A036
On Rough Grids - Convergence and Reproduction
of Uniform Flow
R.A. Klausen* (University of Oslo) & A.F. Stephansen (University of
Bergen)
SUMMARY
In reservoir simulation the flow driven by pressure differences and
gravity is usually approximated by means of the empirically derived
Darcy's law. The elliptic pressure equation that results is preferably
solved with a mass conservative method, for which several candidates are available. We distinguish
between full field methods and raw field methods.
On the one hand we have continuous full field methods which
approximate the velocity with a field that is defined inside the
element itself. On the other hand we have discrete methods that
approximate the flux, and for which no velocity field is defined
inside the elements. We denote this raw field.
We will examine some mass-conservative methods on the basis of two
different characteristics: uniform flow reproduction and convergence
on rough grids.
Analysis apart, we propose an interpolation on polygons that can
reproduce uniform flow by using edge basis functions constructed by
means of barycentric functions. This permits us to obtain an H(div)
field starting from discrete fluxes evaluated at the interfaces of the elements. We also show that a similar
interpolation can be constructed to obtain an H(curl) field on polygons in 2D.
ECMOR XII – 12 th European Conference on the Mathematics of Oil Recovery
6-9 September 2010, Oxford, UK
Introduction
Numerous considerations must be taken into account when choosing a numerical scheme, balancing
accuracy, flexibility and solving efficiency. Our main focus here is the elliptic pressure equation formulated by means of the Darcy velocity. Desirable properties when dealing with oil field simulations are
amongst others mass conservation, explicit local fluxes, possibility of dealing with anisotropies and grid
flexibility. We will here discuss rough grids, polygonal grids and reproduction of uniform flow and what
this implies for the numerical method. It will here be useful to distinguish between methods that define
the velocity field only at the interfaces from those that define a velocity field over the whole domain. We
are particularly interested in the former, which comprises the mimetic finite difference method and the
multi-point flux approximation methods.
Some attention has been paid lately to convergence on rough grids versus convergence on smoothed
grids. A method which converge on rough grids say something more of the robustness of the method
with respect to skew and distorted grid cells, than a method which only converge on meshes with cells
approaching parallelogram. If a method converges on rough grids it is possible to take into account
the geology on a finer scale when refining and be assured that the method gives an improved result.
The definition of rough grids leads us to focus first on quadrilateral grids for then to consider the problems regarding polygonal grids. Approximations of physical processes involving for instance geological
structure further call for a rough non orthogonal mesh. The possibility to apply shape regular polygons
instead of shape regular tetrahedron in layered materials can reduce the need for cells by a factor of 100.
As the flow can be considered constant in a large part of the domain of an oil field, numerical schemes
which are exact for linear pressure fields have been sought. Our discussion shows the importance of
providing an interpolation of the fluxes that is exact for linear flow fields. In the last section of the paper
we therefore present barycentric interpolation on polygonal grids in H(div) which reproduce uniform
flow. A similar interpolation is also presented that reproduces a uniform flow field in H(curl). The
interpolations are presented with edge based basis functions.
Preliminaries
Let Ω be a bounded domain in R2 with polygonal boundary ∂Ω. Let Wkp (E) be the Sobolev space on the
domain E ⊆ Ω, defined to be the subset of Lp such that a function and its weak derivatives up to order
k have a finite Lp norm. In particular we will make use of the space L2 (E) of functions whose square is
1/2
Lebesgue-integrable, with inner product (· , ·)0,E and norm · E = (· , ·)0,E . Also, let H 1 (E) denote
the Sobolev space of first order differentiable functions in L2 (E). We let the operator name curl denote
both the rotation and the curl, dependent on whether it is applied to a scalar or a vector. Furthermore we
define the spaces
H(div; E) = {v ∈ (L2 (E))2 : div v ∈ L2 (E)},
equipped with the norm vdiv,E = (v20,E + div(v)20,E )1/2 , and
H(curl; E) = {v ∈ (L2 (E))2 : curl v ∈ L2 (E)},
equipped with the norm vcurl,E = (v20,E + curl(v)20,E )1/2 . The notation H(div) and H(curl)
will be used when the domain considered is Ω.
The mixed formulation of the elliptic pressure equation is obtained by using both the pressure p and the
Darcy velocity u = −K grad p as variables. We seek (u, p) ∈ H(div) × L2 (Ω) such that
(K −1 u, v)0,Ω − (p, div v)0,Ω = 0,
(div u, q)0,Ω = (f, q)0,Ω
∀v ∈ H(div),
∀q ∈ L2 (Ω),
ECMOR XII – 12th European Conference on the Mathematics of Oil Recovery
6 – 9 September 2010, Oxford, UK
(1)
where homogeneous Dirichlet boundary conditions are imposed for the pressure. For the well posedness
of the system let f ∈ L2 (Ω) and let the permeability tensor K be a symmetric, positive definite field in
1 (Ω)]2,2 whose smallest eigenvalue is bounded from below by a positive constant.
[W∞
Raw field and full field methods
For mass-conservative methods for the mixed pressure equation (1) a classification into K and K −1
methods was introduced by Klausen and Russell [12]. For K methods the flux is calculated as a function
of the pressure differences (or gradient), while for a K −1 method the pressure differences are calculated
as a function of the fluxes. The former methods can be put in relation to finite volume methods, while
the latter can be seen as a discretization of the mixed formulation of the pressure equation. Some
methods can be specified both as one or the other, but the implementation will be different in the two
cases. The K −1 methods we will consider are the mixed finite element (MFE) method, the Mimetic
Finite Difference (MFD) method, the corner velocity interpolation (CVI) finite element method and the
control volume MFE (CVMFEM ) method. The only K methods which we will examine are the MPFA
methods - the symmetric MPFA O-method and the non-symmetric MPFA O- and L-methods.
It is however also interesting to distinguish between what we have termed raw field and full field methods: by a raw field method we intend a method that specifies the velocity only at the interfaces of the
elements and not inside the elements themselves. Raw field methods include the MFD method and the
MPFA methods. Conversely, a full field method will specify basis functions for the elements. Examples
of full field methods are the Raviart-Thomas (RT) and the Arnold-Boffi-Falk (ABF) MFE methods and
the CVI method. The CVMFEM method is originally derived using the RT0 basis functions, but as only
the fluxes are used as unknowns this method can be viewed as a raw field method.
While a full field method can give the velocity field at the interfaces through projection, a raw field
method can give a full velocity field by means of interpolation. The distinction will prove useful when
discussing the characteristics that we are concerned with here, namely definition on quadrilateral and
polygonal grids, rough grid convergence and reproduction of uniform flow. Finally we note that the
classification into raw field and full field methods is not limited to methods that solve the pressure
equation (1), but can be applied in general.
Rough grid convergence and quadrilateral grids
Let {Th } denote a family of partitions of Ω into convex quadrilateral elements, where h is the maximum
element edge. It is possible to refine the mesh uniformly by dividing each cell edge in its midpoint
for each refinement level, so that the quadrilaterals become successively smoother as h tends to zero.
More specifically, the family of meshes created when refining is said to be asymptotic h2 –parallelogram
meshes if there exits a constant c, independent of h, such that
|Fx̂ŷ | ≤ ch2
(2)
where F denotes the mapping from the unit square to the quadrilateral. Other names used on the same
family of refined meshes are smooth meshes, uniformly refined meshes, h2 – meshes, etc. A further
discussion of this class of grids can for instance be found in [14]. General quadrilateral grids without
any asymptotic refinement condition on Th are referred to as rough grids.
To seek a discretized velocity space that is in H(div) leads to restrictions that are more difficult to satisfy
when using a full field method compared to a raw field method. For a raw field method all that is needed
is to assign a unique value to the normal velocity at each interface. For a full field method the choice of
basis functions inside the element must be consistent with the assigned normal direction at the interface.
We also note that as the definition of smooth grids is based on the mapping from the unit square to
the quadrilateral, the lack of rough grid convergence can manifest itself for methods that make use of
this mapping. For raw field methods we can here mention the symmetric MPFA method proposed in
ECMOR XII – 12th European Conference on the Mathematics of Oil Recovery
6 – 9 September 2010, Oxford, UK
x3
(0, 1) (1, 1) x4
Ê
FE
(0, 0) (1, 0) x1
E
x2
Figure 1 Mapping of quadrilateral cell the unit square
[2, 14]. For full field methods the definition on the elementary unit square is used by for instance the
RT elements [4] and the ABF elements [3]. For these methods the basis functions are defined on the
unit square and the Piola mapping is used to define the basis functions on the quadrilateral. To examine
rough grid convergence we therefore take a closer look at the definitions involved.
Let xi = (xi , yi ), i = 1, 2, 3, 4 be the vertices of a quadrilateral, cf. Figure 1. The transformation
FE = F can then be written as
F (x̂, ŷ) = (1 − x̂)(1 − ŷ)x1 + x̂(1 − ŷ)x2 + x̂ŷ x3 + (1 − x̂)ŷ x4 =
4
φi xi
(3)
i=1
for (x̂, ŷ) ∈ (0, 1) × (0, 1). The Jacobian matrix of F is denoted D and J denotes the Jacobian determinant. The vector space is then defined on the reference square, and transferred to physical space by
the Piola mapping. If v̂ is a vector field on the reference space, we define a vector field v by the Piola
transform P, i.e.
1
v(x) = P v̂(x) = Dv̂ ◦ F −1 (x).
J
Let us examine the lowest order Raviart-Thomas mixed finite elements RT 0 × Q0 ⊂ H(div) × L2 (Ω).
ˆ 0 is spanned by {(1, 0)T , (0, 1)T , (x, 0)T , (0, y)T }. As F is a bilinear map,
On the reference space RT
D and J are not constants unless the cell is a parallelogram. This means that div v is not a polynomial
even if v̂ is so, and we do not have the commutative quality div RT 0 ⊂ Q0 , where Q0 are cell wise
constants. This can be seen as the source for the missing convergence in H(div) for the lowest order
Raviart-Thomas mixed finite element methods. As the condition is fulfilled in the reference space the
inf-sup condition is satisfied and both the pressure and the velocity field converge in the L2 norm on
rough grids.
In [3], Arnold, Boffi and Falk give conditions on vector elements defined by the Piola mapping for L2
convergence of the discrete velocity and divergence on quadrilateral grids. For the lowest order Raviartˆ 0 should contain the vector space
Thomas elements, this means that the discrete reference space RT
T
T
T
ˆ 0 should contain {1, x, y} which
spanned by {(1, 0) , (0, 1) , (x, −y) } which is the case, while div RT
is not fulfilled. The ABF0 element proposed in the same article does however satisfy both conditions
and converge in H(div) on rough grids.
As the velocity field of RT0 converges on rough grids, it is possible to postprocess the divergence to
ˆ 0 projection on the reference cell, with Πh = P Π̂P −1 ,
obtain convergence in H(div). Let Π̂ be the RT
cf. [4]. This can also be treated as an interpolation from the raw field. Define the projection Rh on each
cell as Rh = P R̂P −1 , for all v ∈ (H 1 (Ω))2 such that
div Π̂v̂
(J(1, 0) − J(0, 0))x̂(x̂ − 1)
.
R̂E v̂ = Π̂v̂ +
2J(1/2, 1/2) (J(0, 1) − J(0, 0))ŷ(ŷ − 1)
(4)
The Jacobian determinant evaluated in the cell vertices returns the area spanned by the adjacent cell
edges, and J(1/2, 1/2) gives the area of the quadrilateral itself. The construction in (4) therefore ensures div Rh v = div(Π̂v̂)/Jc ∈ P0 (E), ∀E ∈ Th . It can also be shown that the divergence of the
ECMOR XII – 12th European Conference on the Mathematics of Oil Recovery
6 – 9 September 2010, Oxford, UK
reconstruction based on the lowest order Raviart-Thomas elements, div Rh , converges in L2 . More
details can be found in [13].
Methods that are defined directly on the grid without the use of the mapping F would be expected to
converge on rough grids. Amongst raw field methods in this category we have the MFD method and
the non-symmetric MPFA methods. All of these converge on rough grids. As an example of a full field
method that is defined directly on the quadrilateral we have the CVI method [17]. This method however
does not converge on rough grids, as the divergence of the velocity field is not comprised in the pressure
space, neither in the physical space nor (if operating an inverse mapping) in the reference space. A
postprocessing of the divergence would only work if the velocity field itself converged. However, when
the elements are parallelograms the CVI method coincides with RT0 and in this case convergence is
obtained.
Rough grid convergence and polygonal grids
Polygons are flexible and relatively easy to match different geological features. Faults and wells can
be particularly challenging to grid in a satisfactory manner, and polygons can come in handy here too.
A polygonal grid can be seen to have the characteristics of a rough grid. While the grid with general
polygonal cell of more than four vertices can be subdivided into quadrilaterals, the refinement of the
grid will not lead to polygons composed of parallelograms.
The distinction between raw field and full field methods is also interesting in this context, as the former
type of method is in general more easily extended to polygonal grids. Many full field methods are developed for triangular or quadrilateral grids, but their extension to general polygons is often not immediate.
This is due to the necessity of creating an H(div) field that is compatible with element defined basis
functions. One way of solving this problem is to subdivide the polygon into either triangles or quadrilaterals and define the basis functions on the subdivided elements [10, 16]. Alternatively one can use edge
based basis functions like the ones proposed in the last section and which would be the extension of the
CVI method to polygons. In both cases it is desirable to choose a method that converges on rough grids.
Amongst raw field methods for (1) that have been extended successfully to polygons, we find the MFD
method [5, 15] and the MPFA O-method [11] and L-method [20, 21]. For the control volume MFEM
the extension is not immediate as it has been developed for quadrilaterals and employ the mapping of
the element to a basis element. A subdivision of the polygon into quadrilaterals may be sufficient to
extend the method, as indeed it is for the symmetric MPFA method. However, the problem of rough grid
convergence makes such extensions undesirable.
Uniform flow
The definition of reproduction of uniform flow field is usually taken to be that constants are included
in the velocity field. This definition however only makes sense for methods where a velocity field
is specified inside the elements themselves, i.e. for full field methods. For a raw field method the
reproduction of a uniform flow field is intrinsically linked to the definition of the method itself and the
specification of a pressure field. To say that a raw field method reproduces a linear pressure field (or
uniform flow) means that the method applied to this field reproduces the correct fluxes. The fluxes can
then be interpolated to give a constant velocity field. The pressure calculated in the process need not be
linear, however.
For a raw field K −1 method the integration of the velocity field over the element is substituted with a
quadrature term. As the K −1 methods are obtained from a discretization of the divergence theorem, the
methods themselves specify a discrete divergence theorem. The fluxes of a linear pressure field should
therefore fulfill the discrete divergence theorem exactly - i.e. produce a raw field that is consistent with
that of the method.
ECMOR XII – 12th European Conference on the Mathematics of Oil Recovery
6 – 9 September 2010, Oxford, UK
The MFD method consists in finding (uh , ph ) ∈ (Vh , Xh ) such that
E∈Ph [M uh , vh ]E − (ph , divh vh )Xh = 0
(divh uh , qh )Xh = (f, qh )0,Ω
∀vh ∈ Vh ,
∀qh ∈ Xh .
(5)
The velocity space Vh is specified by associating a velocity field at each interface aligned with the
normal component of the interface. The velocity field is continuous at interfaces shared by neighboring
elements. The pressure space Xh is specified as being constant at each element. [M ·, ·]E is a (symmetric)
quadrature that substitutes the integral (K −1 ·, ·)0,E , while divh (·) is defined as the sum of the fluxes out
of the element divided by the area (in 2D) of the element.
The discrete divergence theorem defined by the method is
[M K0 grad q h , vh ]E = −(q, divh vh )0,E + (q, vh · nE )0,∂E
∀vh ∈ Vh
valid for any q linear on the element E.
To assure that a linear flow field induces the correct fluxes is often the basis of construction for the raw
field K-methods. The pressure is specified in a certain number of nodes, one interpolates the pressures
in between the nodes with a (piecewise) linear function, and the fluxes are calculated on the interfaces
based on the gradient of the linear function. Rewriting a raw field K method as a K −1 method we also
find that they satisfy a discrete divergence theorem. For the MPFA O-method the quadrature matrix in
(5) is substituted with a block-diagonal matrix ΛO
E , and the velocity field Vh is (in 2D) doubled, so that
the flux at each half-side is specified. For more details see [11]. The discrete divergence theorem is then
written
E
[ΛO
|Fi |−1 (q, ṽiE )0,Fi
∀vh ∈ Vh .
(6)
E K0 grad q, vh ]E = −(q, divh vh )0,E +
Fi ∈∂E
The fluxe over an interface i is here indicated by ṽiE . For the MPFA L-method, which can be rewritten
as a K −1 method with some restrictions, the quadrature matrix ΛL
E can have entries that are either at the
diagonal or that are block-diagonal (see[21] for details in 2D). The discrete divergence theorem is not
satisfied element by element, but is correct when summing over the whole of the domain
E
[ΛL
K
grad
q,
v
]
=
−
q(x̄E
∀vh ∈ Vh
E
h
E 0
0 )|E| divh vh |E
E∈Ph
E∈Ph
For raw field methods then, the reproduction of a uniform flow field is reduced to getting the right fluxes
and a decent pressure approximation. How the field can be approximated inside the element is not a
matter of concern for raw field methods. The discrete divergence theorem being exact for linear pressure
fields is a measure of correct quadrature being used.
For full field methods the integration is exact in any case, and the divergence theorem says something
about the validity of the choice of spaces. To say that the method is exact for linear flow fields should,
if to be consistent with the raw field case, imply not only that the constants are contained in the velocity
space, but that the pressure is well approximated. If looking at the divergence theorem, this means that
(q − ph , div v) = 0 for all ph ∈ Ph , v ∈ Vh , and q being the exact pressure field. With div v ∈ Ph ,
ph becomes an L2 projection of q onto the space Ph . The compatibility of the divergence space and the
pressure space is indeed a problem for the CVI method, as mentioned earlier.
Interpolation on polygons
We have seen that raw field methods are more easily extended to polygons than full field methods, but
as they only provide the velocity field at the interface of the element an appropriate interpolation is
ECMOR XII – 12th European Conference on the Mathematics of Oil Recovery
6 – 9 September 2010, Oxford, UK
ti+1
xi
Bi
ti
ti−1
xi−1
Figure 2 The vertices and edges of a polygonal cell.
needed. Apart from providing a flow field, a good interpolation is also needed for instance in proving
superconvergence of the pressure (if possible). We will here present an interpolation of fluxes that is a
generalization of the CVI [9] velocity field to polygons.
Let E ⊂ R2 be a convex polygon with vertices x1 , x2 , . . . , xn , n ≥ 3, numbered in an anticlockwise
direction. Define the edge vector ti associated with Fi ∈ ∂E, for i = 1, . . . , n also numbered in an
anticlockwise direction. The length of ti is equal to the length of the edge Fi itself. The numbering of
vertices and edges is shown in Figure 2. Also define the triangle area Bi associated with the vertex xi
of E, spanned by the edges Fi and Fi+1 and equal to 12 ti × ti+1 Assume we have a set of barycentric
functions {φi } on E. These can either be Wachspress coordinates or mean value coordinates for n ≥ 5,
cf. [22, 8]. Then
φi (x) ≥ 0,
(7)
φi (xj ) = δij
with i, j = 1, . . . , n, and
φi (x) = 1,
(8)
for x ∈ E.
(9)
Barycentric coordinates also provide linear precision, ie.
x=
xi φi , for x ∈ E.
(10)
i
i
An example of barycentric coordinates on a quadrilateral are {φi } defined in equation (3).
Interpolation of H(div) raw fields
Definition 1. Let {φi } be a set of barycentric functions on a n-polygon. For every edge Fi ⊂ ∂E define
ψ i (x) =
ti−1
ti+1
φi−1 (x) −
φi (x).
2Bi−1
2Bi
(11)
Lemma 1. Let nj , j = 1, . . . , n, be the outer normal vector to the edge Fj , with length equal to the
length of Fj . Let xFj ∈ Fj . Then
ψ i (xFj ) · nj = δij .
Proof. We have ti−1 · ni = ti−1 × ti = 2Bi−1 , and similarly ti+1 · ni = ti+1 × ti = −2Bi . So
ψ i (xFi ) · ni = φi−1 (xFi ) + φi (xFi ) = 1
Further, the scalar product ti+1 · ni+1 is equal to zero, so
ψ i (xFi+1 ) · ni+1 =
ti−1 · ni+1
φi−1 (xFi+1 ) = 0,
2Bi−1
since φi−1 is zero on all edges except the two edges Fi−1 and Fi adjacent to vertex xi−1 . Similarly
ψ i (xFi+1 ) · nj = 0 for all j = i.
ECMOR XII – 12th European Conference on the Mathematics of Oil Recovery
6 – 9 September 2010, Oxford, UK
Given a discrete set {uj } associated to the edges of Th , we have that the interpolated field ui ψ is in
H(div). We also note that the basis function {ψ i } reproduce constant vector fields exactly on each cell
E.
Lemma 2. Let ek = (δ1k , δ2,k )t , k ∈ {1, 2}, be the unit vector defined for any x ∈ E, and let ni be the
outer normal vector associated with the edge Fi and with length equal to the edge length. Then
ek =
n
fki ψ i ,
(12)
i=1
with flux fki = (ek · ni ).
Proof. Substituting the edge basis functions from the definition (11), for the right hand side we have
n
fki ψ i =
i=1
=
=
n
(ek · ni ) ti−1
i=1
n
i=1
n
2Bi−1
φi−1 −
n
n
(ek · ni ) ti+1
i=1
φi
(ek · ni ) ti+1
(ek · ni+1 ) ti
φi −
φi
2Bi
2Bi
i=1
[ (ek · ni+1 ) ti − (ek · ni ) ti+1 ]
i=1
For e1 =
2Bi
φi
.
2Bi
(1, 0)t
and ni = ((ni )x , (ni )y )t = ((ti )y , −(ti )x )t
2 1
ti+1 ti − t2i t1i+1
(e1 · ni+1 ) ti − (e1 · ni ) ti+1 = 2 2
= (ti × ti+1 ) e1 = 2Bi e1 .
ti+1 ti − t2i t2i+1
A similar result is obtained for e2 . Summing and using equation (9),
n
fki ψ i =
i=1
n
n
2Bi ek
i=1
φi
= ek
φi = ek .
2Bi
i=1
Interpolation of H(curl) raw fields
Definition 2. Let {φi } be a set of barycentric functions on a n-polygon. For every edge Fi ⊂ ∂E, define
ni−1
ni+1
ϕi (x) = −
φi−1 (x) +
φi (x)
(13)
2Bi−1
2Bi
where nj , j = 1, . . . , n, is the outer normal vector to the edge Fj with length equal to the length of Fj .
Lemma 3. Let tj , j = 1, . . . , n, be the tangential vectors to the edge Fj of an element, directed from
Fj−1 to Fj+1 , with length equal to the length of Fj . Let xFj ∈ Fj . Then
ϕi (xFj ) · tj = δij .
Proof. We have ni−1 · ti = −ti−1 × ti = −2Bi−1 , while ni+1 · ti = −ti+1 × ti = 2Bi . So
ϕi (xFi ) · ti = φi−1 (xFi ) + φi (xFi ),
which equals 1 along edge Fi . As ni+1 · ti+1 = 0 we obtain
ϕi (xFi+1 ) · ti+1 = −
ni−1 · ti+1
φi−1 (xFi+1 ) = 0,
2Bi−1
since φi−1 is zero on all edges except the two edges Fi−1 and Fi adjacent to vertex xi−1 . Similarly,
ϕi (xFj ) · tj = 0 for all j = i.
ECMOR XII – 12th European Conference on the Mathematics of Oil Recovery
6 – 9 September 2010, Oxford, UK
Lemma 4. Let ek = (δ1k , δ2,k )t , k ∈ {1, 2}, be the unit vector defined for any x ∈ E. Then
ek =
n
(ek · ti ) ϕi .
(14)
i=1
Proof. For the right hand side, substituting the definition (13), we have
n
(ek · ti )ϕi =
i=1
n
i=1
=
=
n
i=1
n
n
(ek · ti ) ni+1
(ek · ti ) ni−1
−
φi−1 +
φi
2Bi−1
2Bi
i=1
−
(ek · ti+1 ) ni
φi +
2Bi
n
(ek · ti ) ni+1
2Bi
i=1
− [ (ek · ti+1 ) ni − (ek · ti ) ni+1 ]
i=1
φi
φi
.
2Bi
For e1 = (1, 0)t and ni = ((ni )x , (ni )y )t = ((ti )y , −(ti )x )t
1 2
ti+1 ti − t1i t2i+1
= −2Bi e1 .
(e1 · ti+1 ) ni − (e1 · ti ) ni+1 =
−t1i+1 t1i + t1i t1i+1
A similar result is obtained for e2 . Summing and using equation (9),
n
i=1
(ek · ti )ϕi =
n
i=1
n
2Bi ek
φi
= ek
φi = ek .
2Bi
i=1
We note that in the proof we have not made use of the linear precision of the barycentric coordinates,
ie. equation (10). In [18] Nordbotten and Hægland prove that for general hexahedra in 3D with bilinear
faces, both a local reconstruction of velocity based on the six face fluxes and reproduction of uniform
flow cannot be satisfied in H(div). In 2D the reproduction of uniform flow can be obtained, but linear
precision for functions which fullfill Lemma 1 can not be found. Lemma 1 presents basis functions for
the velocity, not the flux, even if these are equal as defined in the Lemma. The flux has one value for
each edge defined as integral of the normal component of the velocity, the velocity itself on the other
hand is defied as a variable vector field. A linear vector field will not be constant along a general edge,
as (x, 0)T only is constant along vertical lines with fixed x-values. So, assuming constant velocity along
any edge which corresponds to a linear flow field is not possible.
Conclusions
We introduced the differentiation between full field methods, which have element basis functions, and
raw field methods that define a velocity field only at the element interfaces. We discuss various properties
in connection with these schemes, namely rough grid convergence, reproduction of uniform flow and
extension to polygons. The element basis functions used in full field methods often have shortcomings
that can be overcome by instead using a raw field method and adopt a suitable interpolation successively.
Finally we present new interpolation basis functions for H(div) and H(curl) which reproduce constant
vector fields on polygons.
References
[1] I. Aavatsmark, T. Barkve, Ø. Bøe, and T. Mannseth. Discretization on unstructured grids for inhomogeneous,
anisotropic media. Part i: Derivation of the methods. Part ii: Discussion and numerical results, SIAM J.
Sci. Comput, 19, pp. 1700–1736, 1998
ECMOR XII – 12th European Conference on the Mathematics of Oil Recovery
6 – 9 September 2010, Oxford, UK
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ECMOR XII – 12th European Conference on the Mathematics of Oil Recovery
6 – 9 September 2010, Oxford, UK