Mimetic MPFA
Runhild A. Klausen† and Annette F. Stephansen∗
†
Centre of Mathematics for Applications, University of Oslo,P.Box 1053 Blindern, N-0316 Oslo, Norway,
r.a.klausen@cma.uio.no
∗
Centre for Integrated Petroleum Research, University of Bergen, N-5007 Bergen, Norway,
annette.stephansen@cipr.uib.no
Abstract
The analysis of the multi point flux approximation (MPFA) method has so far relied
on the possibility of seeing it as a mixed finite element method for which the convergence
is then established. This type of analysis has been successfully applied to triangles and
quadrilaterals, lately also in the case of rough meshes.
Another well known conservative method, the mimetic finite difference method, has
also traditionally relied on the analogy with a mixed finite element method to establish
convergence. Recently however a new type of analysis proposed by Brezzi, Lipnikov and
Shashkov, cf. [3], permits to show convergence of the mimetic method on a general polyhedral mesh.
We propose to formulate the MPFA O-method in a mimetic finite difference framework, in order to extend the proof of convergence to polyhedral meshes. The general nonsymmetry of the MPFA method and its local explicit flux, found trough a splitting of the
flux, are challenges that require special attention. We discuss various qualities needed for a
structured analysis of MPFA, and the effect of the lost symmetry.
Introduction
The multi point flux approximation (MPFA) is a control volume method developed by the oil industry as a reliable discretization of the pressure equation, derived from Darcy’s law, on general
rough meshes. In reservoir simulation the geology of the reservoir, which includes faults and
non parallel layers in the media, is a major challenge. The particularity of the MPFA method is
its ability to provide a local explicit flux with respect to the pressure, which allows for a fully
implicit multiphase simulation. This property however leads in general to a non-symmetric
method.
The MPFA methods are cell oriented finite volume methods that have been essentially developed and investigated in the last decade, cf. [1, 2, 4, 5, 8, 11, 12, 13] and the references
therein. A rigorous analytical study of convergence behavior has so far been based on an equivalence to mixed finite element methods. In [9, 13] a MPFA method which is derived from
a mapping onto an orthogonal reference cell is analyzed. However, the analysis requires asymptotic h2 –parallelogram meshes, or so called smooth meshes. On triangulations symmetric
MPFA methods are available without similar asymptotic mesh-restrictions, and convergence can
be showed, cf. [10]. A limitation in the convergence study based on the analogy with mixed
finite elements is its limitation to triangles and quadrilaterals and the difficulty of extending the
proof to three dimensional space.
The mimetic finite difference method has been successfully applied to diffusion problems
with strong heterogeneities, see for instance [15] and [14], but does not in general allow
11th European Conference on the Mathematics of Oil Recovery — Bergen, Norway,
8 - 11 September 2008
for the definition of an explicit flux as a function of local pressure values. The method is a
symmetric discretization defined through a discrete framework with operators and properties
similar to that of the continuous space. In particular, a discrete divergence theorem is defined,
with discrete divergence and gradient operators. As the discrete velocity field is defined only at
the element faces, the possibility of a local reconstruction at each element has so far been needed
to demonstrate convergence in analogy with the mixed finite element method. In [7] a symmetric
MPFA version as a local mimetic finite difference method was derived, and convergence was
shown on triangulations which can be extended to smooth quadrilateral meshes. In [3] a rigorous
proof of convergence for mimetic finite difference methods was provided for general polyhedra,
where the need for element interpolation was eliminated. In the present paper, we use ideas
from [3] to prove convergence of the classical MPFA O-method as formulated in [10].
Model problem and setting
For simplicity of exposition we will limit our discussion to two dimensional meshes and Dirichlet boundary conditions. Let Ω be a domain in R2 with polygonal boundary ∂Ω. We consider
the following elliptic equation with homogeneous Dirichlet boundary conditions:
(
−∇·(K∇p) = f
in Ω,
(1)
p=0
on ∂Ω.
Viewing (1) as the prototype for the pressure equation in porous medium flow, p is the pressure,
K indicates the permeability tensor and f is a source term.
The mixed formulation of (1) is obtained by introducing the unknown Darcy velocity U =
−K∇p as a new variable. We seek (U, p) ∈ H(div, Ω) × L2 (Ω) such that
(
(K −1 U, V )0,Ω − (p, ∇·V )0,Ω = 0,
∀V ∈ H(div, Ω),
(2)
(∇·U, q)0,Ω = (f, q)0,Ω
∀q ∈ L2 (Ω),
supplied by boundary conditions, where f ∈ L2 (Ω) and K is a symmetric, positive definite field
1 (Ω)]2,2 whose smallest eigenvalue is bounded from below by a positive constant. The
in [W∞
subscript 0, Ω denotes the L2 -scalar product on Ω. Similarly, the subscript 0, R indicates the the
L2 -scalar product and its associated norm on a subset R ⊂ Ω. For s ≥ 1, a norm (semi-norm)
with the subscript s, R designates the usual norm (semi-norm) in H s (R).
Let {Th }h>0 be a shape-regular family of affine polyhedral meshes of the domain Ω. A
generic element in Th is denoted by E, hE denotes the diameter of E and nE its outward unit
normal. The normal on an edge si ⊂ ∂E will be indicated by nEi . Set h = maxE∈Th hE .
Let N designate the number of nodes/corners of the polyhedron E, while k designates the local
counterclockwise numbering of the nodes, 1 ≤ k ≤ N . The letter i will designate the local
counterclockwise numbering of the edges of the polyhedron, 1 ≤ i ≤ N . We partition the
polyhedron E into k quadrilaterals Ek by dividing each edge si (of length |si |) into two halfedges and joining these points with the midpoint, i.e. the barycenter, of the polyhedron, see
Figure 1a. The midpoint will be used as the local origin. Each quadrilateral Ek consists of two
k=3
y
i=3
O
lk2
k=2
k
I
lk2
i=2
k=1
i=1
Fig. 1a: Example of polygon with N = 6
O
lk1
x
I
lk1
Fig. 1b: Edge lengths of the subcells Ek ; cell center in
origin
11th European Conference on the Mathematics of Oil Recovery — Bergen, Norway,
8 - 11 September 2008
I and lI , and two
inner edges (vectors spanning from the origin to the midpoint) denoted lk1
k2
outer (half-)edges (vectors spanning from the midpoint to the node k of the element) denoted
O and lO . An illustration is found in Figure 1b. Note that for neighbouring quadrilaterals we
lk1
k2
I
I . The projection of a vector v on the x direction will be denoted
have the relation l(k+1)1
= lk2
j
by v|xj .
We consider the space Xh of discrete pressures that are constant on each element E. Secondly, we define the space Vh which consists of discrete velocity vectors defined only on the
half-sides of the element. To each element E we associate then 2N unknown scalars (outward
E , 1 ≤ k ≤ N , 1 ≤ j ≤ 2. The discrete velocity fields are thus defined
half-side fluxes) g̃kj
1
E nt , where i = (k + j − 1) using the convention that i = N + 1 ≡ 1. Conas 2 |si |−1 g̃kj
Ei
servation of half-side fluxes imposes that on a half-side shared by two elements the sum of the
corresponding half-side fluxes is zero. Notice that the flux gi over side si is equal to the sum
over the corresponding half-edge fluxes, i.e.
gi = g̃i,1 + g̃i−1,2
1 ≤ i ≤ N,
(3)
where g̃0,2 ≡ g̃N,2 .
The superscript E will henceforward be omitted if there is no ambiguity in order to facilitate
reading. The half-side fluxes can be regrouped into the vector G̃ = (g̃1,1 , g̃1,2 , . . . , g̃N,1 , g̃N,2 )t .
Similarly, G = (g1 , . . . , gN ). To simplify the analysis we will also make use of the vector
G̃k = (g̃k,1 , g̃k,2 )t which contains the the two half-edge (outward) fluxes associated with the
quadrilateral k. Introducing the (N × 2N ) matrix Z the relation (3) can be written as G = Z G̃.
The elements of the matrix Z consist of δik for odd columns and δ(i−1)k for even columns,
where δ indicates Kronecker’s delta and δ0k ≡ δN k .
To comply with the mimetic finite difference setting, we define the discrete divergence of
Gh ∈ Vh as an operator ∇h : Vh 7→ Xh such that
N
∇h ·Gh ≡
2
1 XX
g̃kj
|E|
i = k + j − 1.
k=1 j=1
The MPFA quadrature is defined using a matrix Λ which is sparse and, more specifically,
block-diagonal, which is why the method provides local explicit fluxes. The quadrature over the
element E can then be decomposed into the sum over its quadrilaterals, i.e, for Fh , Gh ∈ Vh :
t
[Fh , Gh ]E = G̃ ΛF̃ =
N
X
G̃tk Λk F̃k .
(4)
∀E ∈ Th .
(5)
k
Set K0 equal to the constant tensor on E such that
kK − K0 kL∞ (E) . hE
The MPFA quadrature matrix Λk for the O-method is defined for each quadrilateral k as
Λk = Rk K0−1 Q−t
k
(6)
where the two matrices Rk and Qk are defined, in 2D, as
O
I
O|
I |
lk1 |y −lk2
lk1 |x lk1
y
y
.
and
Qk =
Rk =
O|
O|
I |
I
−lk1
lk2
lk2
x
x
x lk2 |y
We define the scalar product on Xh as
X
(ph , qh )Xh =
(ph , qh )0,E
∀ph , qh ∈ Xh
E∈Th
11th European Conference on the Mathematics of Oil Recovery — Bergen, Norway,
8 - 11 September 2008
(7)
The discrete version of (2) using the MPFA quadrature is then: find (Fh , ph ) ∈ (Vh , Xh ) such
that
(P
∀Gh ∈ Vh ,
E∈Th [Fh , Gh ]E − (ph , ∇h ·Gh )Xh = 0
(8)
(∇h ·Fh , qh )Xh = (f, qh )0,Ω
∀qh ∈ Xh .
This formulation coincides with the MPFA formulation presented for instance in [2, 1, 10]. Note
that [Fh , Gh ]E is not a scalar product, as Λk is in general non-symmetric. The matrix Λ can be
decomposed into its symmetric and skew-symmetric parts
Λ = 12 (Λ + Λt ) + 12 (Λ − Λt ) = ΛS + ΛA
(9)
and similarly Λk = (ΛS )k + (ΛA )k . We define the following scalar product on Vh :
X
(Fh , Gh )Vh =
(Fh , Gh )ΛS (E) ;
(Fh , Gh )ΛS (E) = G̃t ΛS F̃ .
E∈Th
For this scalar product to be valid, Λs must be positive definite. We make the following assumptions on ΛS : there exists two positive constants c and c̄ independent of hE and E such that
∀Gh ∈ Vh and ∀E ∈ Th
c
2
N X
X
G̃2kj ≤ (G, G)ΛS (E) ≤ c̄
2
N X
X
G̃2kj .
(10)
k=1 j=1
k=1 j=1
The validation and implication of this assumption is discussed in the last section. We can now
define the following norms based on the scalar products:
kGh k2Vh = (Gh , Gh )Vh ,
kGh k2Vh (E) = (Gh , Gh )ΛS (E)
∀Gh ∈ Vh
(11)
kqh k2Xh = (qh , qh )Xh ,
kqh k2Xh (E) = (qh , qh )0,E
∀qh ∈ Xh .
(12)
Finally we define the interpolation operator I of any vector-valued function V ∈ [L2 (Ω)]d ∩
H(div, Ω) so that VI ∈ Vh and
(VI )k,j = 0.5|si |−1 (V · nEi , 1)0,si
∀E ∈ Th ∀si ∈ ∂E, i = k + j − 1.
(13)
An important property of this interpolation is that its discrete divergence is equal to the L2
projection on E of the divergence of the continuous function, i.e.
N
∇h ·VI =
1
1
1 X
(V ·nEi , 1)si =
(V ·nE , 1)0,∂E =
(∇·V, 1)0,E .
|E|
|E|
|E|
(14)
i=1
In particular, setting F = −K∇p,
∇h ·FIh =
1
1
(∇·(−K∇p), 1)0,E =
(f, 1)0,E .
|E|
|E|
(15)
Theoretical results
The MPFA quadrature [·, ·]E can be seen as an approximation of the integral (K −1 ·, ·)0,E ,
which, using the divergence theorem on the element E for v ∈ H 1 and W ∈ H(div) is equal to
(K −1 K∇v, W )0,E = −(v, ∇·W )0,E + (v, W · nE )0,∂E
= −(v, ∇·W )0,E +
N
X
li−1 (v, W · nEi li )0,si
i=1
where we have rewritten the border integral to more easily see the analogy with the discrete
fluxes. A similar relation is thus sought for the MPFA method, indication that the MPFA quadrature is exact on linear fields.
11th European Conference on the Mathematics of Oil Recovery — Bergen, Norway,
8 - 11 September 2008
Lemma 1. For any linear field v the MPFA method satisfies the following discrete divergence
theorem:
[K0 ∇v, Gh ]E = −(v, ∇·h Gh )0,E +
N
X
|si |−1 (v, (Z G̃)i )0,si
∀Gh ∈ Vh .
(16)
i=1
Proof. Observe that the integral (v, ∇h ·Gh )0,E is zero since the divergence is constant on the
element E and the barycenter of the element is the local origin. Set v = xj with 1 ≤ j ≤ 2.
The two half-edge fluxes of K0 ∇v on the quadrilateral k are equal to Qtk K0 ∇v, as K0 ∇v is
constant and each column of the matrix Qk corresponds to the integral of the normal vector of
one of the half-edges. Using the quadrature of the MPFA method (see (4) and (6)), the left-hand
edge of (16) is
[K0 ∇v, Gh ]E =
N
X
G̃tk Λk Qtk K0 ∇xj =
N
X
t
G̃tk Rk K0−1 Q−t
k Qk K0 ∇xj =
G̃tk Rk ∇xj .
k=1
k=1
k=1
N
X
(17)
Regarding the right-hand side of (16) we first note that
I
|xj
|si |−1 (v, 1)0,si = li1
since we have chosen to partition the element into quadrilaterals using the midpoints of the
I | , we rewrite the border integral as
edges. Introducing the vector R whose ith entry is li1
xj
N
X
|si |−1 (v, (Z G̃)i )0,si = G̃t Z t R
i=1
I
I ,
Since l(k+1)1
= lk,2
I
I
I
I
t
Z t R = (l11
|xj , l12
|xj , . . . , lN
1 |xj , lN 2 |xj ) .
Writing the product as a sum over the nodes we obtain
t
t
G̃ Z R =
N
X
I
I
|xj )t = G̃tk Rk ∇xj ,
|xj , lk2
G̃tk (lk1
(18)
k=1
which proves that (18) is equal to (17) for all G̃k , and thus (16) is satisfied for all Gh ∈ Vh .
Theorem 2. Let (p, F ) be the solution of the continuous problem (2), and let (ph , Fh ) be the
solution of the discrete problem (8). Let FI ∈ Vh be the interpolant of F defined by (13). Then,
if p ∈ H 2 (Ω),
kFI − Fh kVh . hkpkH 2 (Ω) .
Proof. We first note that for all Fh , Gh ∈ Vh
[Fh , Gh ]E = (Fh , Gh )ΛS +
N
X
G̃tk (ΛA )k F̃k ≤ (Fh , Gh )ΛS +
N
X
k(ΛA )kL∞ (Ek ) G̃tk F̃k
k=1
k=1
≤ (1 + C)(Fh , Gh )ΛS ≤ (1 + C)kFh kVh (E) kGh kVh (E) ,
where
C = max
k
k(ΛA )kL∞ (Ek )
,
min(λk )
11th European Conference on the Mathematics of Oil Recovery — Bergen, Norway,
8 - 11 September 2008
(19)
(20)
and λk denotes the eigenvalues of (ΛS )k . Furthermore we note that ∀Fh ∈ Vh ,
X
X
[Fh , Fh ]E .
(Fh , Fh )ΛS (E) =
kFh k2Vh =
E∈Th
E∈Th
Then, using the discrete problem (8),
X
X
X
kFI − Fh k2Vh =
[FI − Fh , FI − Fh ]E =
[FI , FI − Fh ]E −
[Fh , FI − Fh ]E
E∈Th
=
X
E∈Th
E∈Th
[FI , FI − Fh ]E − (ph , ∇h ·(FI − Fh ))Xh
E∈Th
where the second term on the right-hand side is zero due to (15). The rest of the proof follows
from interpolation properties, (5) and (19), and retraces the proof proposed by Brezzi, Lipnikov
and Shashkov [3].
Theorem 3. Let (p, F ) be the solution of the continuous problem (2), and let (ph , Fh ) be the
solution of the discrete problem (8). Let Π0 p indicate the L2 projection of p on each element E.
Then, if p ∈ H 2 (Ω),
kph − Π0 pkXh . h(kpk2,(Ω) + hkf k0,Ω )
Proof. The proof is based on the well-posedness of the dual problem
(
−∇·(K∇ψ) = Π0 p − ph
in Ω
ψ=0
on ∂Ω.
(21)
Then, for convex domains,
kψk2,Ω . kΠ0 p − ph kXh
(22)
Setting H = K∇ψ we define Gh ∈ Vh as the interpolant of H as defined by (13). Then, since
Π0 p − ph is a constant on each element E and due to property (14),
∇h ·Gh = ph − Π0 p.
Using (8) and (21), we obtain
X
X
[Fh , Gh ]E + (K∇p, ∇ψ)0,Ω
[Fh , Gh ]E − (p, ∇·(K∇ψ))0,Ω =
kph − Π0 pk2Xh =
E∈Th
E∈Th
=
X
E∈Th
[Fh , Gh ]E + (f, ψ)0,Ω ≤=
X
[Fh , Gh ]E + h2 kf k0,Ω kψk2,Ω .
E∈Th
The rest of the proof relies on interpolation results, equations (5) and (19) and the stability of
the dual problem (22). See [3] for more details.
Discussion and numerical test
An important part of the analysis above is the assumption regarding the symmetric matrix ΛS ,
namely (10), required to establish coercivity and convergence of the method. This condition is
linked to the definition of the MPFA dual mesh and the permeability tensor K. The dual mesh
of the MPFA method creates quadrilateral subcells on all 2D cell shapes, from triangulations to
general polyhedral meshes, see Figure 2a. To fulfill the lower bound of (10), the eigenvalues of
(Λs )k for all k of E and all E ∈ Th have to be strictly positive. To ensure this, det(ΛS ) > λ0 >
0. By dividing Λ into a symmetric and a skew-symmetric part, cf. (9), it can easily be shown
that, in 2D,
det Λ = det(ΛS ) + det(ΛA ).
11th European Conference on the Mathematics of Oil Recovery — Bergen, Norway,
8 - 11 September 2008
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Fig. 2a: Example of polyhedral mesh with the dualmesh
All subcells are quadrilaterals.
Mesh — β
8×8
16 × 16
32 × 32
64 × 64
Conv. rate last step
Percentage of subcells violating (10)
1
0.0242
0.0059
0.0015
0.0004
1.9809
0
10
0.0766
0.0178
0.0040
0.0009
2.0955
25
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig 2b: The 4 × 4 mesh, used for the numerical test
shown in Table 1 and 2
20
0.1082
0.0255
0.0055
0.0012
2.1928
50
50
0.1567
0.0399
0.0087
0.0018
2.3093
75
100
0.1908
0.0549
0.0125
0.0025
2.3128
100
1000
0.2446
0.1035
0.0375
0.0098
1.9299
100
Table 1: The error kph − p(xcell center )kXh , and the influence of violating (10)
Furthermore det Λk = det Rk det K −1 det Qk , where det Rk can be visualised as the area
I and lI , see Figure 1b. Similarly det Q is the area spanned by lO and lO .
spanned by lk1
k
k2
k1
k2
Denote the respective areas AI and AO . On subcell k we then have
det((Λs )k ) =
1
1
AI AO − (l1I K −1 l10 − l2I K −1 l20 )2 .
det K
4
(23)
For K-orthogonal meshes the last term vanishes, the method is symmetric and the lower bound
is always fulfilled. For other meshes the assumptions in equation (10) will be the dominant mesh
restrictions. Unfortunately, this restriction gives a rather conservative condition on the mesh and
K, and is not necessarily optimal. Figure 3 shows different cell shapes where we deform the
cells by pulling one vertex out, see the arrow in Figure 3. For K = I the condition breaks down
for at least one subcell when we have pulled the vertices out by 8.9h for the triangle, 3h for the
quadrilateral and 2.6h for the polyhedral. Here h denotes the original edge length. This shows
that the allowed deformation is unfortunately quite small. Since Λ is an combination of both
the cellshape and K, cf. (6), we can equivalently to deforming the mesh vary the permeability.
In Table 1 and 2 we show a numerical test with K = diag(β, 1) where β is increased from
1 to 1000. The mesh is shown in Figure 2b, where refinement is a replication of the shown
4 × 4 mesh. The data is chosen so that p(x, y) = cos(2πx) cos(2πy) is the exact solution on
the domain which is the unit square. To test convergence when ΛS has negative eigenvalues,
we set ΛS = β −1 I in the norm, where I is the identity matrix. The convergence results and
the percentage of subcells for which the mesh conditions break down are found in Table 1 for
the pressure and Table 2 for the flux. In the chosen norms the flux converges with optimal first
order, while the pressure converges with second order.
Fig. 3: Twisted cells. The cells are fixed in vertices marked with a cross, while the verices marked with red dots are
pulled out untill the condition (10) does not hold anymore.
11th European Conference on the Mathematics of Oil Recovery — Bergen, Norway,
8 - 11 September 2008
Mesh — β
8×8
16 × 16
32 × 32
64 × 64
Conv. rate last step
Percentage subcells violating (10)
1
1.0416
0.5025
0.2479
0.1234
1.0065
0
10
1.3541
0.6124
0.2919
0.1437
1.0229
25
20
1.4895
0.6602
0.3067
0.1490
1.0417
50
50
1.6524
0.7277
0.3276
0.1547
1.0822
75
100
1.7490
0.7797
0.3462
0.1595
1.1181
100
1000
1.8876
0.9123
0.4219
0.1873
1.1714
100
Table 2: The error β −1 kFh − (F(xedge center ))I kVh (Λ=I) , and the influence of violating (10).
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11th European Conference on the Mathematics of Oil Recovery — Bergen, Norway,
8 - 11 September 2008