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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 Fellow Center for Subsurface Modeling Institute for Computational Engineering

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SLIDE 1

Multipoint Flux Mixed Finite Element Method in Porous Media Applications

Part I: Introduction and Multiscale Mortar Extension Guangri Xue (Gary)

KAUST GRP Research Fellow Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin In collaboration with: Mary F. Wheeler, The University of Texas at Austin Ivan Yotov, University of Pittsburgh Acknowledgement: GRP Research Fellowship, made by KAUST KAUST WEP Workshop, Saudi Arabia, 1/30/2010

Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

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SLIDE 2

Modeling Carbon Sequestration 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 Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

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SLIDE 3

Corner Point Geometry

  • General hexahedral grid (with non-planar faces)
  • Fractures and faults
  • Pinch-out
  • Layers
  • Non-matching

Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

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SLIDE 4

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 Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

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SLIDE 5

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 Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

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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) =

  • Pˆ

v|ˆ v ∈ ˆ

V ( ˆ E)

  • ,

Wh(E) =

  • q|ˆ

q ∈ ˆ W( ˆ E)

  • Numerical quadrature rule:

(K−1uh, vh)Q =

  • E∈Th

(K−1uh, vh)Q,E =

  • E∈Th

1

J BTK−1Bˆ

uh, ˆ vh

  • Q, ˆ

E

Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

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SLIDE 7

Multipoint Flux Mixed Finite Element (MFMFE)—2 FEM space on ˆ E:

  • Simplicial element [Brezzi, Douglas, Marini 1985; Brezzi, Douglas, Duran, Fortin 1987]:

ˆ

V( ˆ

E) = P1( ˆ E)d, ˆ W( ˆ E) = P0( ˆ E),

  • 2D square [Brezzi, Douglas, Marini 1985]:

ˆ V ( ˆ E) = BDM1( ˆ E) =

  • α1x + β1y + r1 + rx2 + 2sxy

α2x + β2y + r2 − 2rxy − sy2

  • ˆ

W( ˆ E) = P0( ˆ E)

  • 3D cube [Ingram, Wheeler, Yotov 2009]:

ˆ V ( ˆ E) = BDDF1( ˆ E) + r2curl(0, 0, ˆ x2ˆ z)T + r3curl(0, 0, ˆ x2ˆ yˆ z)T + s2curl(ˆ xˆ y2, 0, 0)T + s3curl(ˆ xˆ y2ˆ z, 0, 0)T + t2curl(0, ˆ yˆ z2, 0)T + t3curl(0, ˆ xˆ yˆ z2, 0)T ˆ W( ˆ E) = P0( ˆ E)

Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

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SLIDE 8

Multipoint Flux Mixed Finite Element (MFMFE)—3 Numerical quadrature rule on ˆ V ( ˆ E): (K−1uh, vh)Q,E =

1

J BTK−1Bˆ

uh, ˆ vh

  • Q, ˆ

E

Symmetric [Wheeler and Yotov 2006]:

1

J BTK−1Bˆ

uh, ˆ vh

  • Q, ˆ

E

= | ˆ E| nv

nv

  • i=1

1

J BTK−1Bˆ

uh · ˆ vh

  • |ˆ

ri

nv: number of vertices of ˆ E.

Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

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Properties of MFMFE—1

  • M

NT N U P

  • =
  • −F
  • Basis functions in ˆ

V ( ˆ E): v11(r1) · n1 = 1, v11(r1) · n2 = 0 v11(ri) · nj = 0, for i = 1, j = 1, 2

1

J BTK−1Bv11, v11

  • Q, ˆ

E

= 0

1

J BTK−1Bv11, v12

  • Q, ˆ

E

= 0

1

J BTK−1Bv11, vij

  • Q, ˆ

E

= 0, i = 1 M is block diagonal. Cell-centered scheme: NM−1NTP = F

Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

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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

Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

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SLIDE 11

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 Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

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SLIDE 12

Multiscale Mortar MFMFE

Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

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SLIDE 13

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≤n

Λi,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 · niΓi, ∀v ∈ Vi, (∇ · u, w)Ωi = (f, w)Ωi, ∀w ∈ Wi,

n

  • i=1

u · ni, µΓi = 0, ∀µ ∈ Λ.

Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

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Multiscale Mortar MFMFE: formulation

Ωi Ωj Γij

spaces

  • lynomial
  • lynomial

)

Vh =

n

  • i=1

Vh,i,

Wh =

n

  • i=1

Wh,i ΛH =

  • 1≤i<j≤n

ΛH,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 · niΓi, ∀v ∈ Vh,i, (∇ · uh, w)Ωi = (f, w)Ωi, ∀w ∈ Wh,i,

n

  • i=1

ΠRuh · ni, µΓi = 0, ∀µ ∈ ΛH.

Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

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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, µΓi.

gH : L2(Γ) → R: gH(µ) =

n

  • i=1

gH,i(µ) =

n

  • i=1

ΠR¯

uh · ni, µΓi,

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 · niΓi,

v ∈ Vh,i,

(∇ · u∗

h(λ), w)Ωi = 0,

w ∈ Wh,i.

Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

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Multiscale Mortar MFMFE: an interface formulation—2 Bar problem: (¯

uh, ¯

ph) ∈ Vh × Wh solve, for 1 ≤ i ≤ n, (K−1¯

uh(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 Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

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SLIDE 17

Weakly Continuous Velocity Sapce

Vh,0 =

  v ∈ Vh :

n

  • i=1

ΠRv|Ωi · ni, µΓi = 0 ∀µ ∈ ΛH

   .

Assumption: For any µ ∈ ΛH, µ0,Γi,j ≤ C

  • QR

h,iµ0,Γi,j + QR h,jµ0,Γi,j

  • ,

1 ≤ i < j ≤ n. (1) Lemma 1 Under assumption (1), there exists a projection operator Π0 :

  • H1/2+ǫ(Ω)

d ∩ V → Vh,0 such that

(∇ · (Π0q − q), w) = 0, w ∈ Wh, Π0q − Πq

n

  • i=1

qr+1/2,Ωihr(h1/2 + H1/2), 0 ≤ r ≤ 1, Π0q − q

n

  • i=1

q1,Ωih1/2(h1/2 + H1/2).

Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

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SLIDE 18

Solvability of Multiscale Mortar MFMFE method (K−1uh, v)Q,Ωi − (ph, ∇ · v)Ωi = − g, ΠRv · ni∂Ωi/ Γ − λH, ΠRv · niΓi, ∀v ∈ Vh,i, (2) (∇ · uh, w)Ωi = (f, w)Ωi, ∀w ∈ Wh,i, (3)

n

  • i=1

ΠRuh · ni, µΓi = 0, ∀µ ∈ ΛH. (4) Lemma 2 Assume that (1) holds. Then, there exists a unique solution

  • f (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) = ph2, implies ph = 0.

  • 3. (2) gives 0 = λH, ΠRv · niΓi = QR

h,iλH, ΠRv · niΓi. ∃v, s.t.

v · ni = QR

h,iλH, implying QR h,iλH = 0. By assumption (1), ΛH = 0.

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SLIDE 19

Velocity Error Analysis Theorem 1 Let K−1 ∈ W 1,∞(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∇ · u1,Ωi, u − uh

n

  • i=1

(Hs−1/2ps+1/2,Ωi + hu1,Ωi + hr(H1/2 + h1/2)ur+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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SLIDE 20

Velocity Error Analysis: Sketch of Proof

  • Divergence error:

∇ · (Πu − uh) = 0 and ∇ · (u − Πu)0,Ωi h∇ · u1,Ωi

  • L2 error:

Let q = Π0u − uh Π0u − uh2 (K−1(Π0u − uh), q)Q =

  • K−1Π0u, q
  • Q −
  • K−1u, ΠRq
  • −

n

  • i=1

p − IHp, ΠRq · niΓi =

  • 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 · niΓi. |(K−1Πu, v − ΠRv)Q| hu1v. |σ(K−1q, v)|

  • E∈Th

hK−11,∞,Eq1,EvE.

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SLIDE 21

Superconvergence of Velocity Theorem 2 Assume that the tensor K is diagonal and K−1 ∈ W 2,∞(Ωi), 1 ≤ i ≤ n. Then, the velocity uh of the mortar MFMFE method (2)-(4)

  • n rectangular and cuboid grids, if (1) holds, satisfies

ΠRu − ΠRuh

n

  • i=1

(hr(H1/2 + h1/2)ur+1/2,Ωi + Hs−1/2ps+1/2,Ωi + h2u2,Ωi), where 0 < s < m + 1, 0 ≤ r ≤ 1. Lemma 3 Assume that K is a diagonal tensor and K−1 ∈ W 1,∞

Th

. Then for all uh ∈ Vh and vh ∈ VR

h 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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SLIDE 22

Pressure Error Analysis Define another weakly continuous space:

VR

h,0 =

  v ∈ VR

h : n

  • i=1

v|Ωi · ni, µΓi = 0 ∀µ ∈ ΛH

   ,

where VR

h : RT 0 space on each subdomain

Lemma 4 Spaces VR

h,0 × Wh satisfy the inf-sup condition: for all w ∈ Wh,

sup

0=v∈VR

h,0

n

  • i=1

(∇ · v, w)Ωi/

n

  • i=1

vdiv,Ωi w, 1 ≤ i ≤ n. Theorem 3 Let K−1 ∈ W 1,∞(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

(hp1,Ωi + hr(H1/2 + h1/2)ur+1/2,Ωi + hu1,Ωi + Hs−1/2ps+1/2,Ωi), where 0 < s ≤ m + 1, 0 ≤ r ≤ 1.

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SLIDE 23

Pressure Error Analysis: Sketch of Proof Qhp − ph sup

0=v∈VR

h,0

n

  • i=1

(∇ · v, Qhp − ph)Ωi/

n

  • i=1

vdiv,Ωi = sup

0=v∈VR

h,0

  • K−1u, v
  • −
  • K−1uh, v
  • Q + n

i=1p − IHp, v · niΓi

n

i=1 vdiv,Ωi

. and

  • K−1u, v
  • −
  • K−1uh, v
  • Q =
  • K−1(u − Πu), v
  • −
  • K−1(uh − Πu), v
  • Q

+ σ(K−1Πu, v)

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SLIDE 24

Superconvergence of Pressure Theorem 4 Assume that K ∈ W 1,∞(Ωi), K−1 ∈ W 2,∞(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)u2,Ωi + Hs(H1/2 + h1/2)ps+1/2,Ωi + hr+1/2(h1/2 + H1/2)2ur+1/2,Ωi), where 0 < s ≤ m + 1, 0 ≤ r ≤ 1.

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SLIDE 25

Superconvergence of Pressure: Sketch of Proof —1

  • Consider an auxiliary problem:

−∇ · (K∇φ) = ph − Qhp, in Ω, φ = 0,

  • n ∂Ω.

By regularity, φ2 Qhp − ph.

  • By definition of Qh, ΠR, Π0,

Qhp − ph2 =

n

  • i=1

(Qhp − ph, ∇ · K∇φ)Ωi =

n

  • i=1

(Qhp − ph, ∇ · ΠRΠ0K∇φ)Ωi =

n

  • i=1

(p − ph, ∇ · ΠRΠ0K∇φ)Ωi

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SLIDE 26

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 · niΓi −

n

  • i=1

g, (v − ΠRv) · ni∂Ωi/Γ, ∀v ∈ Vh,0, get Qhp − ph2 = (K−1u, vh) − (K−1uh, vh)Q +

n

  • i=1

p, vh · niΓi.

  • Use the weak continuity of vh,

Qhp − ph2 =

  • K−1(u − Πu), vh
  • −
  • K−1(uh − Πu), vh
  • Q

+ σ(K−1Πu, vh) +

n

  • i=1

p − PHp, vh · niΓi.

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SLIDE 27

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 1.5 2 H2 1 1 1.75 1.25

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SLIDE 28

Numerical Examples

Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

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SLIDE 29

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

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

pres 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2

X Y

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

pres 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2

Multiscale Mortar MFMFE solution: discontinuous linear (left) and discontinuous quadratic (right) mortars.

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SLIDE 30

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 2.02 32 2.97E-02 1.00 1.27E-01 1.02 8.66E-04 2.00 1.92E-03 2.00 64 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 2.00 256 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 2.02 64 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 1.83 64 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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SLIDE 31

Numerical Example 1: On a rectangular mesh—3

X Y

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

errp 3.5E-03 3.3E-03 3.0E-03 2.8E-03 2.6E-03 2.4E-03 2.1E-03 1.9E-03 1.7E-03 1.5E-03 1.2E-03 1.0E-03

X Y

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

errp 3.5E-03 3.3E-03 3.0E-03 2.8E-03 2.6E-03 2.4E-03 2.1E-03 1.9E-03 1.7E-03 1.5E-03 1.2E-03 1.0E-03

Error in Multiscale Mortar MFMFE solution: discontinuous linear (left) and discontinuous quadratic (right) mortars.

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SLIDE 32

Numerical Example 2: On an h2-parallelogram mesh—1 The map is defined as x = ˆ x + 0.03 cos(3πˆ x) cos(3πˆ y), y = ˆ y − 0.04 cos(3πˆ x) cos(3πˆ y).

X Y

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

pres 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2

X Y

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

pres 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2

Multiscale Mortar MFMFE solution: discontinuous linear (left) and discontinuous quadratic (right) mortars.

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SLIDE 33

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 1.75 32 2.41E-02 1.00 1.03E-01 1.01 4.54E-04 2.01 9.55E-03 1.87 64 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 1.83 256 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 1.70 64 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

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SLIDE 34

Numerical Example 2: On an h2-parallelogram mesh—3

X Y

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

errp 3.2E-03 2.9E-03 2.7E-03 2.4E-03 2.1E-03 1.8E-03 1.6E-03 1.3E-03 1.0E-03 7.5E-04 4.7E-04 2.0E-04

X Y

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

errp 3.2E-03 2.9E-03 2.7E-03 2.4E-03 2.1E-03 1.8E-03 1.6E-03 1.3E-03 1.0E-03 7.5E-04 4.7E-04 2.0E-04

Error in Multiscale Mortar MFMFE solution: discontinuous linear (left) and discontinuous quadratic (right) mortars.

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SLIDE 35

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 z2 + 1 sin(xy) 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 1.99 32 2.71E-02 1.00 1.94E-02 1.00 1.54E-04 2.00 6.50E-05 2.00 64 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 1.92 64 1.35E-02 1.00 9.68E-03 1.00 3.85E-05 2.00 1.67E-05 1.98

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SLIDE 36

Numerical Example 3: On a cubic mesh—2

pres 1.2 1 0.8 0.6 0.4 0.2

  • 0.2
  • 0.4
  • 0.6
  • 0.8
  • 1
  • 1.2

errp 9.0E-04 8.3E-04 7.7E-04 7.0E-04 6.3E-04 5.7E-04 5.0E-04 4.3E-04 3.7E-04 3.0E-04 2.3E-04 1.7E-04 1.0E-04

Discontinuous quadratic mortars and matching grids: Multiscale Mortar MFMFE solution (left) and error (right)

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SLIDE 37

Numerical Example 4: On regular h2-parallelpipeds—1 Mapping: x = ˆ x + 0.03 cos(3πˆ x) cos(3πˆ y) cos(3πˆ z), y = ˆ y − 0.04 cos(3πˆ x) cos(3πˆ y) cos(3πˆ z), z = ˆ z + 0.05 cos(3πˆ x) cos(3πˆ y) cos(3πˆ z).

pres 1.2 1 0.8 0.6 0.4 0.2

  • 0.2
  • 0.4
  • 0.6
  • 0.8
  • 1
  • 1.2

errp 6.5E-03 6.0E-03 5.5E-03 5.0E-03 4.5E-03 4.0E-03 3.5E-03 3.0E-03 2.5E-03 2.0E-03 1.5E-03 1.0E-03 5.0E-04

Discontinuous quadratic mortars and matching grids: Multiscale Mortar MFMFE solution (left) and error (right)

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SLIDE 38

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 1.30 32 2.75E-02 1.00 4.51E-02 0.99 5.24E-04 1.83 5.93E-03 1.82 64 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 0.92 64 1.37E-02 1.00 2.24E-02 1.00 1.35E-04 1.89 1.53E-03 1.89

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SLIDE 39

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 Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA