multipoint flux mixed finite element method in porous media...
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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
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
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
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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
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
The University of Texas at Austin, USA
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
The University of Texas at Austin, USA
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
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
The University of Texas at Austin, USA
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)
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
The University of Texas at Austin, USA
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.
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
The University of Texas at Austin, USA
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
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
The University of Texas at Austin, USA
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
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
The University of Texas at Austin, USA
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.
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
The University of Texas at Austin, USA
Multiscale Mortar MFMFE
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
The University of Texas at Austin, USA
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
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 .
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
The University of Texas at Austin, USA
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.
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
The University of Texas at Austin, USA
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.
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
The University of Texas at Austin, USA
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).
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
The University of Texas at Austin, USA
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.
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.
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.
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‖.
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.
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)
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.
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
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.
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
Numerical Examples
Center for Subsurface ModelingInstitute for Computational Engineering and Sciences
The University of Texas at Austin, USA
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.
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
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.
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.
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
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.
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
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)
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)
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
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