On a two-level domain decomposition preconditioner for 3D flows in - - PowerPoint PPT Presentation

On a two-level domain decomposition preconditioner for 3D flows in anisotropic highly heterogeneous porous media

Iryna Rybak

Institute of Mathematics, Belarus

Oleg Iliev

Fraunhofer ITWM, Germany

Richard Ewing, Raytcho Lazarov

Texas A&M University, USA

Strobl, July 3-7, 2006 17th International Conference on Domain Decomposition Methods

(work in progress)

Contents

• Motivation
• Statement of the problem, applications
• Finite volume discretization
• Two-level DD

– Smoother – Restriction – Coarse grid operator – Prolongation

• Numerical results
• Conclusions
• Future work

Statement of the problem

p K v ∇ ⋅ − =

Continuity equation + Darcy´s law

f v = ⋅ ∇

( )

f p K = ∇ ⋅ ⋅ ∇ −

Pressure equation >           =

zz yz xz yz yy xy xz xy xx

k k k k k k k k k K Permeability tensor Boundary conditions Dirichlet p p = Neumann v n v = ⋅

Applications

p K v f v ∇ ⋅ − = = ⋅ ∇ ,

Saturated flow in anisotropic heterogeneous porous media Two-phase flow in heterogeneous porous media

( )

, , p K S v v

w

∇ ⋅ − = = ⋅ ∇ λ ) ( = ∇ ⋅ + ∂ ∂

w w w

S f v t S

Fine grid – isotropic permeability tensor Coarse grid – full tensor (effective permeability)

Finite volume discretization

FV scheme

f h v v h v v h v v

z in z

• ut

z y in y

• ut

y x in x

• ut

x

= − + − + −

( )

z y x

v v v v , , =

∫∫∫ ∫∫∫

= ∇

V V

f v f v = ∇

in x

v

in z

v

in y

v

• ut

z

v

• ut

y

v

• ut

x

v z y x

Velocity vector in 3D Continuity equation Finite volume

Cell-centered grid

Finite volume discretization

Pressure is continuous at 12 points Pressure is given at 8 points Velocities are continuous along 12 interfaces Polynomials:

8 , 1 , = + + + = i d z c y b x a p

i i i i

12 eqns 12 eqns 8 eqns 32 equations 32 unknowns

Multipoint Flux Approximation

Ware A.F., Parrott A.K., and Rogers C. shifted control volume

N NE

p

NE

p

N

p

P

p

E P E

FV discretization (validation)

Permeability tensor Exact solution

1 2 1

, 1 25 . 25 . 25 . 1 5 . 25 . 5 . 1 K K K α =           =

)) ( cos( ) ( ) ( ) (

2 2 2

z y x z z y y x x p

i i i

+ + − − − = π

K1 K2 K2 K1

xi yi zi 1 1 1

• jump discontinuity

α

FV discretization (validation)

Convergence rate

Grid α α α α = 10-2 α α α α = 10-5 ||p – ph||L2 ||p – ph||C ||p – ph||L2 ||p – ph||C 4 x 4 x 4 0.1709 0.2174 0.1711 0.2174 8 x 8 x 8 0.0395 0.0284 0.0395 0.0284 16 x 16 x 16 0.0087 0.0075 0.0087 0.0075 32 x 32 x 32 0.0020 0.0018 0.0020 0.0018

2

) (h O

doesn‘t depend on jump discontinuity

Two-grid method

Fine grid Coarse grid Extended subdomain

One sweep of TGM

• Smooth with DD (2-3 iterations)
• Calculate the residual
• Restrict the residual in each subdomain
• Discretize and solve on coarse grid
• Prolong coarse grid correction by solving

local problems in shifted subdomains

• Correct the solution
• Post smooth with DD

1 +

→

n n

x x

b Ax =

n h n h

x A b r ~ − =

∑

=

=

m i i h H

r m r

1

1

H H H

r c A =

n

x ~

h

c

n n n

c x x + =

+1

~ ~

1 + n

x

Neuss N., Jäger W., Wittum G.

DD smoothing

Additive Schwarz

With overlapping

n n

x x ~ →

Without overlapping

∑

=

=

m i i h H

r m r

1

1

H

r

• residual on a coarse grid

i h

r

• residual on a fine grid

i

z y x

m m m m =

• number of fine grid blocks in a coarse one

Restriction

Multiplicative Schwarz

Coarse grid operator

Coarse scale Darcy´s law

p K v

eff

∇ ⋅ − =

Local flow problems

. 3 , 1 , = ∇ ⋅ − = i p K v

i eff i

1-0 Dirichlet + Neumann (v = 0) b.c. 1-0 Dirichlet + piecewise linear b.c. RHS = 0 Boundary conditions and RHS for local flow problem

f

• volume average

Prolongation

1D by TDMA at the edges 2D problem for the planes BCs for 2D problem 3D problem inside the cell BCs for 3D problem

=       ∂ ∂ ∂ ∂ x p k x

xx

=         ∂ ∂ ∂ ∂ +       ∂ ∂ ∂ ∂ y p k y x p k x

yy xx

( )

f p K = ∇ ⋅ ⋅ ∇ −

• coarse grid

solution

Numerical results

Convergence of TGM depends on overlapping and number of subdomains

Cubic inclusion L-shaped inclusion Random inclusion

Periodic Non-periodic

          = 1 1 1

1

K           = 10000 10000 10000

2

K

1

K

1

K

1

K

2

K

2

K

2

K

Permeability tensor

One- and two-level DD

Coarse grid 4x4x4 Fine grid 4x4x4 8x8x8 16x16x16 32x32x32 DD iter. -- 95 162 247 TGM iter.

• 4 5

7 Coarse grid 8x8x8 Fine grid 4x4x4 8x8x8 16x16x16 DD iter. 158 266 TGM iter. 3 4 5

64 inclusions, acc=1e-4

• vrlp=2: 1.5h

          = 1 1 1

1

K

1

K

2

K

          = 10000 10000 10000

2

K

DD smoothing (overlapping)

          = 1 1 1

1

K

1

K

2

K

          = 10000 10000 10000

2

K

Coarse grid 8x8x8, fine grid 8x8x8

• vrlp = 1

TGM iter = 13 Coarse grid 8x8x8, fine grid 16x16x16

• vrlp = 2

TGM iter = 7 Acc = 1E-5 Coarse grid 8x8x8, fine grid 16x16x16

• vrlp = 1

TGM iter = 23

• vrlp = 2

TGM iter = 7

• vrlp = 3

TGM iter = 6 2 presmooth. 2 postsmooth.

DD pre- and post-smoothing

1

K

2

K

8x8x8 coarse blocks, 8x8x8 fine blocks Accuracy for TGM = 1E-5 DD smoother: 2-pre, 2-post: 13 TGM iter 0-pre, 2-post: 47 TGM iter 0-pre, 4-post: 24 TGM iter

          = 1 1 1

1

K           = 10000 10000 10000

2

K

DD smoothing

          = 1 1 1

1

K

1

K

2

K

          = 10000 10000 10000

2

K

Additive Schwarz TGM iter = 13 Coarse grid 8x8x8, fine grid 8x8x8

• vrlp = 1

Acc = 1E-5 2 presmooth. 2 postsmooth. Multiplicative Schwarz TGM iter = 7

TGM for different geometries

Periodic cubic inclusion

TGM acc = 1E-5

Coarse grid 8x8x8, fine grid 8x8x8 TGM iter = 13

1

K

2

K

Periodic L-shaped inclusion

Coarse grid 8x8x8, fine grid 12x12x12 TGM iter = 23

Random inclusion

Coarse grid 8x8x8, fine grid 8x8x8 TGM iter = 12

E K =

1

E K 10000

2 =

1

K

1

K

2

K

2

K

TGM for different geometries

TGM acc = 1E-5

Small inclusions

Coarse grid 8x8x8, fine grid 8x8x8

1

K

2

K

TGM iter = 11

1

K

2

K

TGM iter = 11

Larger inclusions

1

K

2

K

Large inclusions

TGM iter = 11 inc = 1x1x1 inc = 2x2x2 inc = 4x4x4

Oversampling

1

K

2

K

          × × × − × − =

− − − −

. 7126 10 1 . 8 10 2 . 1 10 1 . 8 1 . 6569 6 . 192 10 2 . 1 6 . 192 1 . 6569

6 5 6 5 *

K

          = 1 1 1

1

K

          = 10000 10000 10000

2

K

1

K

2

K

          × × × − × − =

− − − −

5 . 6957 10 1 . 8 10 5 . 1 10 1 . 8 6 . 6411 . 256 10 5 . 1 . 256 6 . 6411

6 4 6 4 *

K

TGM iter = 7 TGM iter = 7

TGM acc = 1E-4

3D upscaling

E K E K α = =

2 1

, Effective permeability

5 4 * 5 4 4 4

1.1232 8.66 10 2.12 10 8.66 10 1.1218 4.16 10 2.12 10 4.16 10 1.1219 K

− − − − − −

  ⋅ − ⋅   = ⋅ ⋅     − ⋅ ⋅  

contrast 1:1000 Fine grid permeability tensor Effective permeability

*

44.55 0.14 0.05 0.14 43.30 0.31 0.05 0.31 43.90 K − −     = − −     − −  

contrast 1:3 acc = 10-E5

Foam

Conclusions

• Finite volume discretization for the case of highly varying

anisotropic permeability tensor

• Additive and multiplicative Schwarz as a smoother withing

two-level preconditioner

• Coarse scale operator obtained from numerical upscaling
• Influence of the overlapping, smoother, number of

subdomains on the convergence of TGM

• Applicability for non-periodic media

Future work

• Two-level DD as a preconditioner for Krylov

subspace methods

• Study the influence of cell-problem formulation
• n the convergence of the preconditioned CG
• Develop further approaches for two-phase

flows

• Theoretical analysis