Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems Svetozar Margenov Institute for Information and Communication Technologies, Bulgarian Academy of Sciences, Acad. G. Bonchev Str. Bl. 25-A, 1113 Sofia, Bulgaria Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems – p. 1/37
CONTENTS 1. Introduction 2. The AMLI method 3. Anisotropic problems 4. Heterogeneous problems 5. HC-HF problems Collaboration: Ivan Georgiev, Johannes Kraus Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems – p. 2/37
1. Introduction Consider the weak formulation of a given elliptic b.v.p. in the form a ( u, v ) = F ( v ) , ∀ v ∈ V , and the related FEM problem a h ( u h , v h ) = F h ( v h ) , ∀ v h ∈ V h . 16 Γ 12 N 15 20 19 3 11 7 2 10 18 14 6 1 9 17 13 5 4 8 Γ D We are interested in the efficient solution of the resulting large-scale FEM linear systems A u = f . Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems – p. 3/37
For large-scale problems, the iterative methods have advantages due to their better/optimal computational complexity and storage requirements. The Conjugate Gradient Method is the best iterative solution framework for large scale FEM systems. The construction of robust Preconditioned Conjugate Gradient (PCG) solution methods is addressed to some special properties of the stiffness matrix A , among which are that: A is symmetric and positive definite (SPD); A is large and even very large but sparse. Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems – p. 4/37
The number of iterations for the Conjugate Gradient (CG) method behaves as �� � , where κ ( A ) = λ max n ( CG ) = O κ ( A ) . it λ min Unfortunately, κ ( A ) = O ( h − 2 ) , i.e., in 2D case κ ( A ) = O ( N ) . To accelerate the convergence rate, the Preconditioned CG (PCG) method is used, for which �� � n ( P CG ) κ ( C − 1 A ) = O , it where the SPD matrix C is called preconditioner. Then, the efficient preconditioning strategy is determined by the conditions: κ ( C − 1 A ) << κ ( A ) N ( C − 1 v ) << N ( A − 1 v ) Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems – p. 5/37
The preconditioner is called optimal, if κ ( C − 1 A ) = O (1) N ( C − 1 v ) = O ( N ) It is well known, that preconditioners based on various multilevel extensions of two-level FEM lead to iterative methods which have an optimal order computational complexity with respect to the size of the system. Unfortunately, the stiffness matrix A becomes additionally ill-conditioned when, e.g., the coefficients of the elliptic operator become more anisotropic or, equivalently, when the mesh aspect ratio increases. The condition number κ ( A ) deteriorates also for various important parameter dependent problems like almost incompressible elasticity, Navier-Stokes equations, etc. For such additionally ill-conditioned problems we need specially designed robust preconditioners. Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems – p. 6/37
Examples µ -FEM analysis of bone microstructure The geometry of the solid phase of at a millimeter scale is obtained by a computer tomography (CT) image at a micron scale. CT image of a trabecular bone specimen Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems – p. 7/37
µ -FEM analysis of coal-polyurethane geocomposite The voxel data represent a coal-polyurethane material. The domain is cubic 75x75x75mm, and is non-uniform in all directions 35x110x110 voxels. The mechanical properties used were: Coal = 0.25, E = 4000MPa; Polyurethane [0.1, 0.25], E [200, 2100]MPa. Figure: CT image of a coal-polyurethane geocomposite brick Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems – p. 8/37
Reservoir simulation The model has been originally generated for use in the PUNQ project (http://www.nitg.tno.nl/punq/index.htm). The fine scale model size is 1 . 122 × 10 6 cells. Figure: Porosity for the whole model Figure: Lower fluvial part with clearly visible channels Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems – p. 9/37
2. The AMLI method Consider the SPD matrix A , written in the form: A − 1 A 11 A 12 A 11 I 1 11 A 12 = , A = A 21 A 22 A 21 S I 2 where S states for the Schur complement S = A 22 − A 21 A − 1 11 A 12 . Let the additive ( C A ) and multiplicative ( C M ) two-level preconditioners are defined as follows: A − 1 A 11 A 11 I 11 A 12 , . C A = C M = A 22 A 21 A 22 I Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems – p. 10/37
The following estimates hold � � ≤ 1 + γ C − 1 κ A A 1 − γ , � � 1 C − 1 κ M A ≤ 1 − γ 2 where γ ∈ [0 , 1) stands for the constant in the strengthened Cauchy-Bunyakowski-Schwarz (CBS) inequality, corresponding to the block splitting of A . The presented condition number estimates clearly state the importance of such splittings where the CBS constant γ is uniformly bounded. Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems – p. 11/37
Let us consider the sequence of nested triangulations T 1 ⊂ T 2 ⊂ . . . ⊂ T ℓ , corresponding FE spaces V 1 ⊂ V 2 ⊂ . . . V ℓ and related stiffness matrices A (1) , A (2) , . . . , A ( ℓ ) . The goal is to solve the finest discretization FEM system A ( ℓ ) u ( ℓ ) = f ( ℓ ) . Consider the 2 × 2 block presentation of A ( k +1) corresponding to the splitting of the nodes N ( k +1) from T k +1 into the subsets N ( k +1) \ N ( k ) and N ( k ) . A ( k +1) A ( k +1) A ( k +1) = 11 12 A ( k +1) A ( k +1) 21 22 Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems – p. 12/37
Let us consider the recursive multilevel generalization of the multiplicative two-level method introduced originally by Axelsson and Vassilevski (1989). C (1) = A (1) ; for k = 1 , 2 , . . . , ℓ − 1 A ( k +1) − 1 A ( k +1) A ( k +1) 0 I C ( k +1) = 11 11 12 , A ( k +1) ˜ A ( k ) 0 I 21 where the Schur complement approximation is stabilized by � � C ( k ) − 1 A ( k ) �� A ( k ) − 1 = A ( k ) − 1 . ˜ I − p β Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems – p. 13/37
The acceleration polynomial is explicitly defined by � 1 + α − 2 t � 1 + T β 1 − α � 1 + α � p β ( t ) = , 1 + T β 1 − α were α ∈ (0 , 1) is a properly chosen parameter, and T β stands for the Chebyshev polynomial of degree β . Theorem: There exists α ∈ (0 , 1) , such that the AMLI preconditioner C = C ( ℓ ) has � � C − 1 A optimal condition number κ = O (1) , and the total computational complexity is O ( N ) , if β satisfies the condition 1 4 > β > � 1 − γ 2 . Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems – p. 14/37
For the model problem: γ 2 < 3 / 4 ⇒ β ∈ { 2 , 3 } κ ( A ( k +1) ) = O (1) ⇒ N AMLI = O ( N ) 11 Construction of Robust Algebraic Multilevel Preconditioners: Uniform estimates of the CBS constant with respect to anisotropy and/or possible small parameters. Optimal order preconditioning (approximation) of the first pivot block A ( k +1) with respect to anisotropy and/or possible small parameters. 11 Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems – p. 15/37
3. Anisotropic problems Consider the elliptic problem � � 2 � ∂ ∂u − a ij = f Ω , in ∂x i ∂x j i,j =1 where Ω is a polyhedral domain, with proper boundary conditions on ∂ Ω . Its variational formulation is : seek u ∈ H 1 g (Ω) such that � for all v ∈ H 1 pace2mm a ( u, v ) = fv 0 (Ω) , where Ω � 2 � ∂u ∂v a ( u, v ) = a ij , ∂x i ∂x j Ω i,j =1 and where the function spaces H 1 g (Ω) and H 1 0 (Ω) incorporate the Dirichlet portion of the boundary conditions. Further, the matrix [ a ij ] is assumed to be symmetric and positive definite (SPD). Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems – p. 16/37
Mesh and coefficient anisotropy Mesh anisotropy Coefficient anisotropy Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems – p. 17/37
The analysis for an arbitrary triangle ( e ) with coordinates ( x i , y i ) , i = 1 , 2 , 3 can be done on the reference triangle ( � e ) , with coordinates (0 , 0) (1 , 0) and (0 , 1) . Transforming the finite element function between these triangles, the element bilinear form becomes a e ( u, v ) = a e e ( � u, � v ) = − 1 − 1 � ∂ � � � ∂ � � T � ( x 2 − x 1 )( y 2 − y 1 ) ( x 2 − x 1 )( x 3 − x 1 ) a 11 a 12 x, ∂ � u u x, ∂ � v v ∂ � ∂ � y ∂ � ∂ � y ( x 3 − x 1 )( y 3 − y 1 ) ( y 2 − y 1 )( y 3 − y 1 ) a 21 a 22 e e where 0 < � x, � y < 1 , i.e. it takes the form � � ∂ � u ∂ � v a e e ( � u, � v ) = a ij , � ∂ � x i ∂ � x j e e i,j and where the coefficients � a ij depend on both triangle ( e ) and the coefficients a ij in the differential operator. Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems – p. 18/37
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