multipoint flux mixed finite element method in porous media...
TRANSCRIPT
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Multipoint Flux Mixed Finite Element Method
in Porous Media Applications
Part I: Introduction and Multiscale Mortar Extension
Guangri Xue (Gary)
KAUST GRP Research FellowCenter for Subsurface Modeling
Institute for Computational Engineering and SciencesThe University of Texas at Austin
In collaboration with:Mary F. Wheeler, The University of Texas at AustinIvan Yotov, University of Pittsburgh
Acknowledgement:
GRP Research Fellowship, made by KAUST
KAUST WEP Workshop, Saudi Arabia, 1/30/2010
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
The University of Texas at Austin, USA
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Modeling Carbon Sequestration
CO2 Sequestration Modeling
Key Processes
• CO2/brine mass transfer
• Multiphase flow
• During injection (pressure driven)
• After injection (gravity driven)
• Geochemical reactions
• Geomechanical modeling
Numerical simulations
• Characterization (fault, fractures)
• Appropriate gridding
• Compositional EOS
• Parallel computing capability
Key Processes
• CO2/brine mass transfer
• Multiphase flow
• During injection (pressure driven)
• After injection (gravity driven)
• Geochemical reactions
• Geomechanical modeling
Numerical Simulations
• Characterization (fault, fractures)
• Appropriate gridding
• Compositional EOS
• Parallel computing capability
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
The University of Texas at Austin, USA
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Corner Point Geometry
• General hexahedral grid (with non-planar faces)
• Fractures and faults
• Pinch-out
• Layers
• Non-matching
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
The University of Texas at Austin, USA
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Outline
• Some locally conservative H(div) conforming method
• Multipoint flux mixed finite element method (MFMFE)
• Multiscale Mortar MFMFE
• Numerical examples
• Summary and Conclusions
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The University of Texas at Austin, USA
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Some locally conservative H(div) conforming method
• Mixed Finite Element
Raviart, Thomas 1977; Nedelec 1980; Brezzi, Douglas, Marini 1985;
Brezzi, Douglas, Duran, Fortin 1987; Brezzi, Douglas, Duran, Marini
1985; Chen, Douglas 1989, Shen 1994; Kuznetsov, Repin 2003;
Arnold, Boffi, Falk 2005; Sbout, Jaffre, Roberts 2009...
• Mimetic Finite Difference
Shashkov, Berndt, Hall, Hyman, Lipnikov, Morel, Moulton, Roberts,
Steinberg, Wheeler, Yotov ...
• Cell-Centered Finite Difference
Russell, Wheeler 1983; Arbogast, Wheeler, Yotov 1997; Arbogast,
Dawson, Keenan, Wheeler, Yotov 1998 ...
• Multipoint Flux Approximation
Aavatsmark, Barkve, Mannseth 1998; Aavatsmark 2002; Edwards
2002; Edwards, Rogers 1998, ...
• Multipoint Flux MFE
Wheeler, Yotov 2006; Ingram, Wheeler, Yotov 2009; Wheeler, X.,
Yotov 2009, 2010
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
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Multipoint Flux Mixed Finite Element (MFMFE)—1
Find u ∈ H(div), p ∈ L2,
(K−1u,v)− (p,∇ · v) = 0, ∀v ∈ H(div)
(∇ · u, q) = (f, q), ∀q ∈ L2
MFMFE method: find uh ∈ Vh, ph ∈Wh,
(K−1uh,v)Q − (p,∇ · v) = 0, ∀v ∈ Vh(∇ · u, q) = (f, q), ∀q ∈Wh
Finite element space: Vh(E) and Wh(E)
Vh(E) =Pv|v ∈ V (E)
, Wh(E) =
q|q ∈ W (E)
Numerical quadrature rule:
(K−1uh,vh)Q =∑E∈Th
(K−1uh,vh)Q,E =∑E∈Th
(1
JBTK−1Buh, vh
)Q,E
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Multipoint Flux Mixed Finite Element (MFMFE)—2
FEM space on E:
• Simplicial element [Brezzi, Douglas, Marini 1985; Brezzi, Douglas, Duran, Fortin 1987]:
V(E) = P1(E)d, W (E) = P0(E),
• 2D square [Brezzi, Douglas, Marini 1985]:
V (E) = BDM1(E) =
(α1x+ β1y + r1 + rx2 + 2sxyα2x+ β2y + r2 − 2rxy − sy2
)W (E) = P0(E)
• 3D cube [Ingram, Wheeler, Yotov 2009]:
V (E) = BDDF1(E) + r2curl(0,0, x2z)T + r3curl(0,0, x2yz)T
+ s2curl(xy2,0,0)T + s3curl(xy2z,0,0)T
+ t2curl(0, yz2,0)T + t3curl(0, xyz2,0)T
W (E) = P0(E)
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The University of Texas at Austin, USA
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Multipoint Flux Mixed Finite Element (MFMFE)—3
Numerical quadrature rule on V (E):
(K−1uh,vh)Q,E =(
1
JBTK−1Buh, vh
)Q,E
Symmetric [Wheeler and Yotov 2006]:
(1
JBTK−1Buh, vh
)Q,E
=|E|nv
nv∑i=1
(1
JBTK−1Buh · vh
)|ri
nv: number of vertices of E.
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The University of Texas at Austin, USA
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Properties of MFMFE—1
(M NT
N 0
)(UP
)=
(0−F
)Basis functions in V (E):
v11(r1) · n1 = 1, v11(r1) · n2 = 0
v11(ri) · nj = 0, for i 6= 1, j = 1,2
(1
JBTK−1Bv11, v11
)Q,E6= 0(
1
JBTK−1Bv11, v12
)Q,E6= 0(
1
JBTK−1Bv11, vij
)Q,E
= 0, i 6= 1
M is block diagonal. Cell-centered scheme:
NM−1NTP = F
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Properties of MFMFE—2
• Locally conservative
• Cell-centered scheme, ”solver friendly”
• Equivalent to multipoint flux approximation method
• Accurate for full tensor coefficient, simplicial grids, h2-quadrilateral
grid, and h2-hexahedral grid with non-planar faces
• Superconvergent
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Convergence Results of MFMFE
Symmetric method
Theorem [Wheeler and Yotov 2006, Ingram, Wheeler, and Yotov 2009] On simplicial
grids, h2-parallelograms, and h2-parallelepipeds
‖u− uh‖+ ‖div(u− uh)‖+ ‖p− ph‖ ≤ Ch‖Qhp− ph‖ ≤ Ch2, for regular h2-parallelpipeds
Proposition On h2-parallelogram and K-orthogonal grids,
‖ΠRu−ΠRuh‖ ≤ Ch2
ΠR: RT 0 projection
Open question for non-orthogonal grid.
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Multiscale Mortar MFMFE
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Multidomain variational formulation
Vi = H(div; Ωi), V =n⊕i=1
Vi,
Wi = L2(Ωi), W =n⊕i=1
Wi = L2(Ω).
Λi,j = H1/2(Γi,j), Λ =⊕
1≤i<j≤nΛi,j.
Find u ∈ V, p ∈W , and λ ∈ Λ such that, for 1 ≤ i ≤ n,
(K−1u,v)Ωi− (p,∇ · v)Ωi
= −〈g,v · ni〉∂Ωi/Γ − 〈λ,v · ni〉Γi, ∀v ∈ Vi,
(∇ · u, w)Ωi= (f, w)Ωi
, ∀w ∈Wi,n∑i=1
〈u · ni, µ〉Γi = 0, ∀µ ∈ Λ.
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
The University of Texas at Austin, USA
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Multiscale Mortar MFMFE: formulation
Multiscale Mortar: Mixed Finite Element
Theorem [Arbogast, Pencheva, Wheeler & Yotov 2007]:
Vh =!n
i=1 Vh,i, Wh =!n
i=1 Wh,i, MH =!n
i=1 MH,i,j
!i !j
!ijVh,i: RT, BDM, .., spacesWh,j : piecewise polynomial
MH,i,j : piecewise polynomial
Find uh ! Vh, p ! Wh, and ! ! MH , for i = 1, · · · , n,
(K!1uh,v)!i" (ph,# · v)!i
= " < !H ,v · ni >"i$v ! Vh,i
(# · uh, q)!i = (f, q) $q ! Wh,i!ni=1 < uh · ni, µ >"i= 0 $µ ! MH
!u" uh! = O(Hm+1/2 + hk+1)!p" ph! = O(Hm+3/2 + hk+1)
m: degree of mortar approximation polynomial space MH
k: order of approximation for velocity and pressure
Vh =n⊕i=1
Vh,i, Wh =n⊕i=1
Wh,i
ΛH =⊕
1≤i<j≤nΛH,i,j
Multiscale mortar MFMFE method is defined as: seek uh ∈ Vh, ph ∈Wh,
λH ∈ ΛH such that for 1 ≤ i ≤ n,
(K−1uh,v)Q,Ωi− (ph,∇ · v)Ωi
=− 〈g,ΠRv · ni〉∂Ωi/ Γ
− 〈λH ,ΠRv · ni〉Γi, ∀v ∈ Vh,i,
(∇ · uh, w)Ωi= (f, w)Ωi
, ∀w ∈Wh,i,n∑i=1
〈ΠRuh · ni, µ〉Γi = 0, ∀µ ∈ ΛH .
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Multiscale Mortar MFMFE: an interface formulation—1
Interface problem:
dH(λH , µ) = gH(µ), µ ∈ ΛH ,
dH : L2(Γ)× L2(Γ)→ R for λ, µ ∈ L2(Γ) by
dH(λ, µ) =n∑i=1
dH,i(λ, µ) = −n∑i=1
〈ΠRu∗h(λ) · ni, µ〉Γi.
gH : L2(Γ)→ R:
gH(µ) =n∑i=1
gH,i(µ) =n∑i=1
〈ΠRuh · ni, µ〉Γi,
Star problem: (u∗h(λ), p∗h(λ)) ∈ Vh ×Wh solve, for 1 ≤ i ≤ n,
(K−1u∗h(λ),v)Q,Ωi− (p∗h(λ),∇ · v)Ωi
= −〈λ,ΠRv · ni〉Γi, v ∈ Vh,i,
(∇ · u∗h(λ), w)Ωi= 0, w ∈Wh,i.
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Multiscale Mortar MFMFE: an interface formulation—2
Bar problem: (uh, ph) ∈ Vh ×Wh solve, for 1 ≤ i ≤ n,
(K−1uh(λ),v)Q,Ωi− (ph(λ),∇ · v)Ωi
= −〈g,ΠRv · ni〉∂Ωi/ Γi, v ∈ Vh,i,
(∇ · uh(λ), w)Ωi= 0, w ∈Wh,i.
with
uh = u∗h(λH) + uh, ph = p∗h(λH) + ph.
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Weakly Continuous Velocity Sapce
Vh,0 =
v ∈ Vh :n∑i=1
〈ΠRv|Ωi· ni, µ〉Γi = 0 ∀µ ∈ ΛH
.Assumption: For any µ ∈ ΛH,
‖µ‖0,Γi,j ≤ C(‖QRh,iµ‖0,Γi,j + ‖QRh,jµ‖0,Γi,j
), 1 ≤ i < j ≤ n. (1)
Lemma 1 Under assumption (1), there exists a projection operator
Π0 :(H1/2+ε(Ω)
)d∩V→ Vh,0 such that
(∇ · (Π0q− q), w) = 0, w ∈Wh,
‖Π0q−Πq‖ .n∑i=1
‖q‖r+1/2,Ωihr(h1/2 +H1/2), 0 ≤ r ≤ 1,
‖Π0q− q‖ .n∑i=1
‖q‖1,Ωih1/2(h1/2 +H1/2).
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Solvability of Multiscale Mortar MFMFE method
(K−1uh,v)Q,Ωi− (ph,∇ · v)Ωi
=− 〈g,ΠRv · ni〉∂Ωi/ Γ− 〈λH ,ΠRv · ni〉Γi, ∀v ∈ Vh,i, (2)
(∇ · uh, w)Ωi= (f, w)Ωi
, ∀w ∈Wh,i, (3)n∑i=1
〈ΠRuh · ni, µ〉Γi = 0, ∀µ ∈ ΛH . (4)
Lemma 2 Assume that (1) holds. Then, there exists a unique solutionof (2)-(4).Sketch of Proof:
1. Let f = 0 and g = 0, v = uh, w = ph, and µ = λH,
n∑i=1
(K−1uh,uh)Q,Ωi= 0, thus uh = 0.
2. ∃q ∈ H1(Ω) s.t. ∇ · q = ph Taking v = Π0q in (2),
0 =n∑i=1
(ph,∇ ·Π0q) = (ph,∇ · q) = ‖ph‖2, implies ph = 0.
3. (2) gives 0 = 〈λH ,ΠRv · ni〉Γi = 〈QRh,iλH ,ΠRv · ni〉Γi. ∃v, s.t.
v · ni = QRh,iλH, implying QRh,iλH = 0. By assumption (1), ΛH = 0.
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Velocity Error Analysis
Theorem 1 Let K−1 ∈W1,∞(Ωi), 1 ≤ i ≤ n. For the velocity uh of the
mortar MFMFE method (2)-(4) on simplicial elements,
h2-parallelograms, and h2-parallelpipeds, if (1) holds, then
‖∇ · (u− uh)‖ .n∑i=1
h‖∇ · u‖1,Ωi,
‖u− uh‖ .n∑i=1
(Hs−1/2‖p‖s+1/2,Ωi+ h‖u‖1,Ωi
+ hr(H1/2 + h1/2)‖u‖r+1/2,Ωi),
where 0 < s ≤ m+ 1,0 ≤ r ≤ 1, and m is the order of polynomial degree
for mortar space.
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Velocity Error Analysis: Sketch of Proof
• Divergence error:
∇ · (Πu− uh) = 0 and ‖∇ · (u−Πu)‖0,Ωi. h‖∇ · u‖1,Ωi
• L2 error:
Let q = Π0u− uh
‖Π0u− uh‖2 . (K−1(Π0u− uh),q)Q
=(K−1Π0u,q
)Q−(K−1u,ΠRq
)−
n∑i=1
〈p− IHp,ΠRq · ni〉Γi
=(K−1(Π0u−Πu),q
)Q
+(K−1Πu,q−ΠRq
)Q− σ
(K−1Πu,ΠRq
)+(K−1(Πu− u),ΠRq
)−
n∑i=1
〈p− IHp,ΠRq · ni〉Γi.
|(K−1Πu,v −ΠRv)Q| . h‖u‖1‖v‖.
|σ(K−1q,v)| .∑E∈Th
h‖K−1‖1,∞,E‖q‖1,E‖v‖E.
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Superconvergence of Velocity
Theorem 2 Assume that the tensor K is diagonal and K−1 ∈W2,∞(Ωi),
1 ≤ i ≤ n. Then, the velocity uh of the mortar MFMFE method (2)-(4)
on rectangular and cuboid grids, if (1) holds, satisfies
‖ΠRu−ΠRuh‖ .n∑i=1
(hr(H1/2 + h1/2)‖u‖r+1/2,Ωi
+Hs−1/2‖p‖s+1/2,Ωi+ h2‖u‖2,Ωi
),
where 0 < s < m+ 1, 0 ≤ r ≤ 1.
Lemma 3 Assume that K is a diagonal tensor and K−1 ∈W1,∞Th . Then
for all uh ∈ Vh and vh ∈ VRh on rectangular and cuboid grids,
|(K−1(uh−ΠRuh),vh)Q| . h|||K−1|||1,∞(‖u−uh‖+‖u−ΠRu‖+‖Πu−uh‖)‖vh‖.
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Pressure Error Analysis
Define another weakly continuous space:
VRh,0 =
v ∈ VRh :
n∑i=1
〈v|Ωi· ni, µ〉Γi = 0 ∀µ ∈ ΛH
,where VR
h : RT 0 space on each subdomain
Lemma 4 Spaces VRh,0×Wh satisfy the inf-sup condition: for all w ∈Wh,
sup06=v∈VR
h,0
n∑i=1
(∇ · v, w)Ωi/
n∑i=1
‖v‖div,Ωi& ‖w‖, 1 ≤ i ≤ n.
Theorem 3 Let K−1 ∈W1,∞(Ωi), 1 ≤ i ≤ n. For the pressure ph of the
mortar MFMFE method (2)-(4) on simplicial elements,
h2-parallelograms, and h2-parallelpipeds , if (1) holds, then
‖p− ph‖ .n∑i=1
(h‖p‖1,Ωi+ hr(H1/2 + h1/2)‖u‖r+1/2,Ωi
+ h‖u‖1,Ωi+Hs−1/2‖p‖s+1/2,Ωi
),
where 0 < s ≤ m+ 1,0 ≤ r ≤ 1.
![Page 23: Multipoint Flux Mixed Finite Element Method in Porous Media Applicationsweb.kaust.edu.sa/faculty/shuyusun/FEM2010/slides/... · 2010-01-30 · Multipoint Flux Mixed Finite Element](https://reader031.vdocuments.net/reader031/viewer/2022022118/5cdd6d9788c993400f8e161a/html5/thumbnails/23.jpg)
Pressure Error Analysis: Sketch of Proof
‖Qhp− ph‖ . sup06=v∈VR
h,0
n∑i=1
(∇ · v, Qhp− ph)Ωi/
n∑i=1
‖v‖div,Ωi
= sup06=v∈VR
h,0
(K−1u,v
)−(K−1uh,v
)Q
+∑ni=1〈p− IHp,v · ni〉Γi∑n
i=1 ‖v‖div,Ωi
.
and(K−1u,v
)−(K−1uh,v
)Q
=(K−1(u−Πu),v
)−(K−1(uh −Πu),v
)Q
+ σ(K−1Πu,v)
![Page 24: Multipoint Flux Mixed Finite Element Method in Porous Media Applicationsweb.kaust.edu.sa/faculty/shuyusun/FEM2010/slides/... · 2010-01-30 · Multipoint Flux Mixed Finite Element](https://reader031.vdocuments.net/reader031/viewer/2022022118/5cdd6d9788c993400f8e161a/html5/thumbnails/24.jpg)
Superconvergence of Pressure
Theorem 4 Assume that K ∈W1,∞(Ωi), K−1 ∈W2,∞(Ωi), 1 ≤ i ≤ n,
and full H2 elliptic regularity condition holds. Then, the pressure ph of
the mortar MFMFE method (2)-(4) on simplicial elements,
h2-parallelograms, and regular h2-parallelpipeds, if (1) holds, satisfies
‖Qhp− ph‖ .n∑i=1
(h3/2(H1/2 + h1/2)‖u‖2,Ωi+Hs(H1/2 + h1/2)‖p‖s+1/2,Ωi
+ hr+1/2(h1/2 +H1/2)2‖u‖r+1/2,Ωi),
where 0 < s ≤ m+ 1, 0 ≤ r ≤ 1.
![Page 25: Multipoint Flux Mixed Finite Element Method in Porous Media Applicationsweb.kaust.edu.sa/faculty/shuyusun/FEM2010/slides/... · 2010-01-30 · Multipoint Flux Mixed Finite Element](https://reader031.vdocuments.net/reader031/viewer/2022022118/5cdd6d9788c993400f8e161a/html5/thumbnails/25.jpg)
Superconvergence of Pressure: Sketch of Proof —1
• Consider an auxiliary problem:
−∇ · (K∇φ) = ph −Qhp, in Ω,
φ = 0, on ∂Ω.
By regularity,
‖φ‖2 . ‖Qhp− ph‖.
• By definition of Qh, ΠR, Π0,
‖Qhp− ph‖2 =n∑i=1
(Qhp− ph,∇ ·K∇φ)Ωi=
n∑i=1
(Qhp− ph,∇ ·ΠRΠ0K∇φ)Ωi
=n∑i=1
(p− ph,∇ ·ΠRΠ0K∇φ)Ωi
![Page 26: Multipoint Flux Mixed Finite Element Method in Porous Media Applicationsweb.kaust.edu.sa/faculty/shuyusun/FEM2010/slides/... · 2010-01-30 · Multipoint Flux Mixed Finite Element](https://reader031.vdocuments.net/reader031/viewer/2022022118/5cdd6d9788c993400f8e161a/html5/thumbnails/26.jpg)
Superconvergence of Pressure: Sketch of Proof —2
• Taking vh = ΠRΠ0K∇φ ∈ Vh,0 in the following error equation
(K−1u,v
)−(K−1uh,v
)Q
=n∑i=1
(p− ph,∇ · v)Ωi−
n∑i=1
〈p,v · ni〉Γi
−n∑i=1
〈g, (v −ΠRv) · ni〉∂Ωi/Γ, ∀v ∈ Vh,0,
get
‖Qhp− ph‖2 = (K−1u,vh)− (K−1uh,vh)Q +n∑i=1
〈p,vh · ni〉Γi.
• Use the weak continuity of vh,
‖Qhp− ph‖2 =(K−1(u−Πu),vh
)−(K−1(uh −Πu),vh
)Q
+ σ(K−1Πu,vh) +n∑i=1
〈p− PHp,vh · ni〉Γi.
![Page 27: Multipoint Flux Mixed Finite Element Method in Porous Media Applicationsweb.kaust.edu.sa/faculty/shuyusun/FEM2010/slides/... · 2010-01-30 · Multipoint Flux Mixed Finite Element](https://reader031.vdocuments.net/reader031/viewer/2022022118/5cdd6d9788c993400f8e161a/html5/thumbnails/27.jpg)
Convergence Rates
h : subdomain fine mesh size
H : mortar coarse mesh size. H > h.
m : degree of polynomial for mortar space
‖u− uh‖ = O(Hm+1/2 + h)
‖p− ph‖ = O(Hm+1/2 + h)
‖Qhp− ph‖ = O(Hm+3/2 +H1/2h3/2)
‖ΠRu−ΠRuh‖ = O(Hm+1/2 +H1/2h)
Theoretical convergence rates for linear and quadratic mortars
m h ‖p− ph‖ ‖u− uh‖ ‖Qhp− ph‖ ‖ΠRu−ΠRuh‖1 H/2 1 1 2 1.52 H2 1 1 1.75 1.25
![Page 28: Multipoint Flux Mixed Finite Element Method in Porous Media Applicationsweb.kaust.edu.sa/faculty/shuyusun/FEM2010/slides/... · 2010-01-30 · Multipoint Flux Mixed Finite Element](https://reader031.vdocuments.net/reader031/viewer/2022022118/5cdd6d9788c993400f8e161a/html5/thumbnails/28.jpg)
Numerical Examples
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
The University of Texas at Austin, USA
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Numerical Example 1: On a rectangular mesh—1
Exact solution: p(x, y) = x3y4 + x2 + sin(xy) cos(y)
Full permeability tensor:
K =
((x+ 1)2 + y2 sin(xy)
sin(xy) (x+ 1)2
).
X
Y
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
pres2.22.01.81.61.41.21.00.80.60.40.2
X
Y
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
pres2.22.01.81.61.41.21.00.80.60.40.2
Multiscale Mortar MFMFE solution: discontinuous linear (left) and
discontinuous quadratic (right) mortars.
![Page 30: Multipoint Flux Mixed Finite Element Method in Porous Media Applicationsweb.kaust.edu.sa/faculty/shuyusun/FEM2010/slides/... · 2010-01-30 · Multipoint Flux Mixed Finite Element](https://reader031.vdocuments.net/reader031/viewer/2022022118/5cdd6d9788c993400f8e161a/html5/thumbnails/30.jpg)
Numerical Example 1: On a rectangular mesh—2
continuous linear mortars and matching grids
1/h ‖p− ph‖ ‖u− uh‖ ‖Qhp− ph‖ ‖ΠRu−ΠRuh‖4 2.53E-01 — 1.06E+00 — 5.39E-02 — 1.27E-01 —8 1.21E-01 1.06 5.23E-01 1.02 1.38E-02 1.97 3.10E-02 2.03
16 5.96E-02 1.02 2.57E-01 1.03 3.46E-03 2.00 7.66E-03 2.0232 2.97E-02 1.00 1.27E-01 1.02 8.66E-04 2.00 1.92E-03 2.0064 1.48E-02 1.00 6.34E-02 1.00 2.16E-04 2.00 4.80E-04 2.00
128 7.42E-03 1.00 3.16E-02 1.00 5.41E-05 2.00 1.20E-04 2.00256 3.71E-03 1.00 1.58E-02 1.00 1.36E-05 1.99 3.67E-05 1.71
continuous quadratic mortars and matching grids
1/h ‖p− ph‖ ‖u− uh‖ ‖Qhp− ph‖ ‖ΠRu−ΠRuh‖4 2.53E-01 — 1.06E+00 — 5.39E-02 — 1.27E-01 —
16 5.96E-02 1.04 2.57E-01 1.02 3.46E-03 1.98 7.69E-03 2.0264 1.48E-02 1.00 6.34E-02 1.01 2.16E-04 2.00 5.71E-04 1.88
256 3.71E-03 1.00 1.58E-02 1.00 1.36E-05 1.99 7.61E-05 1.45
discontinuous quadratic mortars and nonmatching grids
1/h ‖p− ph‖ ‖u− uh‖ ‖Qhp− ph‖ ‖ΠRu−ΠRuh‖4 1.97E-01 — 7.54E-01 — 3.64E-02 — 1.45E-01 —
16 4.76E-02 1.02 1.81E-01 1.03 2.32E-03 1.99 1.14E-02 1.8364 1.19E-02 1.00 4.48E-02 1.01 1.45E-04 2.00 8.46E-04 1.88
256 2.97E-03 1.00 1.12E-02 1.00 9.12E-06 2.00 7.75E-05 1.72
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Numerical Example 1: On a rectangular mesh—3
X
Y
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
errp3.5E-033.3E-033.0E-032.8E-032.6E-032.4E-032.1E-031.9E-031.7E-031.5E-031.2E-031.0E-03
X
Y
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
errp3.5E-033.3E-033.0E-032.8E-032.6E-032.4E-032.1E-031.9E-031.7E-031.5E-031.2E-031.0E-03
Error in Multiscale Mortar MFMFE solution: discontinuous linear (left)
and discontinuous quadratic (right) mortars.
![Page 32: Multipoint Flux Mixed Finite Element Method in Porous Media Applicationsweb.kaust.edu.sa/faculty/shuyusun/FEM2010/slides/... · 2010-01-30 · Multipoint Flux Mixed Finite Element](https://reader031.vdocuments.net/reader031/viewer/2022022118/5cdd6d9788c993400f8e161a/html5/thumbnails/32.jpg)
Numerical Example 2: On an h2-parallelogram mesh—1
The map is defined as
x = x+ 0.03 cos(3πx) cos(3πy),
y = y − 0.04 cos(3πx) cos(3πy).
X
Y
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
pres2.22.01.81.61.41.21.00.80.60.40.2
X
Y
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
pres2.22.01.81.61.41.21.00.80.60.40.2
Multiscale Mortar MFMFE solution: discontinuous linear (left) and
discontinuous quadratic (right) mortars.
![Page 33: Multipoint Flux Mixed Finite Element Method in Porous Media Applicationsweb.kaust.edu.sa/faculty/shuyusun/FEM2010/slides/... · 2010-01-30 · Multipoint Flux Mixed Finite Element](https://reader031.vdocuments.net/reader031/viewer/2022022118/5cdd6d9788c993400f8e161a/html5/thumbnails/33.jpg)
Numerical Example 2: On an h2-parallelogram mesh—2
discontinuous linear mortars and nonmatching grids
1/h ‖p− ph‖ ‖u− uh‖ ‖Qhp− ph‖ ‖ΠRu−ΠRuh‖4 1.96E-01 — 8.56E-01 — 3.11E-02 — 3.53E-01 —8 9.66E-02 1.02 4.19E-01 1.03 7.46E-03 2.06 1.17E-01 1.59
16 4.82E-02 1.00 2.08E-01 1.01 1.83E-03 2.03 3.49E-02 1.7532 2.41E-02 1.00 1.03E-01 1.01 4.54E-04 2.01 9.55E-03 1.8764 1.20E-02 1.01 5.13E-02 1.01 1.13E-04 2.01 2.60E-03 1.88
128 6.02E-03 1.00 2.56E-02 1.00 2.82E-05 2.00 7.36E-04 1.83256 3.01E-03 1.00 1.28E-02 1.00 7.04E-06 2.00 2.20E-04 1.74
discontinuous quadratic mortars and nonmatching grids
1/h ‖p− ph‖ ‖u− uh‖ ‖Qhp− ph‖ ‖ΠRu−ΠRuh‖4 1.96E-01 — 8.53E-01 — 3.16E-02 — 3.52E-01 —
16 4.82E-02 1.01 2.07E-01 1.02 1.84E-03 2.05 3.32E-02 1.7064 1.20E-02 1.00 5.12E-02 1.01 1.13E-04 2.01 2.25E-03 1.94
256 3.01E-03 1.00 1.28E-02 1.00 7.05E-06 2.00 1.52E-04 1.94
![Page 34: Multipoint Flux Mixed Finite Element Method in Porous Media Applicationsweb.kaust.edu.sa/faculty/shuyusun/FEM2010/slides/... · 2010-01-30 · Multipoint Flux Mixed Finite Element](https://reader031.vdocuments.net/reader031/viewer/2022022118/5cdd6d9788c993400f8e161a/html5/thumbnails/34.jpg)
Numerical Example 2: On an h2-parallelogram mesh—3
X
Y
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
errp3.2E-032.9E-032.7E-032.4E-032.1E-031.8E-031.6E-031.3E-031.0E-037.5E-044.7E-042.0E-04
X
Y
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
errp3.2E-032.9E-032.7E-032.4E-032.1E-031.8E-031.6E-031.3E-031.0E-037.5E-044.7E-042.0E-04
Error in Multiscale Mortar MFMFE solution: discontinuous linear (left)
and discontinuous quadratic (right) mortars.
![Page 35: Multipoint Flux Mixed Finite Element Method in Porous Media Applicationsweb.kaust.edu.sa/faculty/shuyusun/FEM2010/slides/... · 2010-01-30 · Multipoint Flux Mixed Finite Element](https://reader031.vdocuments.net/reader031/viewer/2022022118/5cdd6d9788c993400f8e161a/html5/thumbnails/35.jpg)
Numerical Example 3: On a cubic mesh—1
Exact solution: p(x, y, z) = x+ y + z − 1.5
Full tensor coefficient:
K =
x2 + y2 + 1 0 00 z2 + 1 sin(xy)0 sin(xy) x2y2 + 1
.discontinuous linear mortars and matching grids
1/h ‖p− ph‖ ‖u− uh‖ ‖Qhp− ph‖ ‖ΠRu−ΠRuh‖4 2.17E-01 — 1.55E-01 — 9.87E-03 — 3.73E-03 —8 1.08E-01 1.01 7.76E-02 1.00 2.47E-03 2.00 1.03E-03 1.86
16 5.41E-02 1.00 3.88E-02 1.00 6.17E-04 2.00 2.60E-04 1.9932 2.71E-02 1.00 1.94E-02 1.00 1.54E-04 2.00 6.50E-05 2.0064 1.35E-02 1.01 9.68E-03 1.00 3.85E-05 2.00 1.66E-05 1.97
discontinuous quadratic mortars and matching grids
1/h ‖p− ph‖ ‖u− uh‖ ‖Qhp− ph‖ ‖ΠRu−ΠRuh‖4 2.17E-01 — 1.55E-01 — 9.87E-03 — 3.73E-03 —
16 5.41E-02 1.00 3.88E-02 1.00 6.17E-04 2.00 2.61E-04 1.9264 1.35E-02 1.00 9.68E-03 1.00 3.85E-05 2.00 1.67E-05 1.98
![Page 36: Multipoint Flux Mixed Finite Element Method in Porous Media Applicationsweb.kaust.edu.sa/faculty/shuyusun/FEM2010/slides/... · 2010-01-30 · Multipoint Flux Mixed Finite Element](https://reader031.vdocuments.net/reader031/viewer/2022022118/5cdd6d9788c993400f8e161a/html5/thumbnails/36.jpg)
Numerical Example 3: On a cubic mesh—2
pres1.210.80.60.40.20-0.2-0.4-0.6-0.8-1-1.2
errp9.0E-048.3E-047.7E-047.0E-046.3E-045.7E-045.0E-044.3E-043.7E-043.0E-042.3E-041.7E-041.0E-04
Discontinuous quadratic mortars and matching grids: Multiscale Mortar
MFMFE solution (left) and error (right)
![Page 37: Multipoint Flux Mixed Finite Element Method in Porous Media Applicationsweb.kaust.edu.sa/faculty/shuyusun/FEM2010/slides/... · 2010-01-30 · Multipoint Flux Mixed Finite Element](https://reader031.vdocuments.net/reader031/viewer/2022022118/5cdd6d9788c993400f8e161a/html5/thumbnails/37.jpg)
Numerical Example 4: On regular h2-parallelpipeds—1
Mapping:
x = x+ 0.03 cos(3πx) cos(3πy) cos(3πz),
y = y − 0.04 cos(3πx) cos(3πy) cos(3πz),
z = z + 0.05 cos(3πx) cos(3πy) cos(3πz).
pres1.210.80.60.40.20-0.2-0.4-0.6-0.8-1-1.2
errp6.5E-036.0E-035.5E-035.0E-034.5E-034.0E-033.5E-033.0E-032.5E-032.0E-031.5E-031.0E-035.0E-04
Discontinuous quadratic mortars and matching grids: Multiscale Mortar
MFMFE solution (left) and error (right)
![Page 38: Multipoint Flux Mixed Finite Element Method in Porous Media Applicationsweb.kaust.edu.sa/faculty/shuyusun/FEM2010/slides/... · 2010-01-30 · Multipoint Flux Mixed Finite Element](https://reader031.vdocuments.net/reader031/viewer/2022022118/5cdd6d9788c993400f8e161a/html5/thumbnails/38.jpg)
Numerical Example 4: On regular h2-parallelpipeds—2
discontinuous linear mortars and matching grids
1/h ‖p− ph‖ ‖u− uh‖ ‖Qhp− ph‖ ‖ΠRu−ΠRuh‖4 2.18E-01 — 2.82E-01 — 1.39E-02 — 7.48E-02 —8 1.10E-01 0.99 1.66E-01 0.76 5.07E-03 1.46 5.15E-02 0.54
16 5.49E-02 1.00 8.96E-02 0.89 1.86E-03 1.45 2.09E-02 1.3032 2.75E-02 1.00 4.51E-02 0.99 5.24E-04 1.83 5.93E-03 1.8264 1.37E-02 1.01 2.23E-02 1.02 1.35E-04 1.96 1.52E-03 1.96
discontinuous quadratic mortars and matching grids
1/h ‖p− ph‖ ‖u− uh‖ ‖Qhp− ph‖ ‖ΠRu−ΠRuh‖4 2.18E-01 — 2.82E-01 — 1.39E-02 — 7.48E-02 —
16 5.49E-02 0.99 8.96E-02 0.83 1.86E-03 1.45 2.09E-02 0.9264 1.37E-02 1.00 2.24E-02 1.00 1.35E-04 1.89 1.53E-03 1.89
![Page 39: Multipoint Flux Mixed Finite Element Method in Porous Media Applicationsweb.kaust.edu.sa/faculty/shuyusun/FEM2010/slides/... · 2010-01-30 · Multipoint Flux Mixed Finite Element](https://reader031.vdocuments.net/reader031/viewer/2022022118/5cdd6d9788c993400f8e161a/html5/thumbnails/39.jpg)
Summary and Conclusions
1. MFMFE method can be viewed as a cell-centered scheme for the
pressure
2. MFMFE method can handle general tensor coefficient
3. A-priori error estimates for pressure and velocity and some
superconvergence estimates.
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
The University of Texas at Austin, USA