Uniform inf-sup condition for the Brinkman problem in...

Uniform inf-sup condition for the Brinkman problem inhighly heterogeneous media

Raytcho Lazarov & Aziz Takhirov

Texas A&M

May 3-4, 2016

R. Lazarov & A.T. (Texas A&M) Brinkman May 3-4, 2016 1 / 30

Outline

1 Outline

2 Motivation

3 Related work

4 The continuous problem

5 The discrete problem

6 References


Motivation

Brinkman equation

−ν∆u + νK−1u +∇p = f in Ω,

∇ · u = 0 in Ω,

u = 0 on ∂Ω,

where- u is the fluid velocity.- p is the pressure.- ν is the viscosity. We assume ν = 1.- ν is the effective viscosity. We assume ν = 1.- f is the external forcing term.- 0 < K (x) <∞ is the permeability of the medium.


Motivation

Brinkman equation

Brinkman can be used to model flows in:Pebble Bed Reactors,filtration,biological flows,oil/water reservoirs.


Motivation

Flows in highly heterogeneous media

Figure: SPE10 3 dimensional permeability distributions, logscale.


Motivation

Flows in highly heterogeneous media

When permeability field K (x) has large variations and jumps, theproblem becomes more challenging.

Contrast of the media: κΩ =maxx∈Ω

K(x)

minx∈Ω

K(x) .

Exact solution has low regularity when κΩ 1.The known iterative methods, converge very slowly or practically donot converge when κΩ 1, due to the dependence of the conditionnumber of the linear system on κΩ.


Motivation

Preconditioners

X - real, separable, Hilbert space,inner product on X is (·, ·), norm on X is ‖ · ‖,X ∗ be the dual of X , 〈·, ·〉 be the duality pairing.

Given A ∈ L (X ,X ∗), symmetric and f ∈ X ∗, find x ∈ X such that

Ax = f ⇔

a(x , y) := 〈Ax , y〉 = 〈f , y〉 ∀y ∈ X .


Motivation

Preconditioners cont-d

Definition[3, Mardal & Winther (2011)] B ∈ L (X ∗,X ) is a preconditioner forA ∈ L (X ,X ∗) if B is symmetric and positive definite in the sense that

〈·,B·〉

is inner product on X ∗.

B is a Riesz operator: Given f ∈ X ∗

(Bf , y) = 〈f , y〉 ∀y ∈ X .


Motivation

Preconditioned system

UsuallyAx = f

is preconditioned asBAx = Bf .

Condition number of the system

cond (BA) := ‖BA‖L(X ,X )‖ (BA)−1 ‖L(X ,X )

Letting

‖a‖ := supx ,y∈X

a(x , y)

‖x‖‖y‖, infx∈X

supy∈Y

a(x , y)

‖x‖‖y‖≥ γ ⇒

cond (BA) ≤ ‖a‖γ.


Motivation

Condition number of the Brinkman problem

Let

A =

(−∆ + IK−1 ∇−∇· 0

).

Brinkman system:

A(

up

)=

(f0

).

If X =(H1

0 , ‖∇ · ‖),Q =

(L2

0, ‖ · ‖), then

B =

((−∆)−1 0

0 I

).

Condition number of the system

cond (BA) ≤ O (κΩ) .


Motivation

Question

Is it possible to establish well-posedness of the Brinkman problem, suchthat cond (BA) is independent of the media contrast κΩ?


Related work

Singularly perturbed Stokes system

In [2, Mardal & Winther (2004)], authors consider

−ε∆u + u +∇p = f in Ω,

∇ · u = 0 in Ω,

u = 0 on ∂Ω.

Authors establish well-posedness in ε-dependent norms, both in continuousand discrete cases.


The continuous problem

Intersection and sum of Hilbert spaces

If X and Y are Hilbert spaces, then X + Y and X ∩ Y are also Hilbertspaces [1, Bergh, Löfström], with the following norms:

‖z‖X∩Y =√‖z‖2X + ‖z‖2Y ∼ max (‖z‖X , ‖z‖Y ),

‖z‖X+Y = infz=x+y

x∈X ,y∈Y

√‖x‖2X + ‖y‖2Y .

If X ∩ Y is dense in both X and Y , then

(X ∩ Y )∗ = X ∗ + Y ∗, and (X + Y )∗ = X ∗ ∩ Y ∗.



The continuous, weak formulation

a(u, v) + b(p, v) = (f, v)

b(q,u) = 0,

a(u, v) := (∇u,∇v) + (u, v)α ,

b(p, v) := (p,∇ · v) ,

where α = K−1.The natural space for the velocity field is

X :=(H1

0 ∩ L2α

)d, ‖u‖X :=

√‖∇u‖2 + ‖u‖2α.



The pressure norm

‖ · ‖Q should be chosen, so that the Brezzi theory for well posedness holds:a(u, v) ≤ ‖a‖‖u‖X‖v‖X ,a(u,u) ≥ a0‖u‖2X ,b(p, v) ≤ ‖b‖‖p‖Q‖v‖X ,infp∈Q

supv∈X

b(p,v)‖p‖Q‖v‖X ≥ β.

For our applications, we need to ensure that cond (BA) is independent ofκΩ.



The pressure norm

Let

Q := L20 + H1

K ∩ L20 =

q ∈ L2

0 : q = q1 + q2, q1 ∈ L20, q2 ∈ H1

K ∩ L20,

with the associated norm

‖q‖Q = infq=q1+q2

q1∈L20,q2∈H1

K∩L20

√‖q1‖2 + ‖∇q2‖2K .



The pressure norm

Lemma

Given q ∈ L20, let q2 ∈ H1

K ∩ L20 be the solution of the following elliptic

problem: −∇ · (K∇q2) + q2 = q in Ω,

K∇q2 · n = 0 on ∂Ω.

Then

‖q‖L20+H1

K∩L20

=√‖q1‖2 + ‖∇q2‖2K

=√‖q − q2‖2 + ‖∇q2‖2K

=√‖q‖2 − ‖q2‖2H1

K∩L20.



Well-posedness: Continuity of b(·, ·)

Lemma

The bilinear form b(·, ·) : Q × X → R is continuous.

Proof.∀q ∈ Q, let q = q1 + q2 with q1 ∈ L2

0 and q2 ∈ H1K ∩ L2

0. Then

b (q, v) = b (q1, v) + b (q2, v)

= − (q1,∇ · v) + (∇q2, v)

= − (q1,∇ · v) +(K

12∇q2, α

12 v)

≤√d‖q1‖‖∇v‖+ ‖∇q2‖K‖v‖α

≤√d√‖q1‖2 + ‖∇q2‖2K‖v‖X .

Taking infimum over all q1, q2 gives b (q, v) ≤√d‖q‖Q‖v‖X .



Well-posedness: Coercivity of b(·, ·)

Lemma

There exists a constant β > 0, independent of 0 < K (x) <∞, such that

infq∈Q

supv∈X

b (q, v)

‖q‖Q‖v‖X≥ β.



Proof of the coercivity of b(·, ·)

Proof.The result of Necas: ∀q ∈ L2

0 (Ω) : ‖∇q‖H−1(Ω) ∼ ‖q‖.∀q ∈ Q with q = q1 + q2 and

∫Ω

qi = 0. Assuming that the duality pairing

〈·, ·〉X∗×X is an extension of the L2 inner product, one obtains that:

supv∈X

b (q, v)

‖v‖X= ‖∇q‖X∗ = ‖∇q‖H−1+L2

K

= infq=q1+q2

∇q1∈H−1,∇q2∈L2K

√‖∇q1‖2H−1 + ‖∇q2‖2K

≥ C infq=q1+q2

q1∈L20,q2∈H1

K∩L20

√‖q1‖2 + ‖∇q2‖2K

= C‖q‖Q .


The discrete problem

Assumption

Assumption. The mesh has resolved the heterogeniety of the medium sothat elementwise contrast

κE =maxx∈E

K (x)

minx∈E

K (x)

is a moderate constant. We will also set

κTh = maxE∈Th

κE .



Inf-sup with conforming subspaces

Fortin’s Lemma for Mini-element:Πh = Πb

h (I − Ch) + Ch.Πbh satisfies

(∇ · Πb

hv, qh)

= (∇ · v, qh).Ch is Clement or Scott-Zhang (quasilocal) interpolant.In particular need, ‖Chv‖α ≤ c‖v‖α, with c independent of κΩ.

For any E ∈ Th:

‖Chv‖2α,E ≤ maxx∈E

α(x)‖Chv‖2E ≤ C maxx∈E

α(x)‖v‖2ΩE

≤ Cmaxx∈E

α(x)

minx∈ΩE

α(x)‖v‖2α,ΩE

.



Non-conforming velocity space

Xh :=(Pdk ⊕ xPk

)∩ H0 (div ,Ω) ,Qh := Pk−1 ⊂ H1 (Ω) .

J (uh, vh) : =∑

e∈Γh∪Γ

σe|e|

∫e

[uh][vh],

ah (uh, vh) : =∑E∈Th

(∇uh,∇vh)E + (αuh, vh) + J (uh, vh)

−∑

e∈Γh∪Γ

∫e

∇uh · n [vh]−∑

e∈Γh∪Γ

∫e

∇vh · n [uh],

bh (ph, vh) : = −∑E∈Th

(ph,∇ · vh)E



Discrete weak formulation

ah (uh, vh) + bh (ph, vh) = (f, vh)

bh (qh,uh) = 0.

‖uh‖Xh:=

√∑E∈Th

‖∇uh‖2E + ‖uh‖2α + J (uh,uh),

‖qh‖Q := infqh=q1+q2

√‖q1‖2 + ‖∇q2‖2K =

√‖qh − q2‖2 + ‖∇q2‖2K .



Discrete inf-sup

Lemma

The following inf-sup condition holds: There exists a constant βh > 0,independent of κΩ and h, such that

infqh∈Qh

supvh∈Xh

bh (qh, vh)

‖qh‖Q‖vh‖Xh

≥ βh.



Proof of discrete inf-sup

Proof.Let ∀qh ∈ Qh. By continuous inf-sup condition, there exists v ∈ X suchthat

b (qh, v) ≥ β‖qh‖Q‖v‖X .

Let vh = πhv ∈ Xh be the Raviart-Thomas interpolant of v. By definitionof the Raviart-Thomas interpolant:

bh (qh, vh) = −∑E∈Th

(∇ · vh, qh)E = −∑E∈Th

(∇ · v, qh)E

= b (qh, v) ≥ β‖qh‖Q‖v‖X .

One can show that

‖vh‖α ≤ CκTh‖v‖α ⇒ ‖vh‖X ≤ C (κTh) ‖v‖X .



Continuous vs. discrete pressure norms

Recall that for ph ∈ L20,

‖ph‖Q =√‖ph‖2 − ‖p2‖2H1

K∩L20,

where

(K∇p2,∇q) + (p2, q) = (ph, q) ∀q ∈ H1K ∩ L2

0.

So in general, p2 /∈ Qh, and therefore ‖ph‖Q is not computable.



Continuous vs. discrete pressure norms

Let

‖ph‖Qh=√‖ph‖2 − ‖p2,h‖2H1

K∩L20,

where

(K∇p2,h,∇qh) + (p2,h, qh) = (ph, qh) ∀qh ∈ Qh,

Then

‖p2,h‖H1K∩L

20≤ ‖p2‖H1

K∩L20⇒

‖ph‖Qh≥ ‖ph‖Q .



The end

THANK YOU!


References

References

J. Bergh, J. Löfström, Interpolation Spaces: An Introduction, SpringerBerlin Heidelberg, 1976.

K-A Mardal, R. Winther, Uniform preconditioners for the timedependent Stokes problem, Numerische Mathematik 98 (2), 305-327.

K-A Mardal, R. Winther, Preconditioning discretizations of systems ofpartial differential equations, Numer. Linear Algebra Appl. (18) 2011,1-40.


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