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

17th International Conference on Domain Decomposition Methods On a two-level domain decomposition preconditioner for 3D flows in anisotropic highly heterogeneous porous media (work in progress) 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

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 Continuity equation + Darcy´s law ∇ ⋅ v = = − ⋅ ∇ f v K p Pressure equation ( ) − ∇ ⋅ ⋅ ∇ = K p f Permeability tensor Boundary conditions   k k k   xx xy xz   = > K k k k 0 xy yy yz Dirichlet   Neumann k k k   xz yz zz p = p ⋅ = v n v 0 0

Applications Saturated flow in anisotropic heterogeneous porous media ∇ ⋅ = = − ⋅ ∇ v f , v K p Two-phase flow in heterogeneous porous media ( ) ∇ ⋅ = = − λ ⋅ ∇ v 0 , v S K p , w ∂ S w + ⋅ ∇ = v f ( S ) 0 w w ∂ t Fine grid – isotropic permeability tensor Coarse grid – full tensor (effective permeability)

Finite volume discretization Finite volume Continuity equation out out v v y z ∫∫∫ ∫∫∫ ∇ = v f ∇ v = f V V in v Velocity vector in 3D out x v x ( ) = v v , v , v x y z in v z in v FV scheme y Cell-centered grid out in − out in out in v v − − y v v v v y y x x z z + + = f z h h h x y z x

Finite volume discretization Multipoint Flux Approximation N NE Pressure is given 8 eqns p at 8 points p N NE Pressure is continuous at 12 points 12 eqns P E Velocities are continuous along 12 interfaces 12 eqns p p E P shifted control volume 32 equations Polynomials: i i i i = + + + = p a x b y c z d , i 1 , 8 32 unknowns Ware A.F., Parrott A.K., and Rogers C.

FV discretization (validation) Permeability tensor 1   1 0 . 5 0 . 25   K 2 K 1   = = α K 0 . 5 1 0 . 25 , K K 1 2 1   y i   0 . 25 0 . 25 1 1 K 1 K 2 z i α - jump discontinuity 0 1 x i Exact solution 2 2 2 = − − − π + + p ( x x ) ( y y ) ( z z ) cos( ( x y z )) i i i

FV discretization (validation) α α α α = 10 -2 α α = 10 -5 α α Grid ||p – p h || L2 ||p – p h || C ||p – p h || L2 ||p – p h || 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 O ( h ) doesn‘t depend on jump discontinuity Convergence rate

Two-grid method Fine grid Coarse grid Extended subdomain

One sweep of TGM Ax = b + n n 1 → x x ~ n x • Smooth with DD (2-3 iterations) ~ n n = − r b A x • Calculate the residual h h m 1 ∑ • Restrict the residual in each subdomain i = r r H h m = i 1 • Discretize and solve on coarse grid = A c r H H H • Prolong coarse grid correction by solving c local problems in shifted subdomains h ~ ~ + 1 n n n = + x x c • Correct the solution + n 1 x • Post smooth with DD Neuss N., Jäger W., Wittum G.

DD smoothing ~ n n → x x Additive Schwarz With overlapping Without overlapping Multiplicative Schwarz Restriction m 1 ∑ i = r r H h m = i 1 r - residual on a coarse grid H i r i - residual on a fine grid h m = m m m -number of fine grid blocks in a coarse one x y z

Coarse grid operator Coarse scale Darcy´s law eff = − ⋅ ∇ f v K p - volume average Local flow problems i i eff = − ⋅ ∇ = v K p , i 1 , 3 . Boundary conditions and RHS for local flow problem 1-0 Dirichlet + Neumann (v = 0) b.c. 1-0 Dirichlet + piecewise linear b.c. RHS = 0

Prolongation     ∂ ∂ ∂ ∂ p p   - coarse grid   + = k k 0    xx  yy ∂ ∂ ∂  ∂  x x y y solution 2D problem for the planes BCs for 3D problem 3D problem 1D by TDMA inside the cell at the edges   ∂ ∂ p ( ) − ∇ ⋅ ⋅ ∇ =   K p f = k 0 BCs for 2D problem xx   ∂ ∂ x x

Numerical results Periodic Non-periodic Cubic inclusion L-shaped inclusion Random inclusion K K 2 2 K 1 K 1 K 1 K 2     10000 0 0 1 0 0         = = K 0 10000 0 K 0 1 0 Permeability tensor 2 1       0 0 10000   0 0 1 Convergence of TGM depends on overlapping and number of subdomains

One- and two-level DD     64 inclusions, 1 0 0 10000 0 0     acc=1e-4     = = 0 10000 0 K 0 1 0 K 1 2 K     1 ovrlp=2: 1.5h     0 0 1 0 0 10000 K 2 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

DD smoothing (overlapping)     10000 0 0 1 0 0 2 presmooth.         = K 0 10000 0 = 0 1 0 K 2 postsmooth. 2 1     K 1     0 0 10000 0 0 1 K 2 Coarse grid 8x8x8, Coarse grid 8x8x8, fine grid 16x16x16 fine grid 8x8x8 Acc = 1E-5 ovrlp = 2 ovrlp = 1 TGM iter = 7 TGM iter = 13 Coarse grid 8x8x8, fine grid 16x16x16 ovrlp = 1 ovrlp = 2 ovrlp = 3 TGM iter = 23 TGM iter = 7 TGM iter = 6

DD pre- and post-smoothing     10000 0 0 1 0 0         = = K 0 1 0 K 0 10000 0 2 1 K     1     0 0 1 0 0 10000 K 2 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

DD smoothing     10000 0 0 1 0 0 2 presmooth.         = K 0 10000 0 = 0 1 0 K 2 postsmooth. 2 1     K 1     0 0 10000 0 0 1 K 2 Coarse grid 8x8x8, fine grid 8x8x8 Acc = 1E-5 ovrlp = 1 Additive Schwarz Multiplicative Schwarz TGM iter = 13 TGM iter = 7

TGM for different geometries Periodic cubic inclusion 2 = K 10000 E K = E 1 K 1 K 2 Random inclusion Coarse grid 8x8x8, fine grid 8x8x8 K TGM iter = 13 1 K 2 Coarse grid 8x8x8, fine grid 8x8x8 Periodic L-shaped inclusion TGM iter = 12 K 2 K 1 Coarse grid 8x8x8, fine grid 12x12x12 TGM iter = 23 TGM acc = 1E-5

TGM for different geometries Small inclusions Coarse grid 8x8x8, TGM acc = 1E-5 fine grid 8x8x8 K 1 K 2 inc = 1x1x1 Large inclusions TGM iter = 11 K Larger inclusions 1 K 2 K 1 K 2 inc = 2x2x2 inc = 4x4x4 TGM iter = 11 TGM iter = 11

Oversampling     10000 0 0 1 0 0       =   K 0 10000 0 = K 0 1 0 TGM acc = 1E-4 2 1         0 0 10000 0 0 1 K 2   − 5 − × 6569 . 1 192 . 6 1 . 2 10   K 1   − * 6 = − × K 192 . 6 6569 . 1 8 . 1 10   − − 5 6 × × TGM iter = 7 1 . 2 10 8 . 1 10 7126 . 0     − 4 − × 6411 . 6 256 . 0 1 . 5 10     − * 6 = − × K 256 . 0 6411 . 6 8 . 1 10 K   2 − − 4 6 TGM iter = 7 × × 1 . 5 10 8 . 1 10 6957 . 5   K 1

3D upscaling Fine grid permeability tensor = = α K E , K E 1 2 contrast 1:3 Effective permeability   − 5 − 4 ⋅ − ⋅ 1.1232 8.66 10 2.12 10   − − * 5 4 = ⋅ ⋅ K  8.66 10 1.1218 4.16 10    − − 4 4 − ⋅ ⋅ 2.12 10 4.16 10 1.1219   Foam contrast 1:1000 Effective permeability   − − 44.55 0.14 0.05   * = − − K 0.14 43.30 0.31      − −  acc = 10-E5 0.05 0.31 43.90

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 on the convergence of the preconditioned CG • Develop further approaches for two-phase flows • Theoretical analysis