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 displacement1 Introduction
We consider the problem of miscible displacement which has considerable and practical importance in petroleum engineering. This problem can be considered as the result of advectivediffusive 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 formulation. Recently, discontinuous Galerkin for miscible displacement has been investigated by numerical experiments and was reported to exhibit good numerical performance [3,4]. In HughesMasudWan [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 semidiscrete discontinuous Galerkin methods with interior penalty are proposed to solve the coupled system of flow and reactive transport in porous media, which arises from many applications including miscible displacement and acidstimulated flow. In [8], stable CrankNicolson 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 [1012] to conservation law and system of conservation 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 of these advantages, the discontinuous Galerkin method has become a very active area of research [47,1318]. 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, we develop a mixed, stabilized, discontinuous Galerkin formulation. To some extent, we develop a more general stabilized formulation and because of the proper choose of the parameters γ and β, this paper includes the methods of [2,6] and [5]. All the schemes are stable for any combination 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 successful 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 modeling 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 velocity 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, combined with a mixed, stabilized and discontinuous Galerkin method. The boundedness and stability of the finite element formulation are studied in Section 6. Error estimates for the incompressible miscible displacement problem are obtained in Section 7.
Throughout the paper, we denote by C a generic positive constant that is independent 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 ∈ R^{d}(d = 2, 3) over time interval J = (0, T] is modeled by the system concentration equation:
Pressure equation:
The initial conditions
The noflow boundary conditions
Dispersion/diffusion tensor
where the unknowns are p (the pressure in the fluid mixture), u (the Darcy velocity of the mixture, i.e., the volume of fluid flowing cross a unit acrosssection 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). ϕ = ϕ(x) is the porosity of the medium, uniformly bounded above and below by positive numbers. The E(u) is the tensor that projects onto the u direction, whose (i,j) component is ; 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 imposed 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 concentration 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) be a sequence of finite element partitions of Ω. Let Γ_{I }denote the set of all interior edges, Γ_{B }the set of the edges e on ∂Ω, and Γ_{h }= Γ_{B }+ Γ_{I}. K^{+}, K^{ }be two adjacent elements of T_{h}; let x be an arbitrary point of the set e = ∂K^{+ }∩ ∂K^{}, which is assumed to have a nonzero (d  1) dimensional measure; and let n^{+}, n^{ }be the corresponding outward unit normals at that point. Let (u, p) be a function smooth inside each element K^{± }and let us denote by (u^{±}, p^{±}) the traces of (u, p) on e from the interior of K^{±}. Then we define the mean values {{·}} and jumps [[·]] at x ∈ {e} as
For e ∈ Γ_{B}, the obvious definitions is {{p}} = p, [[u]] = u·n, with n denoting the outward unit normal vector on ∂Ω. we define the set 〈K, K'〉 as
For s ≥ 0, we define
The usual Sobolev norm on Ω is denoted by ·_{m, Ω }[19]. The broken norms are defined, for a positive number m, as
The discontinuous finite element space is taken to be
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 cutoff operator is defined as
where M is a large positive constant. By a straightforward argument, we can show that the cutoff operator is uniformly Lipschitz continuous in the following sense.
Lemma 3.1 [7] (Property of operator ) The cutoff operator defined as in Equation 3.4 is uniformly Lipschitz continuous with a Lipschitz constant one, that is
We shall also use the following inverse inequalities, which can be derived using the method in [20]. Let K ∈ T_{h}, v ∈ 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
3.2 Discontinuous Galerkin schemes
Let ∇_{h }· v and ∇_{h}v be the functions whose restriction to each element are equal to ∇ · v, ∇v, respectively. We introduce the bilinear form B(c, w; u) and the linear functional L(w; u, c)
with
here c_{11 }> 0 is a constant independent of the meshsize.
We now define the weak formulation on which our mixed discontinuous method is based
Let N be a positive integer, and t_{m }= mΔt for m = 0, 1, ..., N. The approximation 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 }∈ W^{1,∞}(0, T; D_{k1}(T_{h})) satisfying
where with the DG velocity u_{h }defined below
4 A stabilized mixed DG method for the velocity and pressure
4.1 Elimination for the flux variable u
Letting α(c) = a(c)^{1}. For the velocity and pressure, we define the following forms
The discrete problem for the velocity and pressure can be written as: find u_{h }∈ (D_{l2}(T_{h}))^{d}, (l ≥ 2), p_{h }∈ D_{l1}(T_{h}) such as
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}:
Using (4.4) we have
Substituting (4.5) in the first equation of (4.3) we obtain
We introduce the lift operator R:L^{1}(∪∂K) → (D_{l2}(T_{h}))^{d }defined by
From (4.6) and (4.7) we have
We also introduce the L^{2}projection π onto (D_{l2}(T_{h}))^{d}
Equation 4.8 gives now
Noting that ∇_{h}D_{l1}(T_{h}) ⊂ (D_{l2}(T_{h}))^{d}, we have π∇_{h}p_{h }≡ ∇_{h}p_{h }for all p_{h }∈ D_{l1}(T_{h}). The Equation 4.10 gives
Using (4.5) and the lifting operator R defined in (4.7) we have
Substituting (4.12) in the second equation of (4.3) and using (4.11) we have
For future reference, it is convenient to rewrite (4.13) as follows
where
4.2 Stabilization of formulation (4.3)
We write first (4.3) in the equivalent form: find (u_{h}, p_{h}) ∈ (D_{l2}(T_{h}))^{d }× D_{l1}(T_{h}) such that
where
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 jumps of u. In [2], Masud and Hughes introduced a stabilized finite element formulation in which an appropriately weighted residual of the Darcy law is added to the standard mixed formulation. In HughesMasudWan [5], the method was extend within the discontinuous Galerkin framework. A family of mixed finite element discretizations of the Darcy flow equations using totally discontinuous elements was introduced in [6]. In this paper, we consider the following stabilized formulation which includes the methods of [2,6] and [5].
The stabilized formulation of (4.16) is
where
where γ and β are chosen as the following (i) γ = 1, β = 1. (ii) γ = 0, β = 1, δ could assume either the value +1 or the value 1. The definition of θ will be given in the following content.
5 A mixed stabilized DG method for the incompressible miscible displacement problem
By combining (3.8) with (4.18), we have the stabilized DG for the approximating (2.1)(2.5): seek c_{h }∈ W^{1,∞}(0, T; D_{k1}(T_{h})) =: W_{h}, p_{h }∈ W^{1,∞}(0, T; D_{l1}(T_{h})) =: Q_{h }and u_{h }∈ (W^{1,∞}(0, T; D_{l2}(T_{h})))^{d }=: V_{h }satisfying
We define the "stability norm" by
where
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
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
Lemma 6.3 [6]Let be a Hilbert spaces, and λ and μ positive constants. Then, for every ξ and η in we have
Theorem 6.1 (Stability) For δ = 1, problem (4.18) is stable for all θ ∈ (0,1).
Proof Consider first the case γ = 1, β = 1. From the definition of A_{stab}(·,·;·,·;·), we have
We remark that (6.4) can be rewritten as
and the stability in the norm (5.2) follows from .
Consider now the case γ = 0, β =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 γ = 0 can be rewritten as: find u_{h }∈ (D_{l2}(T_{h}))^{d}, p_{h }∈ D_{l1}(T_{h}) such that
From the first equation in (6.6) and (4.9) we have
Substituting the expression (6.7) in the second equation of (6.6) for δ = 1, we have
Denote by B_{1h}(·,·) the bilinear form (6.8), we have
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 γ = 1, β = 1. The problem (4.18) for δ = 1 reads
Using the arithmeticgeometric mean inequality, we have
and since θ < 0 the result follows.
Consider now the case γ = 0, β = 1. From (6.7) the second equation of (6.6) for δ = 1 can be written as
We remark that formulation (6.12) can be rewritten as
where A_{BO}(p_{h}, ψ) is introduced by Baumann and Oden [14], and given by
Denote by B_{2h}(·,·) the bilinear form (6.13), we have
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 solution of (2.1)(2.5) and are essentially bounded, then
provided that the constant M for the cutoff 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
and ∃C > 0 such that
where (6.17) clearly holds for every θ ∈ (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_{l1}(T_{h}) ⊂ (D_{l2}(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
7 Error estimates
Let be an interpolation of the exact solution (u, p, c) such that
Let us define interpolation errors, finite element solution errors and auxiliary errors
It was proven in [18] that
hold for all t ∈ J with the constant C independent only on bounds for the coefficient α(c), but not on c itself.
Theorem 7.1 (Error estimate for the velocity and pressure) Let (u, p, c) be the solution to (2.1)(2.5), and assume p ∈ L^{2}(0, T; H^{l}(T_{h})), u ∈ (L^{2}(0, T; H^{l1}(T_{h})))^{d }and c ∈ L^{2}(0, T; H^{k}(T_{h})). We further assume that p, ∇p, c and ∇c are essentially bounded. If the constant M for the cutoff operator is sufficiently large, then there exists a constant C independent of h such that
Proof For the sake of brevity we will assume in the following content. Consider the case γ = 1, β = 1. From the second equation of (5.1) and (6.16) we have
That is
Choosing v = ξ_{1}, ψ = η_{1 }and splitting e_{p }according e_{p }= η_{1} η_{2}, from (7.1) and we obtain
Let us first consider the left side of error equation (7.5)
We know that (7.2) and quasiregularity that are bounded in L^{∞}(Ω). So the right side of the error equation (7.5) can be bounded from below. Noting that , we have
The second and the third terms of the right side of the error equation (7.5) can be bounded using CauchySchwartz inequality and approximation results,
The fourth term can be bounded in a similar way as that for the first term
The last two terms can be bounded as follows
Substituting all these inequalities into Equation 7.5, we have
The theorem follows from the triangle inequality.
Now consider the case γ = 0, β = 1. The bilinear form (6.8) from the second equation of (5.1) for reads
where
Replacing (6.8) with p_{h }= p and subtracting it from (7.13) we finally obtain
Choosing ψ = η_{1}, we have
Let us first estimate the left side of (7.15). From (6.17) and using the fact , we have
The first and the second terms of the right side of (7.15) can be bounded using Lemma 6.1 and (3.5)
Note that are bounded in L^{∞}(Ω) and , we have
Substituting all these inequalities into the (7.15) and using the triangle inequality we have
We easily deduce,using (7.19)
which completes the proof. □.
From [7], we state two lemmas for the properties of the dispersiondiffusion tensor, which will be used to prove error estimates for the concentration.
Lemma 7.1 [7] (Uniform positive definiteness of D(u)) Let D(u) defined as in Equation 2.6, where ϕd_{m}(x) ≥ 0, d_{l}(x) ≥ 0 and d_{t}(x) ≥ 0 are nonnegative functions of x ∈ Ω. Then
If, in addition, φd_{m}(x) ≥ d_{m,* }> 0 uniformly in the domain Ω, then D(u) is uniformly positive definite in Ω:
Lemma 7.2 [7] (Uniform Lipschitz continuity of D(u)) Let D(u) defined as in Equation 2.6, where d_{m}(x) ≥ 0, d_{l}(x) ≥ 0 and d_{t}(x) ≥ 0 are nonnegative functions of x ∈ Ω, and the dispersivities d_{l }and d_{t }is uniformly bounded, i.e., d_{l}(x) ≤ d_{l}^{* }and d_{t}(x) ≤ d_{t}^{*}. Then
where is a fixed number (d = 2 or 3 is the dimension of domain Ω).
Theorem 7.2 (Error estimate for concentration) Let (u, p, c) be the solution to (2.1)(2.5), and assume p ∈ L^{2}(0, T; H^{l}(T_{h})), u ∈ (L^{2}(0, T; H^{l1}(T_{h})))^{d }and c ∈ L^{2}(0, T; H^{k}(T_{h})). We further assume that p, ∇p, c and ∇c are essentially bounded. If the constant M for the cutoff operator is sufficiently large, then there exists a constant C independent of h and Δt such that
Proof The first equation of (5.1) is
It can be written as
Subtracting the DG scheme equation from the weak formulation, we have for any w ∈ D_{k1}(T_{h})
that is
Choosing w = τ_{1}^{n+1}, we obtain
Let us first consider the left side of the error equation (7.25). The first term can be bounded as
The second term of Equation 7.25 is
The second term of B(·,·;·) can be estimated using the boundedness of u_{M }and :
Thus
Let us bound the right side of the error equation (7.25).
Using Taylor series expansion, we have
The fourth term in the right side of the error equation (7.25) is
Terms T_{1 }through T_{3 }can be bounded by using CauchySchwartz inequality and approximation results,
and
Terms T_{4 }and T_{5 }can be estimated using inverse inequalities,
and
Using CauchySchwartz inequality and the trace inequality, we have
Noting that [[c^{n+1}]] = 0, if the constant M for the cutoff operator is sufficiently large, we write the last two terms in the right side of the error equation (7.25) as
Noting that pointwise if the constant M for the cutoff operator is sufficiently large, we can bound term S_{1 }as
Term S_{2 }can be bounded in a similar way as that for S_{1}
Term S_{3 }can be bounded using the penalty term and continuity of dispersiondiffusion tensor
Combining all the terms in (7.25), we have
Suppose that m is an integer, 0 ≤ m ≤ N  1. Multiplying by 2Δt, summing from n = 0 to n = m, we obtain
The theorem follow from (7.3), the discrete Gronwall's lemma and the triangle inequality. □
Theorem 7.3 (Error estimate for flow in coupled system) Let (u, p, c) be the solution to (2.1)(2.5), and assume p ∈ L^{2}(0, T; H^{l}(T_{h})), u ∈ (L^{2}(0, T; H^{l1}(T_{h})))^{d }and c ∈ L^{2}(0, T; H^{k}(T_{h})). We further assume that p, ∇p, c and ∇c are essentially bounded. If the constant M for the cutoff operator is sufficiently large, then there exists a constant C independent of h and Δt such that
Proof Taking L^{∞ }norm with time in (7.3), we have
Substituting (7.24) into the above inequality, we obtain (7.39). □
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
YL participated in the design and theoretical analysis of 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.
Acknowledgments
The work was supported by National Natural Science Foundation of China (Grant Nos.11101069, Grant Nos.11126105)and the Youth Research Foundation of Sichuan University (no. 2009SCU11113).
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