Download pdf - Multiscale Mixed Finite Element Method for elliptic problems … · Multiscale Mixed Finite Element Method for elliptic problems with oscillating coe cients Rostislav Hrtus 14th November

Multiscale Mixed Finite Element Method for elliptic

problems with oscillating coecients

Rostislav Hrtus

14th November 2011

R. Hrtus RICAM, Special semester 2011

Motivation

Properties

Flow transport in highly heterogenous porous media.

We want to capture large scale behaviour without resolving

the ne scale problem (expensive).

possibility to obtain reasonable results only on a coarse grid.

We construct so called Multi Scale nite element bases which

locally solve Neumann boundary value problem.

Idea

With asumption on locally periodic coecients : incorporate

the local small-scale information of the leading-order

dierential operator into the FE bases (captures small scale

behaviour on the large scale).

Our case

Propose and analyse MsMFEM with an over-sampling

technique. (osc. coe.)


Motivation

Why MsMFEM?

We compute primal and dual variable at the same time.

Less computational eort- solution on a coarse mesh gives

similar results as a much ner mesh without MsMFEM.

Suitable when some composite materials, rock or soil

formations (some special cases) are considered (assumed some

periodicity).


Motivation - rigorous gure


Notation

Basic ones

Ω ⊆ Rd, d = 2, 3 with boundary ∂Ω- outer normal vector n

D ⊆ Ω with Lipschitz boundary Γ

∀m ≥ 0 and 1 ≤ p ≤ ∞denote Sobolev space Wm,p (D) with norm ‖.‖m,p,D and

seminorm |.|m,p,Din special case if p = 2: Wm,2 (D) is denoted by Hm (D) with

norm ‖.‖m,D and seminorm |.|m,DIn following V (D) ⊆ H1 (D) whose functions have zero

average over D


Equations

Second order elliptic equation with locally periodic coecients

Lεuε :=∂

∂xi

(aij

(x,x

ε

) ∂uε∂xj

)= f in Ω,

−a(x,x

ε

)∇uε . n = g on ∂Ω.

Here ε is small and a(x, xε

)=(aij(x, xε

))is a symmetric matrix

(bounded, elliptic).

Furthermore, assume that aij ∈ C1(Ω, C1

p

(Rd)), where C1

p

(Rd)

stands for the collection of all C1(Rd)periodic functions with

respect to the unit cube Y .

Homogenized solution u0 of the problem above

L0u0 :=∂

∂xi

(a∗ij (x)

∂u0

∂xj

)= f in Ω, (1)

−a∗ (x)∇uε . n = g on ∂Ω. (2)


Equations cont. - choice of a∗ (x) =(a∗ij (x)

)

a∗ij (x) =1

|Y |

∫Y

aik (x, y)

(δkj −

∂χj

∂yk(x, y)

)dy,

and χj (x, y) is the periodic solution of the cell problem

∂

∂yi

(aik (x, y)

∂χj

∂yk(x, y)

)=

∂

∂yiaij (x, y) in Y,

∫Y

χj (x, y) dy = 0.

Here δkj means Kronecker delta.


Example of periodical matrix of coecients

ε = 0.5

aij

(x,x

ε

)= a

(xε

)δij , a

(xε

)=

2 + Psin(2π x1ε

)2 + Psin

(2π x2ε

)+2 + Psin

(2π x2ε

)2 + Pcos

(2π x1ε

)f (x) = 2π2cos (πx1) cos (πx2) and g (x) = 0



ε = 0.25



ε = 0.1


Important assumptions and notations

(H1) f ∈ H1 (Ω) , g = q0 . n on ∂Ω for some q0 ∈ H1 (Ω)d .

(H2) Compatibility

∫Ωfdx =

∫∂Ωgds.

Let

L20 (Ω) be a subspace of L2 (Ω) whose functions have zero

average over Ω

H0 (div,Ω) be the subspace of H (div,Ω) given by

H0 (div,Ω) =v ∈ L2 (Ω) , div v ∈ L2 (Ω) , v . n = 0 on ∂Ω

.

H (div,Ω) norm ‖.‖div,Ω.


Variational formulation

Let pε = −a(x, xε

)∇uε and a−1 (x, y) the inverse matrix of

a (x, y), then ∇uε = a−1(x, xε

)pε

Mixed formulation to (1) , (2)

Find a pair (pε, uε) ∈ H (div,Ω)×L20 (Ω) such that pε . n = g on ∂Ω

and

(div pε, v) = (f, v) ∀v ∈ L2 (Ω) ,(a−1

(x,x

ε

)pε, q

)− (uε, div q) = 0 ∀q ∈ H0 (div,Ω) .

( . , . ) stands for the inner product of L2 (Ω) or L2 (Ω)d.The existence of a unique solution to these eq. - Babu²ka-Brezzi th.


Discretization

τh - regular, quasi-uniform partition of Ω

∀ K ∈ τh : hK diameter, |K| it's Lebesgue measure, νK the

unit outer normal to ∂KeKj

d+1

j=1- edges surfaces of ∂K with

∣∣∣eKj ∣∣∣- measure of eKj

∀ K ∈ τh :

RT0 (K) = P0 (K)d + xP0 (K) , x = (x1, . . . , xd)T ∈ Rd,

where P0 (K) is the constant element space

Basis of RT0 (K) :RKid+1

i=1satisfying

RKi . νK =

1

|eKi |on eKi ,

0 on eKj , j 6= i.


Discretization

Since RKi is constant in K: (by Green) div RKi = 1|K| .

Wh ⊂ H (div,Ω) - lowest-order RT FE element space.

interpolation : ∃ rh : H (div,Ω) ∩H1 (Ω)d →Wh s.t.

rK := rh|K statises the relations∫eKj

(rK q − q) . νKds = 0 ∀q ∈ H1 (K)d , j = 1, . . . , d+ 1,

with error estimate

‖q − rK q‖m,K ≤ Ch1−mK |q|1,K ∀q ∈ H1 (K)d , m = 0, 1.


Discretization cont.

Mh ⊂ L20 (Ω) - standard piecewise const. lin. FE space for

approx. uε

Discrete mixed variational formulation to (1) , (2)

Find a pair (ph, uh) ∈Wh ×Mh s.t. ph . n = rhq0 . n on ∂Ω and

(div ph, vh) = (f, vh) ∀vh ∈Mh,(a−1

(x,x

ε

)ph, qh

)− (uh, div qh) = 0 ∀qh ∈Wh ∩H0 (div,Ω) .

Error estimate (Babu²ka,Brezzi)

‖pε − ph‖div,Ω + ‖uε − uh‖0,Ω ≤ Ch(‖pε‖1,Ω + ‖uε‖1,Ω

). (3)

Note: if ‖pε‖1,Ω ≤ Cε−1, h: must be h << ε to obtain accurate

approximations.

We want to get rid of this requirement on the mesh and introduce

multiscale mixed FEM.R. Hrtus RICAM, Special semester 2011

Multiscale Mixed FEM

Formulation of the problem

Let V (K) ⊂ H1 (K), whose functions have zero average over Kand dene wKi ∈ V (K) as the solution of the following Neumann

BVP over K for i = 1, . . . , d+ 1 :∫K

a(x,x

ε

)∇wKi ∇φdx =

1

|K|

∫K

φdx− 1∣∣eKi ∣∣∫eKi

φds ∀φ ∈ H1 (K)

Above equation is the weak formulation of the following BVP.

LεwKi =

1

|K|in K, −a

(x,x

ε

)∇wKi . νK =

1

|eKi |on eKi ,

0 on eKj , j 6= i.



Let pKi = −a(x, xε

)∇wKi and denote by MS (K) the

multiscale FE space spanned by pKi , i = 1, . . . , d+ 1.Recalling that RKi = 1

|K| . We have

div pKi = div RKi in K, and pKi . νK = RKi . νK on ∂K.

Moreover for any qh ∈MS (K), the relations∫eKi

qh . νKds = 0, i = 1, . . . , d+ 1,

Imply qh = 0 in K.

DOFs for qh ∈MS (K) can be chosen as 0th moments of

qh . νK on the sides of faces of K.

In practice: the base functions pKi of MS (K)- approximated

by solving weak form on τh of K with mesh size resolving εusing the lowest-order RT MFEM.



Σ = H (div,Ω) ⊃ Σh

Q = H0 (div,Ω) ⊃ QhMh = L2

0 (Ω)

Σh = qh ∈ H (div,Ω) : qh|K ∈MS (K) ,∀K ∈ τhQh = Σh ∩QBh - collection of all sides of faces of τh, which lie on the

boundary ∂Ω

for any e ∈ Bh s.t. e = ∂K ∩ ∂Ωpe ∈MS (K)- corresponding MS basis

gh = q0,h . n on ∂Ω with q0,h ∈ Σh

q0,h =∑e∈Bh

∫e

gds

pe.


Multiscale Mixed FEM formulation

Discretization of Mixed FEM

nd a pair (ph, uh) ∈ Σh ×Mh s.t. ph . n = gh on ∂Ω and

(div ph, vh) = (f, vh) ∀vh ∈Mh,(a−1

(x,x

ε

)ph, qh

)− (uh, div qh) = 0 ∀qh ∈ Qh. (4)


Multiscale Mixed FEM formulation - convergence theorem

Theorem

The discrete problem (4) has a unique solution (ph, uh) ∈ Σh ×Mh

s.t. ph . n = gh on ∂Ω. Moreover, if the homogenized solution

u0 ∈ H2 (Ω) ∩W 1,∞ (Ω) and the assumptions (H1) and (H2) are

satistied, then there exists a constant C > 0 independent of h and

ε such that

‖qε − ph‖div,Ω + ‖uε − uh‖0,Ω≤ C (h+ ε)

(‖u0‖2,Ω + ‖f‖1,Ω + ‖q‖div,Ω

)(5)

+c

√ε

h‖u0‖1,∞,Ω .


Notes

Mixed FEM

While in (3), where a term hε−1 appears, if ε→ 0 we have troubles

(the whole estimate tends to innity).


We can see in the above shown estimate (5) improvement. It is safer,

because as ε tends to zero, nothing wrong happens(√

εh

).

On the other hand, estimate deterioration as ε is of the same order as

h can be observed - this is caused by boundary layer eect (oscillating

structure on pε, while lin. FE solution ph is on ∂K) and we call it

resonant error.

... but still, a better estimate is desired...


MsMFEM - oversampling idea

With this approach, we can delete square root of εh from estimate

and obtain better approximations.R. Hrtus RICAM, Special semester 2011

MsMFEM - oversampling idea

Instead of red element K, we compute multiscale bases on

green, oversampled domain K.

Once we have solution on green K, we just use restriction of

solution on the red K.


Comments on numerical results with oversampling

Computation took parameters from already presented parameters

matrix aij where analytical solution is unknown. Due to this, only

comparison with standard conforming lin. FEM on very ne mesh

was possible.

In case when εh is chosen xed (0.5) and only the coarse mesh

was varying, results were stable. As ε decreased (as well as h)we still have convergence.

If xed ε was chosen (ε = 1128) we can observe error increase

as h decreases. Strongest resonant error accurs when ε = h

From previous studies: cell resonance error more visible

especially when we have periodic coecients.


Final comments

Some motivation of MsMFEM.

Mixed formulation dened based on homogenisation results.

Multiscale Mixed Finite Element Method with idea to improve

solution.


Thank you for your attention! ... Questions?
