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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



Corner Point Geometry

• General hexahedral grid (with non-planar faces)

• Fractures and faults

• Pinch-out

• Layers

• Non-matching



Outline

• Some locally conservative H(div) conforming method

• Multipoint flux mixed finite element method (MFMFE)

• Multiscale Mortar MFMFE

• Numerical examples

• Summary and Conclusions



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



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




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)




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.



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



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



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.



Multiscale Mortar MFMFE



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.



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 .



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.



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.



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).



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



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.


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


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


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.


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


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


X

Y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


X

Y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


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


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


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


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


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.

