Abstract We investigate the use of discontinuous Galerkin finite element methods in a multiphysics setting involving coupled flow and transport in porous media. We solve an elliptic equation for the fluid pressure using Nitsche’s method and an approximation, Σ, of the exact convection field σ will be constructed by interpolation onto the lowest-order Raviart-Thomas space of functions. We sequentially solve the transport equation, with the convection field Σ, for the fluid saturation by use of the lowest order discontinuous Galerkin method. We also supply numerical evidence of the importance of local conservation in this setting, and furthermore propose a line of argument indicating that if Σ is constructed using conservative fluxes, the modeling error σ − Σ may not have a great impact on the total error in certain quantities of interest. Acknowledgements Many thanks to my thesis supervisors Mats G. Larson and Fredrik Bengzon for their support, encouragement and insightful advice. Thanks to the additional members of the computational mathematics research group at Umeå University for creating a friendly and creative atmosphere to work in. Contents 1 Introduction 1 2 Problem Formulation 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Multiphysics in a Finite Element Setting . . . . . . . . . . . . . . . . . . . 2.3 A Coupled Flow and Transport Problem . . . . . . . . . . . . . . . . . . . 2 2 2 4 3 Discontinuous Galerkin Methods 3.1 dG for the Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . 3.2 dG for the Poisson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6 7 4 Discretization and some Standard Estimates 4.1 Discretization of the dG Transport Method . . . . . . . . . . . . . . . . . 4.2 Discretizising Nitsche’s Method . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Coupled Discretizised Problem . . . . . . . . . . . . . . . . . . . . . . 10 11 12 16 5 Application in Reservoir Simulation 5.1 Immiscible Two-Phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Quarter Five Spot Problem . . . . . . . . . . . . . . . . . . . . . . . . 21 21 23 6 Error Analysis 6.1 Numerical Evidence of the Importance of Local Conservation . . . . . . . . 6.2 Analytical Solution of the Transport Equation . . . . . . . . . . . . . . . . 6.3 An Error Representation Formula for the Transport Equation . . . . . . . 28 28 28 33 7 Conclusions 36 1 Introduction Due to the nature of convection dominated problems the design of stable numerical schemes for their solution is known to be difficult. There are two main reasons for this [1]. The first is that the analytical solutions to such problems may have discontinuities across the characteristics. The second is that the behavior of the solution can be very rich in the neighborhood of these discontinuities. The numerical scheme should thus be able to capture both discontinuities and richness while maintaining stability and accuracy in the approximation. The discontinuous Galerkin family of methods are known to handle this rough terrain quite well. Since the approximating space allows discontinuities across the element boundaries the inherent discontinuities in the exact solution can be captured without pushing the limits of the method too far. Another favorable property of the dG family of methods is that of local conservation. Locally conservative methods are generally regarded to be more stable than non conservative ones. The stability features of locally conservative methods is an open problem in the field, and additional research effort is needed for its full understanding. In this report we shall investigate the use of discontinuous Galerkin methods in a multiphysics setting involving coupled flow and transport in porous media. Our problem will consist of two linear partial differential equations; one elliptic for fluid pressure; and one hyperbolic, convection dominated transport equation for fluid saturation in space and time, where the convection field σ is proportional to the pressure gradient. We shall solve the elliptic equation using Nitsche’s method, and an approximation, Σ, of the pressure field will be constructed by interpolation onto the lowest-order Raviart-Thomas space of functions. The transport equation, with the convection field Σ, will be solved by use of the lowest order discontinuous Galerkin method. We shall furthermore supply numerical evidence of the importance of local conservation, and we shall propose a line of argument based on duality techniques indicating that if Σ is constructed using conservative fluxes, the modeling error σ − Σ may not have a great impact on the total error in certain quantities of interest. The report is organized as follows. We begin by building the scenery and briefly discuss numerical simulation involving several types of physics. We then state the coupled model problem formally, upon which the dG methods for the elliptic and hyperbolic equations are derived and tested individually. We continue by introducing the Raviart-Thomas space of functions which we use to connect the two stand alone solvers while maintaining local conservation. The physical motivation to the model problem from an oil reservoir simulation point of view is then accounted for, and the numerical part of the report is completed by a simulation of incompressible miscible flow. The report is concluded by an investigation of the local conservation property. 1 2 2.1 Problem Formulation Preliminaries We begin by stating some preliminary definitions and notation commonly used in this report. Without loss of generality we throughout regard Ω ⊂ R2 as the unit square Ω = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} and we denote by ∂Ω its boundary. We let K = {K} be a uniform partition of Ω into quadrilaterals K with side length h, and we let E denote the set of edges in K. For X ⊂ Ω we denote the L2 (X) inner product and norm by (·, ·)X and || · ||X respectively. In the case X = Ω we will simply write (·, ·) and || · ||. In order to formulate the discontinuous Galerkin method we also need some suitable spaces of functions. By the broken space on Ω we mean the mesh dependent Hilbert space H p (K) = {v ∈ L2 (Ω) : v|K ∈ H p (K), ∀K ∈ K}. (2.1) We see that H p (K) consists of the functions in L2 that elementwise belong to H p (K). For definitions of the L2 , and H p spaces we refer the reader to [2]. Furthermore we let Vh,k be the finite dimensional subspace of H p (K) consisting of the discontinuous piecewise polynomials of degree k. That is Vh,k = {v ∈ L2 (Ω) : v|K ∈ Pk (K)}. (2.2) Since our aim is to approximate partial differential equations with functions potentially discontinuous over the element boundaries we shall also need a couple of operators dealing with this situation. Following [3], for E ∈ E and v|E we define the average and the jump operators by + (v + v − )/2, E ⊂ Ω◦ , hvi = (2.3) v+, E ⊂ ∂Ω, and [v] = v + − v − , E ⊂ Ω◦ , v+, E ⊂ ∂Ω, (2.4) respectively, where v ± (x) = lim+ v(x ∓ ns), s→0 (2.5) where Ω◦ denotes the interior of Ω, n is the exterior unit normal to E for E ⊂ ∂Ω and a fixed, arbitrary unit normal to E, for E∩ ⊂ Ω◦ . 2.2 Multiphysics in a Finite Element Setting When simulating many real world physical phenomenon, one often has to take into account the interactions between several different types of physics present in the simulated model. For instance, when simulating a thermo elastic material one of the physics present is the heat in the material which leads to an expansion of the same. Such a simulation thus involves solving the heat equation for an input to the linear (or non linear) elasticity 2 n v + v − Figure 2.1: v + and v − equation. This is an example of a so called multiphysics problem. However, a multiphysics problem is not limited to the coupling of merely two equations. The general setting might involve several different physics, all interacting in a more or less complex fashion. Commonly, multiphysics problems are solved by connecting into a network a set of single physics solvers corresponding to the different types of physics in the system. In the basic setting, assume that a given physical phenomenon is described by two partial differential equations A and B, where B somehow depends on the solution a of A. Here we would have a solver SA for the PDE A, and a solver SB(a) for the PDE B = B(a). We now ask, what are the problems associated with the approximation of a solution to a multiphysics problem in this way? Well, for one a is not known. In our possession is however an approximation ā to a from the solver SA . Undoubtedly though, most of the time B(a) 6= B(ā), and thus we get a modeling error in B. This means, that in order to control the error in the solution to B(ā) we also need control over the error a − ā. Now, in the general setting the data exchange in a system of single physics solvers can become quite complex, and with this complexity comes the problem of guaranteeing both accuracy in the solution and efficient use of computational resources. Some of the problems might be very illconditioned for example, and perhaps rely heavily on accurate input data, whereas some may not. See Figure 2.2 for an example of this type of connectivity. Error control for multiphysics problems is an active field of research. See [4] for an abstract framework for error control, where the issues mentioned above are addressed. We will now introduce a multiphysics problem related to the simulation of the flow of liquid (e.g., water, oil) through a porous medium (e.g., sand, rock). 3 F D A E C B Figure 2.2: Nodes A to F represent stand alone solvers. Data is passed in the direction of the arrows. The solution of A depends on B only, whereas the solution of F depends on E, A, and B. 2.3 A Coupled Flow and Transport Problem Consider the following stationary transport problem: find c : Ω −→ R such that ∇ · (σc) = f, x ∈ Ω, c = fD , x ∈ Γ− , (2.6) where n is the exterior unit normal to Ω, Γ− = {x ∈ ∂Ω : n · σ < 0} is the inflow part of the boundary, σ = (σ1 , σ2 )T with σi ∈ C 1 (Ω̄), i = 1, 2 is the flux velocity, and f ∈ L2 (Ω), g ∈ L2 (Γ− ) are given functions. Furthermore, let the flux σ be determined by the equation ∇ · σ = g, σ = −κ∇p, p = gD , x ∈ Ω, x ∈ Ω, x ∈ ∂Ω, (2.7) where κ is a positive definite first order tensor. Together, (2.6) and (2.7) constitute a coupled system of partial differential equations. The non stationary version of the system, somewhat extended, is typically used as a model problem in the study of some incompressible fluid component with concentration c, transported through some porous medium by means of some pressure created by the injection of some other incompressible fluid component. The first equation we will refer to as the transport equation and the second as the pressure equation. The physical motivation for the problem will be discussed in greater detail in Section 5. For now however, we simply state our coupled problem in the abstract finite element setting. 4 Hence, let a(σ; c, v) be some bilinear form and ℓ(v) some linear form corresponding to the left and right hand side of (2.6) respectively. Also let b(p, v) and (v) correspond to (2.7) in the analog way. In a finite element setting the coupled problem now reads: find ch ∈ Vh,k such that a(Σ; ch , v) = ℓ(v), ∀v ∈ Vh,k (2.8) where Σ is an approximation to −K∇p computed from the solution to the problem: find ph ∈ Vh,m such that b(ph , v) = (v), ∀v ∈ Vh,m . (2.9) In the following section we will derive discontinuous Galerkin versions of the forms a, b, ℓ, and . 5 3 3.1 Discontinuous Galerkin Methods dG for the Transport Equation In order to derive the discontinuous Galerkin method for the transport equation we multiply (2.6) by v ∈ H 1 (K) and following [5] we integrate by parts elementwise to get Z Z Z − σc · ∇v dx + (n · σc)v ds = f v dx. (3.1) K ∂K Ω The identity XZ K∈K (n · τ )ν ds = ∂K XZ E∈E hn · τ i[ν] ds + E XZ E∈E ◦ [n · τ ]hνi ds, (3.2) E where E ◦ is the set of interior edges E, holds for vector valued functions τ and scalar valued functions v, piecewise smooth on K. Substituting (3.2) in (3.1), with τ = σc and ν = v, we have Z Z XZ hn · σci[v] ds = f v dx, (3.3) − σc · ∇v dx + K E∈E E Ω since σc = 0 on the internal edges. Splitting the last sum and including the boundary condition from (2.6) yields Z X Z XZ X Z (n · σg)v ds. hn · σci[v] ds = f v dx − − σc · ∇v dx + (3.4) K∈K K E6⊂Γ− E Ω E⊂Γ− E We define a(σ; c, v) = − XZ K∈K ℓ(v) = Z Ω σc · ∇v dx + K f v dx − X Z E6⊂Γ− X Z E⊂Γ− hn · σciu [v] ds, (3.5) E (n · σg)v ds, (3.6) E where hn · σciu is the so called upwind value of n · σc given by n · σc+ , if σ · n > 0, hn · σciu = n · σc− , if σ · n < 0, hn · σci, if σ · n = 0. (3.7) Our variational problem reads: find c ∈ H 1 (K) such that a(σ; c, v) = ℓ(v), 6 ∀v ∈ H 1 (K). (3.8) To formulate the discontinuous Galerkin method of order k we proceed in the standard way by replacing H 1(K) by the finite dimensional subspace Vh,k . The dG(k) method is then formulated by: find ch ∈ Vh,k such that a(σ; ch , v) = ℓ(v), ∀v ∈ Vh,k . (3.9) The reason for using the upwind value of n · σc in (3.5) is to increase the stability of the method [5]. Due to discontinuities in the boundary data, the solution might otherwise locally behave erratically. We note that if H 1 (K) is replaced by H 1 (Ω) ⊂ H 1 (K) above, the bilinear form a(σ; v, w) collapses to the continuous Galerkin bilinear form. This form however is known to be unstable for convection dominated problems [6], and this is one of the main reasons for the popularity of the discontinuous Galerkin methods for convection dominated problems. 3.2 dG for the Poisson Equation We now continue with the derivation of a dG method for the second equation in our coupled problem. We shall in the following derive the classical Nitsche Method for the pressure equation (2.7). For simplicity we let κ be the identity in the following sections until we return to the physical motivation of (5.7) and (2.7) in Section 5. The pressure equation then becomes the standard Poisson equation, which reads −∆p = g, p = gD , x ∈ Ω, x ∈ ∂Ω. (3.10) Multiplying (3.10) by v ∈ H 1 (K) and partially integrating elementwise yields Z Z XZ ∇p · ∇v dx − (n · ∇p)v ds = gv dx. K∈K K ∂K Applying (3.2) with τ = ∇p and ν = v on the second term in (3.11) yields Z XZ XZ hn · ∇pi[v] ds = gv dx. ∇p · ∇v dx − K K∈K On adding α E∈E XZ E∈E E hn · ∇vi[p] ds + µ E (3.11) Ω (3.12) Ω XZ E∈E [p][v] ds = 0, (3.13) E where α, µ ∈ R, to the left hand side of (3.12), we get XZ XZ ∇p · ∇v dx − hn · ∇pi[v] ds K∈K K E∈E +α E XZ E∈E hn · ∇vi[p] ds + µ E XZ E∈E 7 [p][v] ds = E Z Ω gv dx. (3.14) We define b(p, v) = XZ ∇p · ∇v dx − K K∈K +α E∈E XZ E∈E (v) = Z XZ hn · ∇pi[v] ds (3.15) E hn · ∇vi[p] ds + µ E XZ [p][v] ds, E E∈E gv dx. (3.16) Ω Here b(u, v) is a bilinear form, and (v) is a bounded linear form. The variational problem reads: find p ∈ H 1 (K) such that ∀v ∈ H 1 (K). b(p, v) = (v), (3.17) Note that for the particular choice of α = −1, b is a symmetric bilinear form, and this is Nitsche’s Method. To formulate the discontinuous Galerkin method, we replace H 1 (K) by the finite dimensional subspace Vh,m , and the dG method reads: find ph ∈ Vh,m such that b(ph , v) = (v), ∀v ∈ Vh,m . (3.18) The parameter µ in (3.15) we call the penalty parameter since the corresponding term penalizes departure from continuity between the elements. It turns out that if µ = β/h, where β is some sufficiently large constant, Nitsche’s method converges optimally in the L2 - and the H 1-norm. Hence, this will be our chosen value of µ from now on. An important property of Nitsche’s method is that of local conservation of the flux. In the continuous case, for the Poisson equation, local conservation of the flux is expressed by Z Z n · ∇p ds + f dx = 0, for ω ⊂ Ω, (3.19) ∂ω ω where ω ⊂ Ω, and n is a outward pointing unit normal to ω. This follows by applying the divergence theorem to the left hand side of equation (2.7). Nitsche’s method preserves this structure elementwise, and the analogous property follows for K ∈ K, by choosing as the test function in (3.18) the characteristic function to K, 1 K ∈ H 1 (K), defined by 1, x ∈ K, (3.20) 1K = 0, x 6∈ K. This yields, XZ XZ ∇ph · ∇11K dx − hn · ∇ph i[11K ] ds K∈K K E∈E +α E XZ E∈E hn · ∇11K i[ph ] ds + µ E XZ E∈E =− Z [ph ][11K ] ds E hn · ∇ph i ds + µ ∂K 8 Z E [ph ] ds = Z K f dx. (3.21) The conservation property is formalized by Z Z Fn (ph ) ds + f dx = 0, ∂K for K ∈ K, (3.22) K where Fn (ph ) is the so called numerical flux defined by Fn (ph ) = hn · ∇ph i − β [ph ]. h (3.23) The idea is of course that the numerical flux is an approximation of the flux. We note also that by subtracting (3.22) from (3.19) for ω = K we have Z Z n · ∇p ds = Fn (ph ) ds. (3.24) ∂K ∂K The reason for the importance of local conservation is twofold. For one it means that we have an actual physical law that locally remains valid in some average sense in the numerical model. The significance of this is blatantly obvious if we’re interested in approximating a quantity such as the flux in the domain. Secondly, conservation of the fluxes also seems to matter for the numerical stability of solution to the transport problem. This qualitative behavior is not yet fully understood, but in Section 6 we shall look into this question carefully. 9 n3 n3 n6 n4 n1 n5 n1 n2 n4 n2 (a) cG Triangles (b) dG Triangles Figure 4.1: The difference between cG and dG degrees of freedom. 4 Discretization and some Standard Estimates Before we solve the coupled flow and transport problem, we shall discretizise, implement, and verify the convergence properties of the two different methods individually. We shall be seeking the solutions to our problems in finite dimensional polynomial spaces of H 1 (K) of order k and m respectively. Before we get down to the nitty gritty of the individual methods, let us settle the question of exactly what these subspaces will be. We remind ourselves of that Ω = [0, 1] × [0, 1], and that K is a uniform partition om Ω into quadrilaterals K. Choosing an order of the polynomial space is always a weigh off between accuracy in the approximation, size of the linear system, and ease of implementation. Now, a higher polynomial order will result in higher rate of convergence. A drawback with dG however is that increasing the polynomial order rather quickly leads to large systems of equations. This is precisely because we allow discontinuities between the elements, which of course means that Vh,k (K) is a larger space than its continuous counterpart and thus requires a larger basis. In practice this means that no degrees of freedom are shared between neighbouring elements, as is the case in continuous Galerkin for example. Thus a dG method of order k will be more costly computationally than a cG method of the same order. See Figure 4.1 for an illustration of typical degrees of freedom in the respective methods. As far as implementing dG goes, dG of order zero is of course easiest to implement. For example, the variational form (3.18) collapses to XZ b(p, v) = µ [p][v] ds, (4.1) E∈E E and that’s pretty minimalistic. All aspects considered we choose the space Vh,0 , the space of of elementwise constants, for the implementation of the dG method for the transport equation, and the space Vh,1 , the space discontinuous bilinear polynomials, for the implementation of Nitsche’s method. 10 4.1 Discretization of the dG Transport Method The Space of Piecewise Constant Polynomials The space Vh,0 is a N-dimensional Hilbert space with N = |K| and a basis to Vh,0 is given by {τj }, where τj is defined by τj = 1 Kj , for Kj ∈ K. Since ∇v = 0, for c ∈ P0 (K), the discrete forms corresponding to (3.9) collaps to X Z hn · Σciu [v] ds, a(Σ; c, v) = E6⊂Γ− ℓ(v) = Z (4.2) (4.3) E f v dx − Ω X Z E⊂Γ− (n · Σg)v ds. (4.4) E Derivation of the Linear System of Equations Since we have a finite dimensional basis to Vh,0 we can make the ansatz ch = N X ξj τj , (4.5) j which we plug into (3.9). This yields X ξj a(Σ; τj , v) = (f, v), ∀v ∈ Vh,1, (4.6) i = 1, . . . , N, (4.7) j equivalent to X ξj a(Σ; τj , τi ) = (f, τi ), j a linear system of equations. In matrix form we write Uξ = q − r, where ξ = (ξ1 , . . . , ξN )⊤ ; U is a N × N matrix with elements X Z hn · Στj i[τi ] ds, i, j = 1, . . . , N; uij = E6⊂Γ− (4.8) (4.9) E and q = (q1 , . . . , qN )⊤ together with r = (r1 , . . . , rN )⊤ are N-vectors with elements Z qi = f τi dx, i = 1, . . . , N, (4.10) Ω and ri = X Z E6⊂Γ− (n · ΣgD )τi ds, E respectively. 11 (4.11) Convergence Results The dG method for the transport equation converges with optimal order in the L2 norm, defined on the broken space by XZ 2 v 2 dx. (4.12) kvk = K∈K K From interpolation theory we get the a priori estimate kc − ch k ≤ Chk+1 kckH k+1 (K) , (4.13) for the solution ch to (3.9), where k as usual is the polynomial order. Now, taking the logarithm of (4.13) yields log kc − ch k ≤ (k + 1) log h + C(c), (4.14) and for the piecewise constant basis this means that log kc − ch k ≤ log h + C(c). (4.15) When verifying the convergence result (4.13) we should hence expect the logarithm of the L2 -norm of the error to decrease as log h when h −→ 0. For the verification we choose the problem ∇ · (σu) = 0, x ∈ Ω c = sin(2πy), x ∈ Γ− , (4.16) where σ = (1, 0)⊤ , with the known analytical solution u = sin(2πy). In Figure 4.4 we have plotted log kc − ch k against log h. We see that k = 0.9997 and this agrees with theory. 4.2 Discretizising Nitsche’s Method The Space of Piecewise Bilinear Polynomials Since continuity between the elements is imposed weakly in discontinuous Galerkin methods (by the penalty term in Nitsche’s method) we are free to construct the global basis from any element basis we want. The standard nodal basis determined on an element K with nodes x1 , . . . , x4 by ϕj (xi ) = δij is familiar though, and as they say, old habits die hard. We do remark however that the canonical basis xα y β , 0 ≤ α, β ≤ 1, would do equally fine. Now, the standard nodal basis ΦK = {ϕ1 , ϕ2 , ϕ3 , ϕ4 } for the bilinear polynomials on the quadrilateral K = {(x, y) : 0 ≤ x ≤ h, 0 ≤ y ≤ h} is given by x y x y ϕ1 = 1 − 1− ϕ2 = 1− (4.17) h h h h x y xy ϕ4 = 1 − , (4.18) ϕ3 = 2 h h h which is precisely what we get if we take the tensor product of the 1D linear bases over the interval [0, h] in x and y respectively. The function ϕ1 is depicted in Figure 4.3. By transformation of coordinates we can define a similar basis ΦKi , for Ki ∈ K, i = 1, . . . , |K|. The global basis for Vh,1 is then constructed by {ΦK }. 12 u = sin(2πy) 1 0 1 −1.5 1 x y 0 0 Figure 4.2: The solution uh = sin(2πy) to (4.16). 1 φ1 0 0 h h Figure 4.3: A bilinear basis function. 13 Derivation of the Linear System of Equations Let M = 4 |K|. The ansatz ch = M X ξ j ϕj , (4.19) j yields the system of equations M X χj b(ϕj , ϕi ) = (f, ϕi ), i = 1, . . . , M, (4.20) j=1 which we can write in matrix notation as (A − S + αS ⊤ + βP )χ = b, where χ = (χ1 , . . . , χM )⊤ , A, S, and P are M × M matrices with elements X aij = (∇ϕj , ∇ϕi )K , i, j = 1, . . . , M, (4.21) (4.22) K sij = X (hn · ∇ϕj i, [ϕi ])E , i, j = 1, . . . , M, (4.23) E βX ([ϕj ], [ϕi ])E , pij = h E i, j = 1, . . . , M, respectively, and b = (b1 , . . . , bM )⊤ with elements Z bi = f ϕi dx, i = 1, . . . , M. (4.24) (4.25) Ω Convergence Results We shall study the convergence of the method in the energy norm and the L2 -norm. For this we will use the test problem −∆p = 8π 2 sin(2πx) sin(2πy), p = 0, x ∈ ∂Ω, x ∈ Ω, (4.26) with the analytical solution p = sin(2πx) sin(2πy). Remember that Nitsche’s method depends on a penalty parameter µ. In the numerics performed for this report we have throughout adopted the more or less standard value of µ = β/h. It can be shown that Nitsche’s Method converges with optimal order in H 1 and L2 when µ = β/h, where β is some sufficiently large constant [7]. In the simulations performed in this report we have used β = 103 . 14 Now, the energy norm corresponding to (3.17) on H 1 (K) is defined by X X X |||v|||2 = k∇vk2K + kh−1/2 [v]k2E + kh1/2 hn · ∇vik2E . K E∈E (4.27) E∈E This norm however is not a ‘standard’ energy norm in the sense that kvk2 6= b(v, v). In fact b(v, v) is not a norm. Specifically b(v, v) = 0, does not imply v = 0. We have the following error estimate in the energy norm: if β > 0 is some sufficiently large constant, then |||p − ph ||| ≤ Chm kpkp+1 (4.28) H , for some constant C. Taking the logarithm of this estimate yields log |||p − ph ||| ≤ log h + K(p), (4.29) for m = 1. That is, the logarithm of the error should decrease as the logarithm of h. In Figure (4.5) we see that with m = 0.9996 this is indeed the case. Continuing now with convergence in the L2 norm in which we have the error estimate: kp − ph k ≤ Chm+1 kpkL2 (K) . (4.30) For the bilinear basis we have then log kp − ph k ≤ 2 log h + K(p), (4.31) and referring to Figure 4.6 we see that we have m = 1.9999 which agrees with theory. −1.4 log||c−c h|| 10 −1.5 10 −1.6 10 −1.5 10 log h Figure 4.4: log kc − ch kL2 plotted against log h. k = 0.9997 15 −0.6 log|||p−ph||| 10 −0.7 10 −1.6 −1.5 10 10 log h Figure 4.5: log |||p − ph ||| plotted against log h. m = 0.9996 4.3 The Coupled Discretizised Problem Since we know now that the implementations of our methods are doing what they’re supposed to, we can go ahead and connect the two solvers according to (2.8) and (2.9). There is however one remaining ı́ssue. Namely, how to compute a good approximation Σ to σ = −∇p. In order to settle this question we need to establish what properties we require from a good approximation in this context. Our first requirement is that Σ is normal continuous over the edges. That is, Σ should be continuous in the components normal to the edges E ∈ E, which means that n · [Σ]E = 0, (4.32) where n is a fixed normal to E. This is a weaker regularity requirement than continuity over the elements, but looking at the derivation of the dG method for the transport equation in Section 2, we see that normal continuity is in fact enough for its validity. Our second requirement concerns the divergence of the field Σ. Specifically we require that our approximation upholds the conservation property (3.22) of Nitsche’s method. That is, the following relation should be in effect: Z Z Fn (ph ) ds = n · Σ ds. (4.33) ∂K ∂K By (3.24) we would then have the favorable result Z Z n · σ ds = n · Σ ds. ∂K ∂K 16 (4.34) log||p−ph|| −3 10 −1.6 10 −1.5 10 log h Figure 4.6: log kp − ph k plotted against log h. m = 1.9999 With these two requirements in mind we will now introduce a suitable space of functions onto which we shall interpolate σ. The resulting interpolant will satisfy both the requirements specified above and will hence qualify as a good approximation according to our standards. The Raviart-Thomas Space of Order Zero The Raviart-Thomas finite element space of order zero on K, RT0 (K), is the space of vector valued functions defined by RT0 (K) = q ∈ L2 (K) : q(x)|K∈K = a + bx, ∀K ∈ K and [q]E · nE = 0, ∀E ∈ E , (4.35) where x ∈ R2 , a ∈ R2 and b ∈ R, see [8]. If we consider a single quadrilateral element K, a basis for RT0 (K) is given by the functions {ψ1 , ψ2 , ψ3 ψ4 }, determined by the nodal variables Ni ∈ RT0 (K)∗ , defined by Z δij = Ni (ψj ) = ni · ψj ds, i, j = 1, . . . , 4, (4.36) Ei where i = 1, . . . , 4 correspond to edge numbers, and ni is an outward pointing unit normal for i = 1, . . . , 4. On K = [0, h] × [0, h] for example, this yields the basis functions functions y ⊤ x ⊤ ψ1 = 0, − 1 ,0 ψ2 = (4.37) h h y ⊤ ⊤ x ψ3 = 0, − 1, 0 , ψ2 = (4.38) h h 17 h 0 h 0 (a) ψ1 (b) ψ2 (c) ψ3 (d) ψ4 Figure 4.7: RT0 basis functions. 18 depicted in Figure 4.7. By looking at the definition we see that any function in the space indeed is normal continuous, and that is our first requirement. Now, on to the second requirement. Let πK : [L2 (K)]2 −→ RT0 (K) be the local interpolation operator defined by πK v = 4 X Ni (v)ψi . (4.39) i=1 For σ : R2 −→ R2 , we then have an crucial property, namely: Z Z ∇ · σ dx = ∇ · πK σ dx. K (4.40) K That is, the divergence of a vector field is an invariant under the interpolation operator. This follows from the familiar property (4.41) of the interpolant following below, together with our chosen degrees of freedom. Let us prove this claim on the quadrilateral K. First of all, from the definition of the nodal variables {Ni } we have Nj (πK σ) = Nj ( 4 X Ni (v)ψi ) = i=1 4 X Ni (v)Nj (ψi ) = Nj (v), (4.41) i=1 and we see that the interpolant πK σ has the same nodal values as σ. From the divergence theorem now follows: Z Z XZ XZ ∇ · σ dx = n · σ ds = ni · σ ds = ni · πK σ ds (4.42) K ∂K = Z Ei Ei n · πK σ ds = ∂K Z Ei ∇ · πK σ dx, Ei (4.43) K which is what we wanted. Remember though, the exact fluxes ni · σ are unknown in practice, so in order to compute our approximation Σ of σ we use the numerical flux (3.23) as coefficients in the interpolant. For this reason let’s define the local numerical interpolant of σ by NK σ = 4 Z X i=1 Ei Fni (ph ) ds ψi , (4.44) where Fni (ph ) is the edgewise numerical flux hni · ∇ph i − β/h[ph ]. Now, in Section 2 we established equality between the integral of the flux and the integral of the numerical flux over ∂K. The edgewise flux of the numerical interpolant is however precisely the edgewise numerical flux. This follows from Z 4 Z X i Nj (NK σ) = Fn (ph ) ds Nj (ψi ) = Fnj (ph ) ds. (4.45) i=1 Ei Ej 19 This means that we also have equality between the integral of the flux and the integral of the flux of the numerical interpolant. Hence, if we let ΣK = NK σ we have a computable approximation of the gradient field σ = −∇p with elementwise preserved divergence, and that is our second requirement established. A minor detail in the argumentation still remains unclear however. Strictly speaking we have only defined the local numerical interpolant NK . What about the global numerical interpolant from the broken space onto the RT0 space? Well, that’s easy. We simply stitch it together from the local numerical interpolants. More precisely we set NK v|Ki = NKi v, ∀Ki ∈ K, (4.46) and we’re home free. The conservation of the flux for the global interpolant now follows trivially. The Coupled Problem Again We now have all the components needed to solve the coupled problem (2.6), (2.7). To sum up, the solution procedure is the following: 1. Find ph such that b(ph , v) = (v), for all v ∈ Vh,1 . 2. Compute Σ = NK σ. 3. Find ch such that a(Σ; ch , v) = ℓ(v), for all v ∈ Vh,0 . One area of application where locally conservative fluxes is highly valuable is that of oil reservoir simulation. It is actually from here we get our coupled model problem, although transport in porous media is found in several other areas as well, like for example simulation of ground water flow and pollutant transport. Now, oil reservoir simulation is a notoriously difficult type of simulation. In a realistic model, there are complex physics present on many scales, the computational domain is vastly irregular, and the equations are non linear. In addition there are also often many unknowns. We shall in the following section give a short account for oil reservoir simulation and give a short physical motivation to our model problem. 20 5 Application in Reservoir Simulation reservoir simulation1 A computer run of a reservoir model over time to examine the flow of fluid within the reservoir and from the reservoir. Reservoir simulators are built on reservoir models that include the petrophysical characteristics required to understand the behavior of the fluids over time. Usually, the simulator is calibrated using historic pressure and production data in a process referred to as ‘history matching.’ Once the simulator has been successfully calibrated, it is used to predict future reservoir production under a series of potential scenarios, such as drilling new wells, injecting various fluids or stimulation. Efficient and accurate methods for the simulation of flow through porous media are vital in the field of petroleum engineering where reservoir simulations are important tools used by the oil companies to aid the evaluation of production strategies like well placement etc. The theory of reservoir simulation is a massive one, encompassing research from multiple scientific fields such as geochemistry, geophysics and of course mathematics. We can not say much on the subject here. Nonetheless, for the sake of completeness, we shall give short record of the physics involved in a certain type of reservoir model. The material in this section is in great extent comprised from that in [10], and we refer the interested reader to this text and the references therein for an actual treatment of the subject. 5.1 Immiscible Two-Phase Flow Layer upon layer of organic material piling up over a period of millions of years has created a sedimentary region in the bedrock some 100 to 1000 meters thick a couple of thousands of meters below the sea bed of the North Sea. By geological activities like earthquakes and volcanoes the layers was shifted and twisted and the previously smooth layered bedrock developed into a highly anisotropic structure. High pressures and temperatures with time turned some materials into hydrocarbons (i.e., petroleum and natural gases) that traveled in the pores of the bedrock towards the surface. At some sites, bent layers of non permeable rock, trapped the hydrocarbons on their way to the surface. These sites are the oil reservoirs of the North Sea. See Figure 5.1 for an illustration. The reservoir is initially at an equilibrium established over millions of years. When a well is drilled through the non permeable upper layer this equilibrium is immediately disturbed and the pressure in the reservoir drives the hydrocarbons towards the production facility at the surface. By this process, motored by nature it self, about 20 percent of the hydrocarbons are produced until a new equilibrium is reached. The oil company might then start a second production process by injecting water or gas into the reservoir aiming to rebuild pressure in the reservoir and to push more hydrocarbons out with the water. By this process perhaps an other 20 percent is produced. As a third and final measure, different types of solvents and foams might be injected into the reservoir. 1 From ‘The Schlumberger Oilfield Glossary’, see [9]. 21 Oil Water Reservoir Non-permeable rock Figure 5.1: Illustration of a North-Sea reservoir setting. Remember, in Section 1 we loosely related our coupled PDE’s to precisely the type of fluid flow described in the second production process above. The modeling of this flow can be done in several ways, each building upon a specific set of physical assumptions. These assumptions usually concern the number of phases present (e.g., gas, oil, water); the composition of each phase (e.g., the gaseous phase contains butane, ethane etc); how the permeability is modeled, and so on. Common to all of them is however the assumption of the validity of a constitutive relation called Darcy’s law. This law is an analog of Fourier’s law of heat transfer and basically says that the filtering velocity is proportional to the negative gradient of the pressure. That is, the flow is directed from high pressure to low pressure. Explicitly, an extended, multi-phase version of Darcy’s law related to phase α, with α = g, o, w is given by: κrα vα = −κ ∇pα , (5.1) µα where κ is the permeability tensor describing the permeability of the computational domain, κrα is the relative permeability describing how α flows in the presence of the two other phases, µα is the viscosity of α, and pα is the pressure in α. The so called Black-Oil Models is a large class of models widespread in reservoir simulations. Common for these is the assumption that hydrocarbon is made up of only two components, oil and gas, and that the hydrocarbon composition remains constant, i.e., no phase transitions in the hydrocarbon take place. It is furthermore assumed that the total void in the porous rock is filled up by either two or three phases. Now, if the assumption is that of two phases, one phase is considered to consist of pure water and the other is considered to be a hydrocarbon phase made up of two components, oil and dissolved gas. 22 By further assuming incompressibility of the fluids, the rock, and immiscibility of the fluids, one ends up with the Incompressible Immiscible Two-Phase Flow model, which in two dimensions has the following pressure equation: ∇ · v = g, v = −κλ∇p. (5.2) (5.3) Here λ = λw + λo , where λα = κrα /µα is the so called mobility of the phase α, p is the is a certain global pressure as defined in [10], and g a source term. The transport equation, or saturation equation, in the model is given by φ ∂c + ∇ · (fw (c) [v + d(c, ∇c)]) = f, ∂t (5.4) where c is the saturation of water, φ, 0 ≤ φ ≤ 1 is the porosity (the void volume fraction) of the rock, fw (c) = λw /λ measures the water fraction of the total flow, f is a source term, and fw (c)d(c, ∇c) represents capillary forces. So, that’s the Incompressible Immiscible Two-Phase Flow model. For practical purposes we set λ = 1; fw = c; and we also assume that all capillary forces can be neglected. What we’re then left with are the equations: ∇ · v = g, v = −κ∇p, (5.5) (5.6) and ∂c + ∇ · (vc) = f, (5.7) ∂t which seem quite familiar. The system is closed by the addition of some appropriate initial and boundary conditions. It is common practice to assume that the reservoir is closed, i.e., that no mass flows in or out of the domain besides at the designated production wells. This is normally modeled by adding the no-flow boundary conditions n · κ∇p = 0 in the pressure equation. The solution to this problemRis however only unique up to a constant. Hence some other constraint, like for instance Ω p dx = 0, is also added to the system. The resulting equation is then solved with the method of Lagrange multipliers. To the transport equation (5.7) we add the initial condition c(0) = 0, reflecting an assumption that no water, only hydrocarbon, is present in the domain at t = 0. We keep the boundary condition from (2.6). 5.2 The Quarter Five Spot Problem A standard test case in oil reservoir simulation is the so called quarter five spot problem. In this setting the reservoir is considered to be located in the domain Ω = [0, 1] × [0, 1]. Typically the injection well is located in a neighborhood of one of the corners of Ω, say (0, 0), and the production well is located in a neighborhood of the opposite diagonal corner, that 23 is (1, 1). The injector and the producer are represented by a source and a sink respectively in σ. We take the permeability κ in our simulation from the Tenth SPE Comparative Solution Project [11]. The original permeability data is represented as a piecewise constant function κSP E defined on a three dimensional grid with 220 × 60 × 85 cells. We use a 60 × 60 × 1 slice of this data. Specifically we let κ = κSPE (i, j, 1), for 1 ≤ i, j ≤ 60. See Figure 5.2. We note in passing that we have κ ∈ Vh,0. 60 50 40 30 20 10 0 0 10 20 30 40 50 60 Figure 5.2: The logarithm of the piecewise constant function κSPE representing the permeability in Ω. Lighter areas represents more permeable material. The source terms g and f in (5.2) and (5.7) are commonly used to model the injection well and the production well. It turns out however, this approach is not fully compatible with the approach on coupling taken here. The reason for this is the combination of weak boundary conditions, together with the elementwise conservation of the fluxes in the pressure solver. Consider some element K ∈ K containing a source, say f (x0 ) = 1, where x0 ∈ K. Suppose furthermore that we have no-flow conditions on one of the edges in K. Since the flux is conserved, there will be a flux out of K equal to the area of K. The issue here is the weakly imposed Neumann conditions which might let some mass sipper through the no-flow edge. This means that not only is the boundary condition not satisfied, the vector field may locally actually be pointing the wrong way! Such an irregularity in the convection field may have serious consequences in the transport solver, effectively extinguishing any kind of regularity in the solution. Hence, we take on a slightly different approach here by setting f = g = 0 and instead we 24 KN K1 Figure 5.3: Boundary fluxes modeling the injector and producer. model the injector and the producer with appropriately chosen boundary conditions in the respective neighborhoods. Referring to Figure 5.3 for the labels, we set −κ∇p = (1, 1)T , for x ∈ ∂K1 ∩ ∂Ω and x ∈ ∂KN ∩ ∂Ω. In Figure 5.5 we see the solution to the pressure equation using the permeability κSPE , and in Figure 5.4 the corresponding flow velocity field. In Figure 5.6 we see the solution to the transport problem using the RT0 representation of the convection field v. We see that the water follows the vector field in Figure 5.4 from the injector to the producer, as should be expected. 25 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 Figure 5.4: The flow velocity Σ. 30 20 10 0 -10 -20 -30 1 0.8 1 0.6 0.8 0.6 0.4 0.4 0.2 0.2 0 0 Figure 5.5: The pressure ph . 26 Figure 5.6: Fifteen time steps in the interval 0.2 ≤ t ≤ 8.0 from the solution to the quarter five spot problem. Lighter areas are more saturated by water. 27 6 Error Analysis Now then, what is the actual deal with local conservation? We have gone through some trouble in order to keep the property intact, but what’s the deal with it really? Well, we’ve mentioned more than once that the local conservation property is important for the numerical stability in the transport equation. As promised we shall now do a careful investigation of this statement. We begin by presenting some numerical evidence that justifies our claim. 6.1 Numerical Evidence of the Importance of Local Conservation Consider the convection field | sin(2πx)|sgn(y − 1) β= , −2πsgn(sin(2πx)) cos(2πx)|y − 1| (6.1) depicted in Figure 6.1. Let Vh be the space of vector valued bilinear polynomials. By interpolating β onto RT0 and Vh respectively, we create a scenario in which we have one approximation πRT0 β of β with preserved fluxes, and one, πVh β, in which the fluxes are not preserved. The two fields are shown in Figures 6.1 and 6.1 respectively. We shall now study the solutions produced by the transport solver when the two different approximations are used. We solve the following euqation: ut + ∇ · (πβu) = 0, u(x, t) = 0, x ∈ Ω, t > 0, x ∈ Γ− , t > 0, −100((x−0.75)2 +(y−0.8)2 ) u(x, 0) = e , (6.2) x ∈ Ω, where πβ is πRT0 β or πVh β. Comparing the two approximative fields one might suspect quite different solutions to the problem. Rightly so. Since the fluxes are not preserved elementwise in the bilinear approximation we see that for x = 0 we have πVh β = 0. This is effectively an impermeable wall in the flow field, causing blow up in the solution. In Figures 6.1 and 6.1 we see the L2 -norm of the respective solutions plotted against time. It is clear from the graph corresponding to the bilinear field when the blow up occurs. 6.2 Analytical Solution of the Transport Equation In order to understand what may happen in the solution when conservation fails we shall study the analytical solution to the transport equation along its characteristics. Thus, consider the equation ut + ∇ · (σu) = 0, u(x, t) = gD , u(x, 0) = 0, 28 x ∈ Ω, t > 0, x ∈ Γ− , t > 0, x ∈ Ω, (6.3) 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 6.1: The vector field β. 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 6.2: The approximated vector field πRT0 β. 29 1 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 6.3: The approximated vector field πVh β. 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Figure 6.4: L2 -norm of solution corresponding to πRT0 β. 30 0.13 0.12 0.11 0.1 0.09 0.08 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Figure 6.5: L2 -norm of solution corresponding to πVh β. where Ω ⊂ R2 , u = u(x, t) : R2 × [0, ∞) −→ R, σ(x) = (σ1 (x), σ2 (x)) : R2 −→ R2 is Lipschitz continuous in Ω. Typical for this equation is that an initial disturbance at x0 ∈ Γ− will propagate along the characteristic of σ passing through x0 . By choosing a coordinate system following the characteristic, we can solve a reduced equation along it. The characteristic passing through x0 is given by the solution x(ξ) = (x1 (ξ), x2(ξ)) to the system dxi = σi , ξ > 0, i = 1, 2, dξ (6.4) 0 xi (0) = xi , where existence and uniqueness follow from the premise σ Lipschitz. Now, the chain-rule yields σ · ∇u = ∂u dx1 ∂u dx2 ∂u + = . ∂x1 dξ ∂x2 dξ ∂ξ (6.5) Hence, by writing (6.3) in its equivalent form ut + σ · ∇u + γ(x)u = 0, (6.6) where γ(x) = ∇ · σ, we have by (6.5), the reduced equation ut + uξ + γ(x)u = 0, u(x0 , t) = gD (x0 ), u(ξ, 0) = 0, 31 ξ > 0, t > 0, ξ > 0, t > 0, (6.7) along the characteristic, which is precicelly the 1D convection equation. This equation can be transformed into an ODE by the Laplace transform. Applying the transform yields Z ∞ Z ∞ Z ∞ −st −st L [ut + uξ + γ(x)u](s) = ut e dt + uξ e dt + γ(x)ue−st dt (6.8) 0 0 0 Z ∞ Z ∞ Z ∞ ∂ −st −st ue dt + γ(x)ue−st dt (6.9) =s ue dt + ∂ξ 0 0 0 ∂ Ū (ξ) + γ(x)Ū (ξ) (6.10) = sŪ (ξ) + ∂ξ ∂ Ū (ξ) = (s + γ(x))Ū (ξ) + = 0, (6.11) ∂ξ and 0 L [u(x , t)](s) = Z ∞ u(x0 , t)e−st dt = Ū (0) = 0 gD (x0 ) . s (6.12) That is, we get the equation (s + γ(x))Ū (ξ) + ∂ Ū (ξ) = 0, ∂ξ Ū(0) = gD (x0 ) , s (6.13) which has the solution −ξs gD (x0 ) −(s+γ(x))ξ 0 −ξγ(x) e Ū (ξ) = e = gD (x )e . s s (6.14) Applying the inverse Laplace transform on the solution then yields u(ξ, t) = L −1 [Ū ] = gD (x0 )e−ξγ(x) θ(t − ξ), where θ is the Heaviside function. That is, along the characteristic we have 0, if t < ξ, u(ξ, t) = 0 −ξγ(x) gD (x )e , if t ≥ ξ. (6.15) (6.16) We see that the sign of γ will have a tremendous impact on the nature of the solution u. Say for instance that γ < 0 for x ∈ ω ⊂ Ω. Since the divergence is preserved in the numerical solution it is reasonable to assume that uh will behave approximately like u in ω, that is grow exponentially. This is of course good. Now, consider a scenario in which we have an analytical convection field σ such that ∇ · σ = 0, for x ∈ Ω. In particular, given K ∈ K, we have ∇ · σ = 0, for x ∈ K. Assume furthermore that we have a numerical approximation Σ̄ such that ∇ · Σ̄ 6= 0. What will happen with the numerical solution uh in K? Why, it will grow of course, or decline, depending on the sign of ∇ · Σ̄. That is, the approximative solution uh does not behave in the same way as u, not even qualitatively. This is the numerical instability we’re talking about. 32 6.3 An Error Representation Formula for the Transport Equation In the following we will derive an error representation formula for the transport equation in the multi physics setting. The analysis will be performed using standard duality arguments, for simplicity in a cG setting. See [12] for an overview of duality based techniques in error analysis of hyperbolic problems. Assume that the vector field ΣR is an approximation of σ ∈ {v ∈ H(div; Ω) : n · σ = 0 on ∂Ω}, such that for K ∈ K, K ∇ · (σ − Σ) dx = 0 on K, and that n · Σ = 0 on ∂Ω. Consider again the transport equation (6.3). By multiplying (6.3) by ϕ ∈ H 1 (Ω) and partially integrating, we derive the following dual problem −ϕt − σ · ∇ϕ = ψ, ϕ(x, t) = 0, ϕ(x, T ) = 0, x ∈ Ω × [0, T ], x ∈ Γ+ × [0, T ], x ∈ Ω, (6.17) where ψ ∈ H −1(Ω). Let Vh be the space of continuous piecewise polynomials of order k. Following the approach taken in [13] we let U : (0, T ] −→ Vh be a solution to: Z Z (Ut , v) + (∇ · (ΣU), v) dt = (g, v) dt, ∀v ∈ Vh , (6.18) Ik Ik and k = 1, . . . , n where Ik = [tk−1 , tk ], tk−1 < tk , t0 = 0, and tn = T . Let e = u − U. Subtracting (6.18) from the weak form of (6.3) yields Z 0 = (et , v) + (∇ · (σu), v) − (∇ · (ΣU), v) dt Z Ik = (et , v) + (∇ · (σu), v) − (∇ · (ΣU), v) − (∇ · (Σu), v) + (∇ · (Σu), v) dt Z Ik = (et , v) + (∇ · (Σe), v) + (∇ · ((σ − Σ)u), v) dt, ∀v ∈ Vh . (6.19) (6.20) (6.21) Ik We have the relation Z Z (et , v) + (∇ · (Σe), v) dt = − (∇ · ((σ − Σ)u), v) dt, Ik ∀v ∈ Vh , (6.22) Ik which is not quite what we want since we’ve defined the dual problem with σ. To get around this little glitch we add (∇ · (σe), v) to both sides of equation (6.22). Rearanging we get Z Z Z (et , v) + (∇ · (σe), v) dt = − (∇ · ((σ − Σ)u), v) dt + (∇ · (σe), v) dt (6.23) Ik Ik Ik Z − (∇ · (Σe), v) dt, ∀v ∈ Vh , Ik 33 and upon expanding and simplifying the right hand side we have the Galerkin orthogonality Z Z (et , v) + (∇ · (σe), v) dt = ((σ − Σ)U, ∇v) dt, ∀v ∈ Vh . (6.24) Ik Ik Continuing in the standard way by taking the L2 inner product of the error e and the dual problem (6.17) and partially integrating in time and space yields Z T (e, ψ) dt = 0 Z T (e, −ϕt − σ · ∇ϕ) dt = 0 Z T (et , ϕ) + (∇ · (σe), ϕ), (6.25) 0 where the boundary terms in the partial integration disappear due to the definition of the dual problem, the assumption on σ, and the assumption u(0) = U(0). From the Galerkin orthogonality we have Z T (e, ψ) dt = 0 = n Z X k=1 Ik n Z X k=1 + (et , ϕ) + (∇ · (σe), ϕ) dt (6.26) (et , ϕ − πϕ) + (∇ · (σe), ϕ − πϕ) dt (6.27) Ik n Z X k=1 ((σ − Σ)U, ∇πϕ) dt, Ik where π : L2 (Ω) → Vh is the Scott-Zhang interpolation operator [14]. By making use of the fact that u satisfies (6.3) we have the error representation formula Z T (e, ψ) dt = 0 n Z X k=1 + (g − U − ∇ · (σU), ϕ − πϕ) dt (6.28) Ik n Z X k=1 ((σ − Σ)U, ∇πϕ) dt. Ik We see that the last term on the right hand side exists because of the modeling error σ −Σ. However, if we consider a scenario in which f = ψ = 0 in the interior of Ω, which may be the case in reservoir simulation, with the source term f and the goal functional ψ living on the boundary ∂Ω only, it turns out that if the modeling error has zero divergence it may not have a very big impact on the error at all. Let us see what we can make of the modulus of the last term in (6.28) on a single element K ∈ K, in such a setting. Partial integration yields |((σ − Σ)U, ∇πϕ)K | = | − (∇ · ((σ − Σ)U), πϕ)K + (U, n · (σ − Σ)πϕ)∂K | ≤ |(∇ · (σ − Σ)U, πϕ)K | + |((σ − Σ) · ∇U, πϕ)K | + |(n · (σ − Σ)U, πϕ)∂K |. 34 (6.29) (6.30) (6.31) The first and third term in (6.30) are zero because |(∇ · (σ − Σ)U, πϕ)K | ≤ C|(∇ · (σ − Σ), 1)K | = 0, |(n · (σ − Σ)U, πϕ)∂K | ≤ C|(n · (σ − Σ), 1)∂K | = 0, (6.32) (6.33) since U ∈ C(K), and πϕ ∈ C(K) and K is compact. We argue now that the second term also is small due to conservation. In the case ∇ · σ = 0 the solution u is constant along the characteristics of σ. Hence σ · ∇u = 0, and σ · ∇U ≈ 0. Since we have conservation of the fluxes, the characteristics of Σ should approximately follow those of σ in the interior of Ω, and hence we argue that Σ · ∇U ≈ 0 as well. It furthermore seems reasonable that σ · ∇U ≈ Σ · ∇U on the K also when ∇ · σ 6= 0. A rigorous justification of these claims is however beyond the scope of this report and subject to further analysis. 35 7 Conclusions Motivated by the local conservation property of Nitsche’s method in conjunction with that of the Raviart-Thomas element, we have surveyed the use of discontinuous Galerkin methods in the context of coupled flow and transport problems. We have pinpointed some previously unnoted difficulties with this approach due to the weak boundary conditions in the dG formulation, but have navigated passed these by the specification of an alternative boundary condition. The proposed methodology has been successfully applied to a model problem stemming from reservoir simulation. Furthermore, we have presented both numerical evidence and analytical arguments indicating the importance of local conservation for the numerical stability in this context. In conclusion, additional research effort is needed to fully understand the local conservation property, but we strongly believe that our proposed line of argument is a step in the right direction. 36 References [1] Bernardo Cockburn, George E. Karniadakis, and Chi-Wang Shu. The development of discontinuous galerkin methods. In Bernardo Cockburn, George E. Karniadakis, and Chi-Wang Shu, editors, Discontinuous Galerkin Methods, Theory, Computation and Applications, volume 11 of Lecture Notes in Computational Science and Engineering. Springer, 2000. [2] D. H. Griffel. Applied Functional Analysis. Dover, 2002. [3] Mats G. Larson and A. Jonas Niklasson. Analysis of a nonsymmetric discontinuous galerkin method for elliptic problems: Stability and energy error estimates. Chalmers Finite Element Center Preprint, 2001. [4] Mats G. Larson and Fredrik Bengzon. Adaptive finite element approximation of multiphysics problems: A MEMS device. Communications in Numerical Methods in Engineering, 2007. To appear. [5] F. Brezzi, L. D. Marini, and E. Süli. Discontinuous Galerkin methods for first-order hyperbolic problems. Mathematical models and methods in applied sciences, 14:1893– 1904, 2004. [6] Claes Johnsson. Numerical solutions of partial differential equations by the finite element method. Studentlitteratur, 1987. [7] Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini. Unified analysis of discontinuous galerkin methods for elliptic problems. SIAM Journal on Numerical Analysis, 39(5):1749–1779, 2002. [8] C. Bahriawati and C. Carstensen. Three Matlab implementations of the lowest-order Raviart-Thomas MFEM with a posteriori error control. Computational methods in applied mathematics, 5:333–361, 2005. [9] The Schlumberger Oilfield Glossary. http://www.glossary.oilfield.slb.com. [10] J.E. Aarnes, T. Gimse, and K.-A. Lie. An introduction to the numerics of flow in porous media using Matlab. Geometrical Modelling Numerical Simulation and Optimization, Industrial Mathematics at SINTEF. Springer Verlag, 2005. [11] SPE Comparative Solution Project. http://www.spe.org/csp/. [12] Endre Süli. A posteriori error analysis and adaptivity for finite element approximations of hyperbolic problems. In M. Ohlberger D. Kroner and C. Rhode, editors, An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, Lecture Notes in Computational Science and Engineering, volume 5, pages 123–194. Springer, Berlin, 1999. 37 [13] Mats G. Larson and Axel Målqvist. Goal oriented adaptivity for coupled flow and transport problems with application in oil reservoir simulation. Computer Methods in Applied Mechanics and Engineering, 2007. [14] Alexandre Ern and Jean-Luc Guermond. Theory and Practice of Finite Elements, volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004. 38

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