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