Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic … · 2012. 1. 11. · 3....

Robust Multilevel Methods forAnisotropic Heterogeneous

Elliptic ProblemsSvetozar 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


1. Introduction

Consider the weak formulation of a given elliptic b.v.p. in t he form

a(u, v) = F(v), ∀v ∈ V,

and the related FEM problem

ah(uh, vh) = Fh(vh), ∀vh ∈ Vh.

Γ

ΓN

D

1

2

3

45

7

8

9

10

11

12

13

14

15

16

17

18

19 20

6

We are interested in the efficient solution of the resulting l arge-scale FEM linear

systems

Au = f .


For large-scale problems, the iterative methods have advan tages due to their

better/optimal computational complexity and storage requ irements.

The Conjugate Gradient Method is the best iterative solutio n framework for large

scale FEM systems.

The construction of robust Preconditioned Conjugate Gradi ent (PCG) solution

methods is addressed to some special properties of the stiff ness matrix A,

among which are that:

A is symmetric and positive definite (SPD);

A is large and even very large but sparse .


The number of iterations for the Conjugate Gradient (CG) met hod behaves as

n(CG)it = O

(√κ(A)

), where κ(A) =

λmax

λmin

.

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(PCG)it = O

(√κ(C−1A)

),

where the SPD matrix C is called preconditioner .

Then, the efficient preconditioning strategy is determined by the conditions:

κ(C−1A) << κ(A)

N (C−1v) << N (A−1

v)


The preconditioner is called optimal, if

κ(C−1A) = O(1)

N (C−1v) = O(N)

It is well known, that preconditioners based on various mult ilevel extensions of

two-level FEM lead to iterative methods which have an optima l order

computational complexity with respect to the size of the sys tem.

Unfortunately, the stiffness matrix A becomes additionally ill-conditioned when,

e.g., the coefficients of the elliptic operator become more a nisotropic or,

equivalently, when the mesh aspect ratio increases. The con dition number

κ(A) deteriorates also for various important parameter depende nt problems

like almost incompressible elasticity, Navier-Stokes equ ations, etc.

For such additionally ill-conditioned problems we need spe cially designed

robust preconditioners.


Examples

µ-FEM analysis of bone microstructure

The geometry of the solid phase of at a millimeter scale is obt ained by a

computer tomography (CT) image at a micron scale.

CT image of a trabecular bone specimen


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 35x1 10x110

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


Reservoir simulation

The model has been originally generated for use in the PUNQ pr oject

(http://www.nitg.tno.nl/punq/index.htm). The fine scale model size is

1.122 × 106 cells.

Figure: Porosity for the whole model

Figure: Lower fluvial part with clearly visible channelsRobust Multilevel Methods for Anisotropic Heterogeneous Elliptic Problems – p. 9/37

2. The AMLI method

Consider the SPD matrix A, written in the form:

A =

A11 A12

A21 A22

=

A11

A21 S

I1 A−1

11 A12

I2

,

where S states for the Schur complement

S = A22 − A21A−111 A12.

Let the additive ( CA) and multiplicative ( CM ) two-level preconditioners are

defined as follows:

CA =

A11

A22

, CM =

A11

A21 A22

I A−1

11 A12

I

.


The following estimates hold

κ(C−1

A A)≤ 1 + γ

1 − γ,

κ(C−1

M A)≤ 1

1 − γ2

where γ ∈ [0, 1) stands for the constant in the strengthened

Cauchy-Bunyakowski-Schwarz (CBS) inequality, correspon ding 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.


Let us consider the sequence of nested triangulations T1 ⊂ T2 ⊂ . . . ⊂ Tℓ,

corresponding FE spaces V1 ⊂ V2 ⊂ . . .Vℓ 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 Tk+1 into the subsets N (k+1) \ N (k) and N (k).

A(k+1) =

A

(k+1)11 A

(k+1)12

A(k+1)21 A

(k+1)22


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

C(k+1) =

A

(k+1)11 0

A(k+1)21 A(k)

I A

(k+1)−1

11 A(k+1)12

0 I

,

where the Schur complement approximation is stabilized by

A(k)−1

=[I − pβ

(C(k)−1

A(k))]

A(k)−1

.


The acceleration polynomial is explicitly defined by

pβ(t) =

1 + Tβ

(1 + α − 2t

1 − α

)

1 + Tβ

(1 + α

1 − α

) ,

were α ∈ (0, 1) is a properly chosen parameter, and Tβ stands for the

Chebyshev polynomial of degree β.

Theorem:

There exists α ∈ (0, 1), such that the AMLI preconditioner C = C(ℓ) has

optimal condition number κ(C−1A

)= O(1), and the total computational

complexity is O(N), if β satisfies the condition

4 > β >1√

1 − γ2.


For the model problem:

γ2 < 3/4 ⇒ β ∈ 2, 3κ(A

(k+1)11 ) = O(1) ⇒ NAMLI = O(N)

Construction of Robust Algebraic Multilevel Precondition ers :

Uniform estimates of the CBS constant with respect to anisot ropy and/or

possible small parameters.

Optimal order preconditioning (approximation) of the first pivot block

A(k+1)11 with respect to anisotropy and/or possible small parameter s.


3. Anisotropic problems

Consider the elliptic problem

−2∑

i,j=1

∂

∂xi

(aij

∂u

∂xj

)= f in Ω,

where Ω is a polyhedral domain, with proper boundary conditions on ∂Ω.

Its variational formulation is : seek u ∈ H1g (Ω) such that

pace2mm a(u, v) =

∫

Ω

fv for all v ∈ H10 (Ω), where

a(u, v) =

∫

Ω

2∑

i,j=1

aij

∂u

∂xi

∂v

∂xj

,

and where the function spaces H1g (Ω) and H1

0 (Ω) incorporate the Dirichlet

portion of the boundary conditions. Further, the matrix [aij ] is assumed to be

symmetric and positive definite (SPD).


Mesh and coefficient anisotropy

Mesh anisotropy

Coefficient anisotropy


The analysis for an arbitrary triangle (e) with coordinates (xi, yi), 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 tri angles, the

element bilinear form becomes

ae(u, v) = aee(u, v) =

∫

ee

[∂u

∂x,∂u

∂y

](x2 − x1)(y2 − y1)

(x3 − x1)(y3 − y1)

−1

a11 a12

a21 a22

(x2 − x1)(x3 − x1)

(y2 − y1)(y3 − y1)

−1[∂v

∂x,∂v

∂y

]T

where 0 < x, y < 1, i.e. it takes the form

aee(u, v) =

∫

ee

∑

i,j

aij

∂u

∂xi

∂v

∂xj

,

and where the coefficients aij depend on both triangle (e) and the coefficients

aij in the differential operator.


Corollary:

It suffices for the analysis of uniform bounds for the finite el ement method to

consider the reference triangle and arbitrary coefficients [aij ], or alternatively,

for the operator −∆ and an arbitrary triangle e.

Given a coarse triangle, it can be subdivided in four congrue nt triangles by

joining the mid-edge nodes, called the h−version .

Alternatively, we can use piecewise quadratic basis functi ons in the added

node points with support on the whole triangle, called the p−version .

(3)

(5) (4)

(1) (6) (2)

Support for the piece-wise linearbasis function ϕ6


Theorem:

For any finite element triangular mesh, where each element ha s been refined

into congruent elements, it holdsγ22 =

4

3γ21 ,

where γ1, γ2 are the CBS constants for the piecewise linear and piecewise

quadratic finite elements, respectively.

Theorem:

Maitre and Musy [1982], Axelsson [1999], Blaheta, Margenov , Neytcheva [2004]

The following estimate holds uniformly with respect to coef ficient and mesh

anisotropyγ21 <

3

4

for both, conforming and nonconforming linear finite elemen ts.

Corollary:γ2 < 1


Theorem: Axelsson, Padiy [1999], Axelsson, Margenov [2003]

The additive preconditioner [A] of A11 has an optimal order convergence rate

with a relative condition number uniformly bounded by

κ(B11

−1A11

)<

1

4(11 +

√105) ≈ 5.31,

which holds independent on shape and size of each element and on the

coefficients in the differential operator.

Connectivity pattern for the additive preconditioner [A].


Numerical tests

Table: Number of PCG/GCG iterations for C.-R. FEM elliptic p roblem

TP1 (a) TP1 (b) TP1 (c)

ℓ V W2 W3 V W2 W3 V W2 W3DA variant of the preconditioner

1 8 11 11 10 14 14 7 10 92 12 13 12 19 15 14 14 18 173 15 13 12 34 16 14 26 22 174 19 13 11 62 18 14 51 26 165 26 13 12 114 18 14 102 25 16

FR variant of the preconditioner1 8 8 8 9 9 9 6 7 62 11 8 8 15 10 9 10 7 63 14 8 8 20 10 9 16 7 64 16 8 7 25 9 9 21 6 65 20 8 8 29 9 9 19 6 6

The robustness with respect to mesh anisotropy is tested for : (a) θ1 = 90o, θ2 = θ3 = 45o; (b)

θ1 = 156o, θ2 = θ3 = 12o; (c) θ1 = 177o, θ2 = 2o, θ3 = 1o.


4. Heterogeneous problems

In the case of highly heterogeneous porous media the finite vo lume and

mixed finite element methods have proven to be accurate and lo cally mass

conservative. While applying the mixed FEM the continuity c ould be

enforced by Lagrange multipliers.

Arnold and Brezzi have demonstrated that after the eliminat ion of the

unknowns representing the pressure and the velocity from th e algebraic

system the resulting Schur system for the Lagrange multipli ers is

equivalent to a discretization by Galerkin method using Cro uzeix-Raviart

linear finite elements. Further, such a relationship betwee n the mixed and

nonconforming finite element methods has been studied for va rious finite

element spaces.

Galerkin methods based on Crouzeix-Raviart and Rannacher- Turek finite

elements has been also used in the construction of locking-free

approximations for parameter-dependent problems.


Rannacher-Turek nonconforming elements:

1 2

4

3

5

6

Figure 4. The reference element

For the variant MP (mid point), φi6i=1 are found by the interpolation condition

φi(bjΓ) = δij , where bj

Γ, j = 1, 6 are the centers of the faces of the cube e:

φ1 = (1 − 3x + 2x2 − y2 − z2)/6 φ2 = (1 + 3x + 2x2 − y2 − z2)/6

φ3 = (1 − x2 − 3y + 2y2 − z2)/6 φ4 = (1 − x2 + 3y + 2y2 − z2)/6

φ5 = (1 − x2 − y2 − 3z + 2z2)/6 φ6 = (1 − x2 − y2 + 3z + 2z2)/6


Alternatively, the variant MV (mean value) corresponds to t he 3D integral

mean-value interpolation condition |Γje|−1

∫Γj

e

φidΓje = δij , where Γj

e are the

faces of the reference element e:

φ1 = (2 − 6x + 6x2 − 3y2 − 3z2)/12 φ2 = (2 + 6x + 6x2 − 3y2 − 3z2)/12

φ3 = (2 − 3x2 − 6y + 6y2 − 3z2)/12 φ4 = (2 − 3x2 + 6y + 6y2 − 3z2)/12

φ5 = (2 − 3x2 − 3y2 − 6z + 6z2)/12 φ6 = (2 − 3x2 − 3y2 + 6z + 6z2)/12

Figure 5. Uniformly refined macroelement


Robust multilevel solvers

Kraus, Margenov, Robust Algebraic Multilevel Methods and A lgorithms, de Gruyter [2009]

The multiplicative AMLI preconditioner is defined as follow s,

H(0) = A(0), and for k = 1, . . . , ℓ,

H(k) =

B

(k)11 0

A(k)21

[S(k)

]

I

(k)1 B(k)−1

11 A(k)12

0 I(k)2

,

where[S(k)

]denotes that certain stabilization technique is performed .

One particular stabilization is via a matrix polynomial, na mely,

[S(k)

]≡ S(k) = A(k−1)

[I − P k

νk(H(k−1)−1

A(k−1))]−1

.

In this case, the following optimality condition for νk holds true:

1√1 − γ2

< νk < ρ

where γ < 1 stands for the constant in the strengthened CBS inequality.


Table 1: Multilevel behavior of γ2 for FR algorithm: 3D case

variant ℓ ℓ − 1 ℓ − 2 ℓ − 3 ℓ − 4 ℓ − 5

MP 0.38095 0.39061 0.39211 0.39234 0.39237 0.39238

MV 0.5 0.4 0.39344 0.39253 0.39240 0.39238

MV

MP

0.39238

0.38095

0.5

l l−5l−4l−3l−2l−1

Figure 6. Multilevel behavior of γ2 for FR algorithm: 3D case


Numerical tests

Georgiev, Kraus, Margenov, Computing [2008]

I. Jump in coefficients aligned with the coarse mesh

The considered second order elliptic problem has a jumping c oefficient a(e)

which is elementwise constant.

Table: Linear AMLI W-cycle: number of PCG iterations: 3D tes ts

MP h−1 8 16 32 64 128DA ε = 1 9 10 10 10 10

ε = 10−3 9 10 10 10 10FR ε = 1 8 9 9 9 9

ε = 10−3 8 9 9 9 9

MV h−1 8 16 32 64 128DA ε = 1 12 15 15 16 16

ε = 10−3 12 15 16 16 16FR ε = 1 10 12 12 12 12

ε = 10−3 10 12 12 12 12


II. Random distribution of jump in coefficients

For the case of “binary material”, the coefficient a(e) is ini tialized randomly,

taking either of the values 1 or ε, where 1 occurs with a probability p. In the last

case, the coefficient on each element is a uniformly distribu ted random number

in (0,1).

Table 3: Non-linear AMLI W-cycle: GCG iterations for proble m with random coefficients: 3D tests

p = 1/2FR-MV h−1 8 16 32 64 128

ε = 10−1 9 9 9 9 9ε = 10−2 21 22 22 21 21ε = 10−3 42 59 58 56 54

p = 1/10FR-MV h−1 8 16 32 64 128

ε = 10−1 9 9 9 9 9ε = 10−2 17 22 22 22 22ε = 10−3 29 60 55 50 50

Random coefficient:

FR-MV h−1 8 16 32 64 128α(e) ∈ (0, 1) 22 39 43 34 35


III. Numerical tests: Towards µFEM analysis of bone structures

The bone specimen is considered as a composite linear elasti c material of solid

and ”fluid” phases. The related coefficients are given by the b ulk moduli

ks = 14 GPa and kf = 2.3 GPa, and the Poisson ratios νs = 0.325 and

νf = 0.5. In our tests we vary νf ∈ 0.4, 0.45.0.49.

Table 4: Convergence results for nonlinear AMLI W-cycle

# voxels νf = 0.4 νf = 0.45 νf = 0.49163 14 19 46323 14 20 50643 14 21 53


5. HC-HF problems

Kraus, Margenov [2009], Georgiev, Kraus, Margenov [2010,2 011], Kraus [2011]

Robust two-level results for the case of high-contrast and high-frequency

(HC-HF) elliptic problems are presented in this section. The assump tion is that

the strong coefficient jumps can be resolved on the finest mesh only!

Let us consider the symmetric interior penalty discontinuo us Galerkin (IP-DG)

finite element method: Find uh ∈ V such that

Ah(uh, v) = (f, v), ∀ v ∈ V,

where

Ah(uh, v) ≡ (a∇uh,∇v)h + α⟨h−1F

[[uh]], [[v]]⟩F0∪FD

−〈a∇uh, [[v]]〉F0∪FD

−〈[[uh]], a∇v〉F0∪FD

.


HC-HF problems - DG

In the setting of IP-DG discretizations the construction of a face-based

hierarchical splitting is proposed. The idea originates in the observation that the

related global stiffness matrix can also be assembled from l ocal matrices

associated with the element faces.

Overlap of macro superelements and coarsening

Theorem: Let us consider a hierarchical basis two-level transformat ion based on

interpolation with limiting weights. If α ≥ α0 = 25 then the estimate γ2G ≤ 0.75

is uniform with respect to arbitrary jumps of the coefficient s a(e) ∈ (0, 1].

Similar results were recently obtained for the case of bilin ear conforming

finite elements.


Let us consider the IP-DG approximation in the form:

Find uh ∈ Vh such that for all vh ∈ Vh the following relation holds:

A(uh, vh) = L(vh),

A(uh, vh) = (a∇huh,∇hvh)T −⟨a∇huhβe

, [[vh]]⟩

F

+θ⟨[[uh]], a∇hvhβe

⟩

F

+ αeke

⟨h−1F

[[uh]], [[vh]]⟩F

,

where βe = a−

a++a−, ke := 2a+a−

a++a−, and auβe

= κeu.

The related IP-DG blinear form is continuous and coercive with constants

independent of the mesh size h and coefficient a.

M. Dryja [2003], B. Ayuso, M. Holst, Y. Zhu and L. Zikatanov [2 010]


Numerical study of CBS constant

The conditions to be met for optimal order multilevel method s, which are based

on the additive and the multiplicative two-level precondit ioners, are

√1 + γ

1 − γ< β < ,

1√1 − γ2

< β < .

Since the reduction factor of the number of DOF in our case is = 4, we can

afford up two third degree stabilization ( β = 3) in this setting.

For the problem with jumps in the coefficient ( a(x) = a(x)I), γ2 < 0.75.

The multiplicative method will be of optimal order for β = 2.

For the problem with anisotropic coefficient matrix γ2 < 0.8. The additive

and the multiplicative methods will be of optimal order for β = 3.

I. Georgiev, J. Kraus, S. Margenov [2011]


HC-HF problems - DD

Table: Number of iterations and estimated condition number for the PCG: ǫ = 10−10, H=1/10,

h=1/100, a = diag( η , 1), see the left figure; MS - multiscale coarse space, EMF - en ergy minimizing

coarse space, and LSM - local spectral multiscale coarse spa ce.

η LIN MS EMF LSM (bilin. χi) LSM (MS χi)

103 113(1.48e+2) 122(1.51e+2) 115(1.81e+2) 53(23.21) 55(26.9)104 257(1.35e+3) 258(1.28e+3) 231(9.70e+2) 41(53.63) 28(5.82)105 435(1.34e+4) 483(1.26e+4) 416(9.64e+3) 28(5.642) 29(6.02)106 627(1.34e+5) 709(1.27e+5) 599(9.63e+4) 30(5.753) 29(6.04)

Dim 81=0.79% 81=0.79% 81=0.79% 732=7.19% 497=4.87%

Efendiev, Galvis, Lazarov, Margenov, Ren [2011]


Discussions on coarse space dimension reduction:

The strongly anisotropic channels cause a substantial incr ease of the size

of the coarse space and the complexity of the method.

To avoid this, we can replace the coarse solve RT0 A−1

0 R0 by

RT0 A−1

0 R0 + RTanA−1

an Ran.

The matrix A0 is a small dimensional coarse matrix. The matrix Aan is

acting on the fine-mesh DOFs restricted to the high-anisotro py channels,

locally constructed by preserving the strongest off-diago nal entries.


TWO RELATED OPEN PROBLEMS:

Robust preconditioning of higher order FEM anisotropic pro blems.

High-contrast and high-frequency problems: from two-leve l to multilevel.

THANK YOU !


Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic … · 2012. 1. 11. · 3....

Documents

Transcript of Robust Multilevel Methods for Anisotropic Heterogeneous Elliptic … · 2012. 1. 11. · 3....