Multiscale Finite Elements Basic methodology and theory for periodic - - PowerPoint PPT Presentation

Multiscale Finite Elements

Basic methodology and theory for periodic coefficients for second-order elliptic equations Markus Kollmann October 18th, 2011

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Outline

1

Motivation

2

Introduction to MsFEM

3

Analysis in 2D

4

Reducing boundary effects

5

Generalization of MsFEM

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Outline

1

Motivation

2

Introduction to MsFEM

3

Analysis in 2D

4

Reducing boundary effects

5

Generalization of MsFEM

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Motivation

Many scientific and engineering problems involve multiple scales, particularly multiple spatial and (or) temporal scales (e.g. composite materials, porous media,...) Difficulty of direct numerical solution: size of the computation From an application perspective: sufficient to predict the macroscopic properties of the multiscale systems ⇒ Multiscale modeling: calculation of material properties or system behaviour on the macroscopic level using information or models from microscopic levels (capture the small scale effect on the large scale, without resolving the small-scale features)

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Outline

1

Motivation

2

Introduction to MsFEM

3

Analysis in 2D

4

Reducing boundary effects

5

Generalization of MsFEM

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Introduction

Capture the multiscale structure of the solution via localized basis functions Basis functions contain information about the scales that are smaller than the local numerical scale (multiscale information) Basis functions are coupled through a global formulation to provide a faithful approximation of the solution ⇒ Two main ingredients of MsFEM:

Global formulation of the method Construction of basis functions

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Problem Formulation

Consider the linear elliptic equation Lu = f in Ω, u = 0

• n

∂Ω, (1) where Lu := −div (k(x)∇u) . Ω ... domain in Rd (d = 2, 3) k(x) ... heterogeneous field varying over multiple scales Additionally assume: k(x) = (kij(x)) is symmetric α|ξ|2 ≤ kijξiξj ≤ β|ξ|2 ∀ξ ∈ Rd (0 < α < β)

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Problem Formulation, contd.

Variational formulation of (1): Find u ∈ H1

0(Ω) such that

a(u, v) = f , v, ∀v ∈ H1

0(Ω),

where a(u, v) =

• Ω

k∇u · ∇vdx and f , v =

• Ω

fvdx.

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Basis Functions

Let Th be a partition of Ω into finite elements K (coarse grid which can be resolved by a fine grid). Let xi be the interior nodes of Th and φ0

i be the nodal basis of the standard

finite element space Wh = span{φ0

i }.

Definition of multiscale basis functions φi: Lφi = 0 in K, φi = φ0

i

• n

∂K, ∀K ∈ Th, K ⊂ Si, (2) where Si = supp(φ0

i ).

Denote by Vh the finite element space spanned by φi Vh = span(φi). ((2) is solved on the fine grid)

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Basis Functions, contd.

Computational regions smaller than K are used if one can use smaller regions (Kloc) to characterize the local heterogeneities within the coarse-grid block (e.g. periodic heterogeneities). Such regions are called Representative Volume Elements (RVE). Definition of multiscale basis functions φi: Lφi = 0 in Kloc, φi = φ0

i

• n

∂Kloc, ∀Kloc ∈ Th, Kloc ⊂ Si;

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Source: [Y. Efendiev and T.Y. Hou, Multiscale Finite Element Methods: Theory and Applications, Springer, New York, 2009]

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Global Formulation

The representation of the fine-scale solution via multiscale basis functions allows reducing the dimension of the computation. When the approximation of the solution uh =

i uiφi is substituted into the fine-scale equation, the

resulting system is projected onto the coarse-dimensional space to find ui. The MsFEM reads: Find uh ∈ Vh such that:

• K
• K

k∇uh · ∇vhdx =

• Ω

fvhdx ∀vh ∈ Vh (3)

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Global Formulation, contd.

E.g. (3) is equivalent to Aunodal = b, (4) where A = (aij) with aij =

• K
• K

k∇φi · ∇φjdx, unodal = (u1, ..., ui, ...) are the nodal values of the coarse-scale solution and b = (bi) with bi =

• Ω

f φi.

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Outline

1

Motivation

2

Introduction to MsFEM

3

Analysis in 2D

4

Reducing boundary effects

5

Generalization of MsFEM

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Repetition

Consider (periodic case) Lǫu = f in Ω, u = 0

• n

∂Ω, (5) where Lǫu := −div (k(x/ǫ)∇u) , with kij(y), y = x/ǫ smooth periodic in y in a unit square Y (ǫ is a small parameter), f ∈ L2(Ω) and Ω a convex polygonal domain. Looking for expansion: u = u0(x, x/ǫ) + ǫu1(x, x/ǫ) + ...

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Repetition, contd.

u0 = u0(x) satisfies the homogenized equation: L0u0 := −div (k∗∇u0) = f in Ω, u0 = 0

• n

∂Ω, (6) where k∗

ij =

1 |Y |

• Y

kil(y)

• δlj − ∂χj

∂yl

• dy,

and χj is the periodic solution of divy

• k(y)∇yχj

= ∂ ∂yi kij(y) in Y ,

• Y

χj(y)dy = 0. In addition we have u1(x, y) = −χj(y)∂u0 ∂xj (x). (7)

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Repetition, contd.

Note that u0(x) + ǫu1(x, y) = u

• n

∂Ω, therefore we introduce a first order correction term Θǫ: LǫΘǫ = 0 in Ω, Θǫ = u1(x, y)

• n

∂Ω, (8) then u0(x) + ǫ(u1(x, y) − Θǫ) satisfies the boundary condition of u. Now we have the following homogenization result: Lemma 1 Let u0 ∈ H2(Ω) be the solution of (6), Θǫ ∈ H1(Ω) be the solution of (8) and u1 be given by (7). Then there exists a constant C independent of u0, ǫ and Ω such that u − u0 − ǫ(u1 − Θǫ)1,Ω ≤ Cǫu02,Ω. (9)

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM Convergence for h < ǫ

Multiscale method and standard linear finite element method are closely related. First we have Céa´s lemma Lemma 2 Let u and uh be the solutions of (1) and (3) respectively. Then u − uh1,Ω ≤ C inf

vh∈Vh

u − vh1,Ω, (10) and the regularity estimate |u|2,Ω ≤ C ǫ f 0,Ω (11) (1/ǫ is due to small-scale oscillations in u).

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM Convergence for h < ǫ

Lagrange interpolation operator: Πh : C(¯ Ω) → Wh Πhu(x) :=

J

• j=1

u(xj)φ0

j (x)

Interpolation operator defined through multiscale basis functions: Ih : C(¯ Ω) → Vh Ihu(x) :=

J

• j=1

u(xj)φj(x) From (2) we have Lǫ(Ihu) = 0 in K, Ihu = Πhu

• n

∂K, ∀K ∈ Th (12)

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM Convergence for h < ǫ

Lemma 3 Let u ∈ H2(Ω) be the solution of (1). Then there exist constants C1 > 0 and C2 > 0, independent of h and ǫ, such that u − Ihu0,Ω ≤ C1 h2 ǫ f 0,Ω, u − Ihu1,Ω ≤ C2 h ǫ f 0,Ω. (13) Theorem 4 Let u ∈ H2(Ω) and uh be the solutions of (1) and (3) respectively. Then there exists a constant C, independent of h and ǫ, such that u − uh1,Ω ≤ C h ǫ f 0,Ω. (14)

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM Convergence for h > ǫ

Convergence result uniform in ǫ as ǫ tends to zero (main feature of the MsFEM

• ver the traditional FEM).

Main result: Theorem 5 Let u ∈ H2(Ω) and uh be the solutions of (1) and (3) respectively. Then there exist constants C1, C2, independent of h and ǫ, such that u − uh1,Ω ≤ C1(h + ǫ)f 0,Ω + C2 ǫ h 1/2 . (15) Lemma 6 Let u ∈ H2(Ω) be the solution of (1) and uI = Ihu0 ∈ Vh. Then there exist constants C1, C2, independent of h and ǫ, such that u − uI1,Ω ≤ C1(h + ǫ)f 0,Ω + C2 ǫ h 1/2 , (16) where u0 ∈ H2(Ω) ∩ W 1,∞(Ω) is the solution of the homogenized equation (6).

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM Convergence for h > ǫ

From (12) LǫuI = 0 in K, uI = Πhu0

• n

∂K, ∀K ∈ Th. Thus (in K) uI = uI0 + ǫuI1 − ǫΘIǫ + ... where L0uI0 = 0 in K, uI0 = Πhu0

• n

∂K, uI1(x, y) = −χj(y)∂uI0 ∂xj (x), LǫΘIǫ = 0 in K, ΘIǫ = uI1(x, y)

• n

∂K. Note that uI0 = Πhu0 in K, (17) because Πhu0 is linear on K.

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM Convergence for h > ǫ

Lemma 7 There exists a constant C, independent of h and ǫ, such that uI − uI0 − ǫuI1 + ǫΘIǫ1,Ω ≤ Cǫf 0,Ω. (18) proof: By standard approximation theory uI02,K ≤ uI0 − u02,K + u02,K ≤ Cu02,K. Using (9) uI − uI0 − ǫuI1 + ǫΘIǫ1,K ≤ CǫuI02,K ≤ ˜ Cǫu02,K. Summing over K and using the regularity estimate u02,Ω ≤ Cf 0,Ω (19) finishes the proof.

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM Convergence for h > ǫ

Now we have u − uI1,Ω ≤u − (u0 + ǫ(u1 − Θǫ))1,Ω + (uI0 + ǫ(uI1 − ΘIǫ)) − uI1,Ω + (u0 + ǫ(u1 − Θǫ)) − (uI0 + ǫ(uI1 − ΘIǫ))1,Ω. Using (9), the last Lemma, the regularity estimate (19) and the triangle inequality we get: u − uI1,Ω ≤Cǫf 0,Ω + u0 − uI01,Ω + ǫ(u1 − uI1)1,Ω + ǫ(Θǫ − ΘIǫ)1,Ω. (20)

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM Convergence for h > ǫ

Lemma 8 We have u0 − uI01,Ω ≤ Chf 0,Ω, ǫ(u1 − uI1)1,Ω ≤ C(h + ǫ)f 0,Ω. (21) Lemma 9 We have ǫΘǫ1,Ω ≤ C√ǫ. (22) Lemma 10 We have ǫΘIǫ1,Ω ≤ C ǫ h 1/2 . (23)

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Outline

1

Motivation

2

Introduction to MsFEM

3

Analysis in 2D

4

Reducing boundary effects

5

Generalization of MsFEM

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Motivation

Boundary conditions for basis functions play a crucial role If local boundary conditions do not reflect the nature of the underlying heterogeneities, MsFEM can have large errors

ǫ h term in the convergence rate is large when h ≈ ǫ (resonance effect) ǫ h term comes from ǫΘIǫ

Remember: ǫΘIǫ term is due to the mismatch between the fine-scale solution and MsFEM solution along the boundaries of the coarse-grid block (MsFEM solution is linear there) Mismatch propagates into the interior of the coarse-grid block But boundary layer is thin (oscillations decay quickly as we move away from the boundaries)

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Oversampling

Sample in a domain with size larger than h and use only the interior sampled information to construct the basis functions Doing this, the influence of the boundary layer in the larger sample domain

• n the basis functions is reduced

Let φE

j satisfying

LǫφE

j = 0

in KE ⊃ K, φE

j = φ0 j

• n

∂KE, then we form φi by φi =

• j

cijφE

j

where cij are determined by φi(xj) = δij for nodal points xj. = ⇒ Nonconforming MsFEM

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Source: [Y. Efendiev and T.Y. Hou, Multiscale Finite Element Methods: Theory and Applications, Springer, New York, 2009]

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

Outline

1

Motivation

2

Introduction to MsFEM

3

Analysis in 2D

4

Reducing boundary effects

5

Generalization of MsFEM

Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM

General Framework of MsFEM

Lu = f (24) where L : X → Y is an operator. Multiscale basis functions are replaced by multiscale maps E MsFEM : Wh → Vh. For each vh ∈ Wh, vr,h = E MsFEMvh is defined as: Lmapvr,h = 0 in K. Lmap captures the small scales. Solving (24): Find ur,h ∈ Vh such that: Lglobalur,h, vr,h = f , vr,h, ∀vr,h ∈ Vh. Correct choices of Lmap and Lglobal are the essential part of MsFEM (can be different) and guarantee the convergence of the method.

Markus Kollmann Multiscale Finite Elements