numerical performance of reduction numerical methods for single-phase flow in rough grids and with...
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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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Reference Reduction Methods Physical Reduction Methods Numerical Examples Summary and Conclusions
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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Reference Reduction Methods Physical Reduction Methods Numerical Examples Summary and Conclusions
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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Reference Reduction Methods Physical Reduction Methods Numerical Examples Summary and Conclusions
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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Reference Reduction Methods Physical Reduction Methods Numerical Examples Summary and Conclusions
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
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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Reference Reduction Methods Physical Reduction Methods Numerical Examples Summary and Conclusions
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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Reference Reduction Methods Physical Reduction Methods Numerical Examples Summary and Conclusions
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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Reference Reduction Methods Physical Reduction Methods Numerical Examples Summary and Conclusions
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
Reference Reduction Methods Physical Reduction Methods Numerical Examples Summary and Conclusions
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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
Reerence Reduction MethodsMulti-Point Flux Mixed Finite Element MethodMixed-Hybrid Finite Element Method
Physical Reduction Methods
MPFA O-MethodLocal 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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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
Reerence Reduction MethodsMulti-Point Flux Mixed Finite Element MethodMixed-Hybrid Finite Element Method
Physical Reduction Methods
MPFA O-MethodLocal ux mimetic MPFA method
Numerical ExamplesComparisons o robustness on quadrilateral grids
Discontinuous experiments on triangular and quadrilateral grids
Summary and Conclusions
Reference Reduction Methods Physical Reduction Methods Numerical Examples Summary and Conclusions
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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
Reerence Reduction MethodsMulti-Point Flux Mixed Finite Element MethodMixed-Hybrid Finite Element Method
Physical Reduction Methods
MPFA O-MethodLocal ux mimetic MPFA method
Numerical ExamplesComparisons o robustness on quadrilateral gridsDiscontinuous experiments on triangular and quadrilateral grids
Summary and Conclusions
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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
(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)
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
(b) Velocity error and convergence rate
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
(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 h -parallelogram 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 3.4: 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
(a) Pressure error and convergence rate
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
(b) Velocity error and convergence rate
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
(a) Pressure error and convergence rate
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
(b) Velocity error and convergence rate
Figure 3.8: Error and convergence rate on smooth mesh
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Outline
Reerence Reduction MethodsMulti-Point Flux Mixed Finite Element MethodMixed-Hybrid Finite Element Method
Physical Reduction Methods
MPFA O-MethodLocal ux mimetic MPFA method
Numerical ExamplesComparisons o robustness on quadrilateral gridsDiscontinuous experiments on triangular and quadrilateral grids
Summary and Conclusions
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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
(a) Pressure error and convergence rate
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
(b) Velocity error and convergence rate
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
(a) Pressure error and convergence rate
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
(b) Velocity error and convergence rate
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
(a) Pressure error and convergence rate
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
(b) Velocity error and convergence rate
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
(a) Pressure error and convergence rate
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
(b) Velocity error and convergence rate
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
(a) Pressure error and convergence rate
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
(b) Velocity error and convergence rate
Figure 3.14: =1.47865601 on the triangular meshes
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Summary and Conclusions
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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