇c are essentially bounded If the constant M for the cut-off operator is sufficiently large, then there exists a constant C independent of h and Δt such that max (u − uh , p − ph ) 0≤t≤T stab (t) ≤ . spaces. Based on our results, we can assert that the mixed stabilized discontinuous Galerkin formulation of the incompressible miscible displacement problem is mathematically viable, and we also. advective-diffusive equation for concentrations and the Darcy flow equation. The more popular approach in application so far has been based on the mixed formulation. In a previous work, Douglas and Roberts [1]. an appropriately weighted residual of the Darcy law is added to the standard mixed for- mulation. Recently, discontinuous Galerkin for mi scible displacement has been investi- gated by numerical