     # Multiscale Mixed Finite Element Method for elliptic problems Multiscale Mixed Finite Element Method

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• Multiscale Mixed Finite Element Method for elliptic

problems with oscillating coe�cients

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 coe�cients : incorporate

the local small-scale information of the leading-order

di�erential 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�.)

R. Hrtus RICAM, Special semester 2011

• Motivation

Why MsMFEM?

We compute primal and dual variable at the same time.

Less computational e�ort- 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).

R. Hrtus RICAM, Special semester 2011

• Motivation - rigorous �gure

R. Hrtus RICAM, Special semester 2011

• Notation

Basic ones

Ω ⊆ Rd, d = 2, 3 with boundary ∂Ω- outer normal vector n̄ 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,D in special case if p = 2: Wm,2 (D) is denoted by Hm (D) with norm ‖.‖m,D and seminorm |.|m,D In following V (D) ⊆ H1 (D) whose functions have zero average over D

R. Hrtus RICAM, Special semester 2011

• Equations

Second order elliptic equation with locally periodic coe�cients

L�u� := ∂

∂xi

( aij

( x, x

) ∂u� ∂xj

) = f in Ω,

−a ( x, x

) ∇u� . n̄ = g on ∂Ω.

Here � is small and a ( x, x�

) = ( aij ( x, x�

)) is a symmetric matrix

(bounded, elliptic).

Furthermore, assume that aij ∈ C1 ( Ω̄, C1p

( Rd )) , where C1p

( 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)∇u� . n̄ = g on ∂Ω. (2)

R. Hrtus RICAM, Special semester 2011

• 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)

) =

∂yi aij (x, y) in Y,

∫ Y

χj (x, y) dy = 0.

Here δkj means Kronecker delta.

R. Hrtus RICAM, Special semester 2011

• Example of periodical matrix of coe�cients

� = 0.5

aij

( x, x

) = a

(x �

) δij , a

(x �

) =

2 + Psin ( 2π x1�

) 2 + Psin

( 2π x2�

)+2 + Psin (2π x2� ) 2 + Pcos

( 2π x1�

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

R. Hrtus RICAM, Special semester 2011

• Example of periodical matrix of coe�cients

� = 0.25

R. Hrtus RICAM, Special semester 2011

• Example of periodical matrix of coe�cients

� = 0.1

R. Hrtus RICAM, Special semester 2011

• Important assumptions and notations

(H1) f ∈ H1 (Ω) , g = q̄0 . n̄ on ∂Ω for some q̄0 ∈ H1 (Ω)d .

(H2) Compatibility

∫ Ω fdx =

∫ ∂Ω gds.

Let

L20 (Ω) be a subspace of L 2 (Ω) whose functions have zero

average over Ω

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

H0 (div,Ω) = { v̄ ∈ L2 (Ω) , div v̄ ∈ L2 (Ω) , v̄ . n̄ = 0 on ∂Ω

} .

H (div,Ω) norm ‖.‖div,Ω.

R. Hrtus RICAM, Special semester 2011

• Variational formulation

Let p̄� = −a ( x, x�

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

a (x, y), then ∇u� = a−1 ( x, x�

) p̄�

Mixed formulation to (1) , (2)

Find a pair (p̄�, u�) ∈ H (div,Ω)×L20 (Ω) such that p̄� . n̄ = g on ∂Ω and

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

( x, x

) p̄�, q̄

) − (u�, div q̄) = 0 ∀q̄ ∈ 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.

R. Hrtus RICAM, Special semester 2011

• Discretization

τh - regular, quasi-uniform partition of Ω

∀ K ∈ τh : hK diameter, |K| it's Lebesgue measure, ν̄K the unit outer normal to ∂K{ eKj

}d+1 j=1

- edges surfaces of ∂K with ∣∣∣eKj ∣∣∣- measure of eKj

∀ K ∈ τh :

RT0 (K) = P0 (K) d + x̄P0 (K) , x̄ = (x1, . . . , xd)

T ∈ Rd,

where P0 (K) is the constant element space

Basis of RT0 (K) : { RKi }d+1 i=1

satisfying

RKi . ν̄ K =

{ 1

|eKi | on eKi ,

0 on eKj , j 6= i.

R. Hrtus RICAM, Special semester 2011

• Discretization

Since RKi is constant in K: (by Green) div R K i =

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 statis�es the relations∫ eKj

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

with error estimate

‖q̄ − rK q̄‖m,K ≤ Ch 1−m K |q̄|1,K ∀q̄ ∈ H

1 (K)d , m = 0, 1.

R. Hrtus RICAM, Special semester 2011

• Discretization cont.

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

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

Find a pair (p̄h, uh) ∈Wh ×Mh s.t. p̄h . n̄ = rhq̄0 . n̄ on ∂Ω and

(div p̄h, vh) = (f, vh) ∀vh ∈Mh,( a−1

( x, x

) p̄h, q̄h

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

Error estimate (Babu²ka,Brezzi)

‖p̄� − p̄h‖div,Ω + ‖u� − uh‖0,Ω ≤ Ch ( ‖p̄�‖1,Ω + ‖u�‖1,Ω

) . (3)

Note: if ‖p̄�‖1,Ω ≤ C�−1, h: must be h

• Multiscale Mixed FEM

Formulation of the problem

Let V (K) ⊂ H1 (K), whose functions have zero average over K and de�ne wKi ∈ V (K) as the solution of the following Neumann BVP over K for i = 1, . . . , d+ 1 :∫ K

a ( x, x

) ∇wKi ∇φdx =

1

|K|

∫ K

φdx− 1∣∣eKi ∣∣ ∫ eKi

φds ∀φ ∈ H1 (K)

Above equation is the weak formulation of the following BVP.

L�w K i =

1

|K| in K, −a

( x, x

) ∇wKi . ν̄K =

{ 1

|eKi | on eKi ,

0 on eKj , j 6= i.

R. Hrtus RICAM, Special semester 2011

• Multiscale Mixed FEM

Let p̄Ki = −a ( x, x�

) ∇wKi and denote by MS (K) the

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

1 |K| . We have

div p̄Ki = div R K i in K, and p̄

K i . ν̄

K = RKi . ν̄ K on ∂K.

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

q̄h . ν̄ Kds = 0, i = 1, . . . , d+ 1,

Imply q̄h = 0 in K.

DOFs for q̄h ∈MS (K) can be chosen as 0th moments of q̄h . ν̄

K on the sides of faces of K.

In practice: the base functions p̄Ki of MS (K)- approximated by solving weak form on τh of K with mesh size resolving � using the lowest-order RT MFEM.

R. Hrtus RICAM, Special semester 2011

• Multiscale Mixed FEM

Σ = H (div,Ω) ⊃ Σh Q = H0 (div,Ω) ⊃ Qh Mh = L

2 0 (Ω)

Σh = {q̄h ∈ H (div,Ω) : q̄h|K ∈MS (K) ,∀K ∈ τh} Qh = Σh ∩Q Bh - collection of all sides of faces of τh, which lie on the boundary ∂Ω

for any e ∈ Bh s.t. e = ∂K ∩ ∂Ω p̄e ∈MS (K)- corresponding MS basis gh = q̄0,h . n̄ on ∂Ω with q̄0,h ∈ Σh

q̄0,h = ∑ e∈Bh

∫ e

gds

 p̄e. R. Hrtus RICAM, Special semester 2011

• Multiscale Mixed FEM formulation

Discretization of Mixed FEM

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

(div p̄h, vh) = (f, vh) ∀vh ∈Mh,( a−1

( x, x

) p̄h, q̄h

) − (uh, div q̄h) = 0 ∀q̄h ∈ Qh. (4)

R. Hrtus RICAM, Special semester 2011

• Multiscale Mixed FEM formulation - convergence theorem

Theorem

The discrete problem (4) has a unique solution (p̄h, uh) ∈ Σh ×Mh s.t. p̄h . n̄ ##### Multiscale Mixed Finite-Element Methods for Simulation .Multiscale Mixed Finite-Element Methods for
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