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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