numerical performance of reduction numerical methods for single-phase flow in rough grids and with...

7/27/2019 Numerical performance of reduction numerical methods for single-phase flow in rough grids and with discontinuous permeability

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Reference Reduction Methods Physical Reduction Methods Numerical Examples Summary and Conclusions

Numerical perormance o reduction numerical methods or

single-phase fow in rough grids and with discontinuous

permeability

Jiangyong Hou

Center or Computational Geosciences, School o Mathematics and Statistics, Xian JiaotongUniversity

October 7, 2013
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Outline

Reerence Reduction MethodsMulti-Point Flux Mixed Finite Element MethodMixed-Hybrid Finite Element Method

Physical Reduction MethodsMPFA O-Method

Local ux mimetic MPFA method

Numerical ExamplesComparisons o robustness on quadrilateral grids

Discontinuous experiments on triangular and quadrilateral grids

Summary and Conclusions
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Reerence Finite element spaces on Quadrilaterals

a1 a2

a3a4

1 2

34

n1 n4

n2

n7

n3

n6

n8 n5

a1 a2

a3a4

n4n1

n3

n6

n8n5

n7

n2

Figure 1.1: A bilinear transormation on quadrilaterals.

1() = 1()2 span{curl(2), curl(2)}

=

1 + 1 + 1 +

2 + 22 + 2 + 2 2

2

and

1

2 (), =

(1 + 1, 1 + 1) (, ) 1(1 + 1, 2 + 2) (, ) 2(2 + 2, 2 + 2) (, ) 3(2 + 2, 1 + 1) (, ) 4
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Reerence Finite element spaces on Triangles

a1 a2

a3

1

2 3

4

56

n1 n4

n2

n5n3

n6

a1 a2

a3

n4n1

n3

n6

n

5

n2

I II

III

Figure 1.2: A linear transormation on triangles.

1() = 1()2 =

{1 + 1 + 12 + 2 + 2

,

1, 1, 1, , R2, 2, 2 R

}and

12 (), =

(1 + 2 + 3)(, ) (2, 1) (, ) 1(1 + 2 + 6)(, ) (2, 1) (, ) 2(3 + 4 + 5)(, ) (5, 4) (, ) 3(2 + 3 + 4)(, ) (2, 4) (, ) 4(1 + 5 + 6)(, ) (5, 1) (, ) 5(4 + 5 + 6)(, ) (5, 4) (, ) 6

R f R d i M h d Ph i l R d i M h d N i l E l S d C l i
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Outline







R f R d ti M th d Ph i l R d ti M th d N i l E l S d C l i
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Multi-Point Flux Mixed Finite Element Method

Find (u, ) (div, ) 2()

(1u, v) (, v) = 0 v

( u, ) = (, )

In quadrilateral meshes, (1u, v) can be dened by symmetric and non-

symmetric quadratures or both 1 and 12

In triangular meshes, the nonsymmetric quadrature o (1u, v) is only

dened in physical 12

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On quadrilateral meshes

SQ1 = 14

4=1

( 1

1 )(a)u(a) v(a)

SQ2 = 14

4=1

( 1

1 )(x,)u(a) v(a)

NSQ1 = 14

4=1

( 1

,

1

)(a)u

(a

) v(a

)

NSQ2 = 14

4=1

( 1

1 ,)(a)u(a) v(a)

On triangular meshes

SQ = 16

3

=1( 1,

,

1 ,)u(a) v(a)

NSQ =

=I,II,III((a3)(a3)(a1)1 )u(a3) v(a3)

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Outline







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y p y

Mixed-Hybrid Finite Element Method

Find (u, , )

(1 u, v) =

(, v)

4=1

(v , )

v

4=1

(u ,) = 0

( u, ) = (, )

where

= {2() | v| 1() or

12 (), }

= e1(e) or (e), (e) is piecewise constant with discontinuity at midpointo e.

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Nonsymmetric rules or Hybrid 1

2

For quadrilaterals

Rule1 :

4=1

1

,1

Rule2 : 11

,1 1

+ 3

1

,3 1

+2

1 ,4 1 +4

1 ,2 1

Rule3 :1

1

,3 1

+

3

1

,1 1

+2

1

,2 1

+

4

1

,4 1

For physical triangles

=I,II,III

,

1

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Outline


Physical Reduction Methods

MPFA O-MethodLocal ux mimetic MPFA method




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MPFA O-Method

Step1. Compute the sub-edge uxes in each subcells o the dual mesh

a

x4x4

x2

x3

x3

x1 x1x2n2

n4

n3

n1

= () = | n

| = 1

{( )

() + ( )

() }

where = (x), = (x), is twice the measure o triangle xxx, () is the outer

normal vector lying on the connection lines between the cell center x and the boundarymidpoints x which is opposite to the midpoint x, and having the same length with theedge on which it lies.

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Step2. Considering the continuity o uxes between two adjacent elements

1 = (1)

1=

(2)

12 =

(1)2 =

(3)2

3 = (2)3 =

(4)3

4 = (3)4 =

(4)4

and eliminating the temporary pressure at the midpoint o sub-edges,

a

= a

a = a + a}

= a = (1+ )a

Step3. Assemble the entire edge uxes, and insert the our uxes into the localbalance equation, that is 1 + 2 + 3 + 4 =

12

22

14

24

13 23

11 21

24

3

1

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Outline









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Local fux mimetic MPFA method

The discretization o the local ux mimetic MPFA in the discrete operator sense:

u =

u =

and in an equivalent variational orm:

Find (u, )

[u, v] [, v] = 0 v

[ u, ] = [, ]



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an2

e2

n1

e1

x2,

x1,

x

Figure 2.1: a = (n1,n2), a =(

12|e1|(x1, x),

12|e2|(x x2,)

)

Symmetric quadrature

[u, v],,a =||

1a

1 a

u,a v,a

Nonsymmetric quadrature

[u, v],,a = a1

a

u,a v,a

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Outline




Numerical ExamplesComparisons o robustness on quadrilateral gridsDiscontinuous experiments on triangular and quadrilateral grids


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Comparisons o robustness on quadrilateral grids

(a) slightly perturbed square mesh (b) 2-parallelogram mesh

(c) -perturbed quadrilateral mesh (d) smooth mesh

Figure 3.1: Quadrilateral meshes used in numerical tests

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Three quality measures dened or the general quadrilaterals

1() =

2 ,

2() =

|e|,|e|,

3() = 1

|e|, |e|,

|e|,

2

12

and

= {() : },

=

1

(), = 1, 2, 3. (3.1)

Mesh 1 2 3 1 2 3

Square 0.7071 1 1 0.7071 1 1Slightly perturbed 0.2333 0.1848 0.2094 0.5643 0.6637 0.62082-parallelogram 0.3187 0.3333 0.3261 0.4863 0.6186 0.5617

Heavily perturbed 0.0052 0.0155 0.0283 0.4512 0.4871 0.4725Smooth 0.4036 0.4933 0.4595 0.5040 0.6898 0.6131

Table 3.1: The qualities o each meshes at the last refnement step

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The problem

() = in

= on

Analytical Solution:

(, ) = ( 1)( 1)(32 + ( 1)sin(2)cos(0.5 1 + ))

The tensor coecient:

(, ) =

( + 1)2 + 2

( + 1)2

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1 2 3 4 5 6 7-14

-13

-12

-11

-10

-9

-8

-7

log2(1/h)

log2

(eL2)

L2 Pressure Error on square mesh

MPFA

MFMFES1

BDM

MFMFENS1

BDM

MFMFES2

BRT

MFMFENS2

BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS1

BRT

MHNS2

BRT

MHNS3

BRT

(a) Pressure error and convergence rate

1 2 3 4 5 6 7-13

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

log2(1/h)

log2

(eL2)

L Velocity Error on square mesh

MPFA

MFMFES1

BDM

MFMFENS1

BDM

MFMFES2

BRT

MFMFENS2

BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS1

BRT

MHNS2

BRT

MHNS3

BRT

(b) Velocity error and convergence rate

Figure 3.2: Error and convergence rate on square mesh

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1 2 3 4 5 6 7-14

-13

-12

-11

-10

-9

-8

-7

log2(1/h)

log2

(eL2)

L2 Pressure Error on slightly perturbed square mesh

MPFA

MFMFES1

BDM

MFMFENS1

BDM

MFMFES2

BRT

MFMFENS2

BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS1

BRT

MHNS2

BRT

MHNS3

BRT


1 2 3 4 5 6 7-13

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

log2(1/h)

log2

(eL2)

L2

Velocity Error on slightly perturbed square mesh

MPFA

MFMFES1

BDM

MFMFENS1

BDM

MFMFES2

BRT

MFMFENS2

BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS1

BRT

MHNS2

BRT

MHNS3

BRT


Figure 3.3: Error and convergence rate on slightly perturbed square mesh

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1 2 3 4 5 6 7-14

-13

-12

-11

-10

-9

-8

-7

log2(1/h)

log2

(eL2)

L Pressure Error on h -parallelogram mesh

MPFA

MFMFES1

BDM

MFMFENS1

BDM

MFMFES2

BRT

MFMFENS2

BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS1

BRT

MHNS2

BRT

MHNS3

BRT


1 2 3 4 5 6 7-13

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

log2(1/h)

log2

(eL2)

L Velocity Error on h -parallelogram mesh

MPFA

MFMFES1

BDM

MFMFENS1

BDM

MFMFES2

BRT

MFMFENS2

BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS1

BRT

MHNS2

BRT

MHNS3

BRT


Figure 3.4: Error and convergence rate on 2-parallelogram mesh

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1 2 3 4 5 6 7-14

-13

-12

-11

-10

-9

-8

-7

log2(1/h)

log2

(eL2)

L Pressure Error on heavily h-perturbed quadrilateral mesh

MPFA

MFMFES1

BDM

MFMFENS1

BDM

MFMFES2

BRT

MFMFENS2

BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS1

BRT

MHNS2

BRT

MHNS3

BRT


1 2 3 4 5 6 7-11

-10

-9

-8

-7

-6

-5

-4

-3

log2(1/h)

log2

(eL2)

L Velocity Error on heavily h-perturbed quadrilateral mesh

MPFA

MFMFES1

BDM

MFMFENS1

BDM

MFMFES2

BRT

MFMFENS2

BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS1

BRT

MHNS2

BRT

MHNS3

BRT


Figure 3.5: Error and convergence rate on heavily -perturbed quadrilateral mesh

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(a) Global (b) Local

Figure 3.6: The last refnement o-perturbed quadrilateral mesh

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5.6 5.7 5.8 5.9 6 6.1 6.2

-13.1

-13

-12.9

-12.8

-12.7

-12.6

-12.5

-12.4

-12.3

log2(1/h)

log2

(eL2

)

L2

Pressure Error on heavily h-perturbed quadrilateral mesh

MPFA

MFMFES1

BDM

MFMFENS1

BDM

MFMFES2

BRT

MFMFENS2

BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS1

BRT

MHNS2

BRT

MHNS3

BRT

Figure 3.7: Partial enlargement o error and convergence o pressure in Figure 3.5a

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1 2 3 4 5 6 7-14

-13

-12

-11

-10

-9

-8

-7

log2(1/h)

log2

(eL2)

L Pressure Error on smooth mesh

MPFA

MFMFES1

BDM

MFMFENS1

BDM

MFMFES2

BRT

MFMFENS2 BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS1

BRT

MHNS2

BRT

MHNS3

BRT


1 2 3 4 5 6 7-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

log2(1/h)

log2

(eL2)

L Velocity Error on smooth mesh

MPFA

MFMFES1

BDM

MFMFENS1

BDM

MFMFES2

BRT

MFMFENS2 BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS1

BRT

MHNS2

BRT

MHNS3

BRT


Figure 3.8: Error and convergence rate on smooth mesh

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Outline




Numerical ExamplesComparisons o robustness on quadrilateral gridsDiscontinuous experiments on triangular and quadrilateral grids


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Considering the the ollowing discontinuous examples which are taken rom [Eigestad,2005,Aavatsmark,2006].

2

3

1102

1 102

(a) =0.13448835

2

3

110

1 10

(b) =0.41033296

2

3

103103

103 1

(c) =1.47865601

For (a) and (b), considering the solution

(, ) = (cos() + sin()),

For (c), the solution is

(, ) =

cos( /3) or [0, 2/3],cos(/3)

cos(2/3) cos(4/3 ) or [2/3, 2].

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Discontinuous experiments on quadrilateral grids

0 1 2 3 4 5 6-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

log2(1/h)

log2

(eL2)

L2 Pressure Error of discontinuous solution

MPFA

MFMFES1

BDM

MFMFENS1

BDM

MFMFES2

BRT

MFMFENS2

BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS1

BRT

MHNS2

BRT

MHNS3

BRT

(d) Pressure error and convergence rate

0 1 2 3 4 5 64

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

5.8

6

log2(1/h)

log2

(eL2)

L Velocity Error of discontinuous solution

MPFA

MFMFES1

BDM

MFMFENS1

BDM

MFMFES2

BRT

MFMFENS2

BRT

LFMFD

SLFMFD

NS

MHS

BDM

MHS

BRT

MHNS1

BRT

MHNS2

BRT

MHNS3

BRT

(e) Velocity error and convergence rateFigure 3.9: =0.13448835 on quadrilateral grids

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0 1 2 3 4 5 6-7

-6

-5

-4

-3

-2

-1

log2(1/h)

log2

(eL2)

L2 Pressure Error of discontinuous solution

MPFA

MFMFES1

BDM

MFMFENS1

BDM

MFMFES2

BRT

MFMFENS2 BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS1

BRT

MHNS2

BRT

MHNS3

BRT


0 1 2 3 4 5 6-0.5

0

0.5

1

1.5

2

2.5

3

3.5

log2(1/h)

log2

(eL2)

L2 Velocity Error of discontinuous solution

MPFA

MFMFES1

BDM

MFMFENS1

BDM

MFMFES2

BRT

MFMFENS2 BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS1

BRT

MHNS2

BRT

MHNS3

BRT


Figure 3.10: =0.41033296 on quadrilateral grids

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0 1 2 3 4 5 6-8

-7

-6

-5

-4

-3

-2

log2(1/h)

log2

(eL2)

L Pressure Error of discontinuous solution

MPFA

MFMFES1

BDM

MFMFENS1

BDM

MFMFES2

BRT

MFMFENS2 BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS1

BRT

MHNS2

BRT

MHNS3

BRT


0 1 2 3 4 5 6-8

-6

-4

-2

0

2

4

6

log2(1/h)

log2

(eL2)

L Velocity Error of discontinuous solution

MPFA

MFMFES1

BDM

MFMFENS1

BDM

MFMFES2

BRT

MFMFENS2

BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS1

BRT

MHNS2

BRT

MHNS3 BRT


Figure 3.11: =1.47865601 on quadrilateral grids

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Discontinuous experiments on triangular grids

0 1 2 3 4 5 6-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

log2(1/h)

log2

(eL2)

L Pressure Error on triangular mesh

MFMFES

BDM

MFMFES

BRT

MFMFENS BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS

BRT


0 1 2 3 4 5 64.5

4.6

4.7

4.8

4.9

5

5.1

5.2

5.3

5.4

5.5

log2(1/h)

log2

(eL2)

L2 Velocity Error on t riangular mesh

MFMFES

BDM

MFMFES

BRT

MFMFENS BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS

BRT


Figure 3.12: =0.13448835 on triangular mesh

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0 1 2 3 4 5 6-7

-6

-5

-4

-3

-2

-1

log2(1/h)

log2

(eL2)

L2 Pressure Error on triangular mesh

MFMFES BDM

MFMFES

BRT

MFMFENS

BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS

BRT


0 1 2 3 4 5 6-1

-0.5

0

0.5

1

1.5

2

2.5

3

log2(1/h)

log2

(eL2)

L2 Velocity Error on triangular mesh

MFMFES BDM

MFMFES

BRT

MFMFENS

BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS

BRT


Figure 3.13: =0.41033296 on triangular mesh

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0 1 2 3 4 5 6-8

-7

-6

-5

-4

-3

-2

log2(1/h)

log2

(eL2)

L Pressure Error on triangular mesh

MFMFES BDM

MFMFES

BRT

MFMFENS

BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS

BRT


0 1 2 3 4 5 6-8

-6

-4

-2

0

2

4

6

log2(1/h)

log2

(eL2)

L Velocity Error on triangular mesh

MFMFES BDM

MFMFES

BRT

MFMFENS

BRT

LFMFDS

LFMFDNS

MHS

BDM

MHS

BRT

MHNS

BRT


Figure 3.14: =1.47865601 on the triangular meshes

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MFMFE

Quadrature NS2 just do little better that S1 and S2, and is not as robustas NS1.

Mixed-Hybrid 1

2

On quadrilaterals:The velocity o MHNS1,MHNS2 and MHNS3 still could converge in thelast step o heavily perturbed grids and have rate close to 0.35(NS1),0.43(NS2) and 0.32 (NS3) respectively. Although these methods haverobust convergence order, their errors are not the best.

On triangles:

The NS quadrature o Mixed-Hybrid 1

2 dened on physical trianglesis not robust on the distorted meshes.

Mixed-Hybrid 1 It is the most accurate and robust among these methods, however, all

the hybrid methods are less simple and computationally ecient.

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MPFA O-method

For the extreme case o solution with singularity =0.13448835, theerror o normal uxes o MPFA-O dened on quadrilateral grids is thebest.

Local ux mimetic MPFA

The error o its normal ux is larger than other methods on both trian-gular and quadrilateral grids.

The diferences o the convergence behaviors o pressures in all these methodsare relatively small.

References
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[Aavatsmark,2002] I. AavatsmarkAn introduction to multipoint ux approximations or quadrilateral grids,Comput Geosci 6(2002), 405-432.

[Eigestad,2005] G. T. Eigestad and R. A. KlausenOn the convergence o the multi-point ux approximation O-method: Numerical ex-periments or discontinuous permeability.

Numer Methods Partial Dif Eqns 21(2005), 1079-1098.[Wheeler,2006] M. F. Wheeler and I. Yotov,

A multipoint ux mixed nite element method,SIAM J Numer Anal 44(2006), 2082-2106.

[Klausen,2006] R. A. Klausen and R. Winther,Robust convergence o multipoint ux approximations on rough grids,

Numer Math 104(2006), 317-337.

References
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[Aavatsmark,2006] I. Aavatsmark, G.T. Eigestad, R.A. Klausen,Numerical convergence o the MPFA O-method or general quadrilateral grids in twoand three dimensions,IMA Vol Math Appl, Springer, New York, 142(2006), 1-21.

[Klausen,2008] R.A. Klausen, F.A. Radu, G.T. Eigestad,

Convergence o MPFA on triangulations and or Richards equation,Int.J.Numer.Methods Fluids, 1-25(2008).

[Lipnikov,2009] K. Lipnikov, M. Shashkov and I. Yotov,Local ux mimetic nite diference methods,Numer Math 112(2009), 115-152.

References
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numerical performance of reduction numerical methods for single-phase flow in rough grids and with...

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