Abstract We investigate the use of discontinuous Galerkin finite element methods... tiphysics setting involving coupled flow and transport in porous media....

Abstract We investigate the use of discontinuous Galerkin finite element methods... tiphysics setting involving coupled flow and transport in porous media....
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
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement