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 Motivation 1 Introduction to MsFEM 2 Analysis in 2D 3 Reducing boundary effects 4 Generalization of MsFEM 5 Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM Outline Motivation 1 Introduction to MsFEM 2 Analysis in 2D 3 Reducing boundary effects 4 Generalization of MsFEM 5 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 Motivation 1 Introduction to MsFEM 2 Analysis in 2D 3 Reducing boundary effects 4 Generalization of MsFEM 5 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 , (1) u = 0 on ∂ Ω , where Lu := − div ( k ( x ) ∇ u ) . Ω ... domain in R d ( d = 2, 3) k ( x ) ... heterogeneous field varying over multiple scales Additionally assume: k ( x ) = ( k ij ( x )) is symmetric α | ξ | 2 ≤ k ij ξ i ξ j ≤ β | ξ | 2 ∀ ξ ∈ R d (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 ∈ H 1 0 (Ω) such that ∀ v ∈ H 1 a ( u , v ) = � f , v � , 0 (Ω) , where � � a ( u , v ) = k ∇ u · ∇ vdx and � f , v � = fvdx . Ω Ω Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM Basis Functions Let T h be a partition of Ω into finite elements K (coarse grid which can be resolved by a fine grid). Let x i be the interior nodes of T h and φ 0 i be the nodal basis of the standard finite element space W h = span { φ 0 i } . Definition of multiscale basis functions φ i : φ i = φ 0 L φ i = 0 in K , on ∂ K , ∀ K ∈ T h , K ⊂ S i , (2) i where S i = supp ( φ 0 i ) . Denote by V h the finite element space spanned by φ i V h = span ( φ i ) . ((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 ( K loc ) 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 φ i : φ i = φ 0 L φ i = 0 in K loc , on ∂ K loc , ∀ K loc ∈ T h , K loc ⊂ S i ; i 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 u h = � i u i φ i is substituted into the fine-scale equation, the resulting system is projected onto the coarse-dimensional space to find u i . The MsFEM reads: Find u h ∈ V h such that: � � � k ∇ u h · ∇ v h dx = fv h dx ∀ v h ∈ V h (3) K Ω K 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 Au nodal = b , (4) where A = ( a ij ) with � � a ij = k ∇ φ i · ∇ φ j dx , K K u nodal = ( u 1 , ..., u i , ... ) are the nodal values of the coarse-scale solution and b = ( b i ) with � b i = f φ i . Ω Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM Outline Motivation 1 Introduction to MsFEM 2 Analysis in 2D 3 Reducing boundary effects 4 Generalization of MsFEM 5 Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM Repetition Consider (periodic case) L ǫ u = f in Ω , u = 0 on ∂ Ω , (5) where L ǫ u := − div ( k ( x /ǫ ) ∇ u ) , with k ij ( y ) , y = x /ǫ smooth periodic in y in a unit square Y ( ǫ is a small parameter), f ∈ L 2 (Ω) and Ω a convex polygonal domain. Looking for expansion: u = u 0 ( x , x /ǫ ) + ǫ u 1 ( x , x /ǫ ) + ... Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM Repetition, contd. u 0 = u 0 ( x ) satisfies the homogenized equation: L 0 u 0 := − div ( k ∗ ∇ u 0 ) = f in Ω , u 0 = 0 on ∂ Ω , (6) where δ lj − ∂χ j � � 1 � k ∗ ij = k il ( y ) dy , | Y | ∂ y l Y and χ j is the periodic solution of ∂ � � k ( y ) ∇ y χ j � χ j ( y ) dy = 0 . div y = ∂ y i k ij ( y ) in Y , Y In addition we have u 1 ( x , y ) = − χ j ( y ) ∂ u 0 ∂ x j ( x ) . (7) Markus Kollmann Multiscale Finite Elements

Motivation Introduction to MsFEM Analysis in 2D Reducing boundary effects Generalization of MsFEM Repetition, contd. Note that u 0 ( x ) + ǫ u 1 ( x , y ) � = u on ∂ Ω , therefore we introduce a first order correction term Θ ǫ : L ǫ Θ ǫ = 0 in Ω , Θ ǫ = u 1 ( x , y ) on ∂ Ω , (8) then u 0 ( x ) + ǫ ( u 1 ( x , y ) − Θ ǫ ) satisfies the boundary condition of u . Now we have the following homogenization result: Lemma 1 Let u 0 ∈ H 2 (Ω) be the solution of ( 6 ), Θ ǫ ∈ H 1 (Ω) be the solution of ( 8 ) and u 1 be given by ( 7 ). Then there exists a constant C independent of u 0 , ǫ and Ω such that � u − u 0 − ǫ ( u 1 − Θ ǫ ) � 1 , Ω ≤ C ǫ � u 0 � 2 , 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 u h be the solutions of ( 1 ) and ( 3 ) respectively. Then � u − u h � 1 , Ω ≤ C inf � u − v h � 1 , Ω , (10) v h ∈ V h 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: Π h : C (¯ Ω) → W h J � u ( x j ) φ 0 Π h u ( x ) := j ( x ) j = 1 Interpolation operator defined through multiscale basis functions: I h : C (¯ Ω) → V h J � I h u ( x ) := u ( x j ) φ j ( x ) j = 1 From (2) we have L ǫ ( I h u ) = 0 in K , I h u = Π h u on ∂ K , ∀ K ∈ T h (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 ∈ H 2 (Ω) be the solution of ( 1 ). Then there exist constants C 1 > 0 and C 2 > 0 , independent of h and ǫ , such that � u − I h u � 0 , Ω ≤ C 1 h 2 ǫ � f � 0 , Ω , (13) � u − I h u � 1 , Ω ≤ C 2 h ǫ � f � 0 , Ω . Theorem 4 Let u ∈ H 2 (Ω) and u h be the solutions of ( 1 ) and ( 3 ) respectively. Then there exists a constant C, independent of h and ǫ , such that � u − u h � 1 , Ω ≤ C h ǫ � f � 0 , Ω . (14) Markus Kollmann Multiscale Finite Elements