Fast solvers for steady incompressible flow

Transcription

1 ICFD 25 p.1/21 Fast solvers for steady incompressible flow Andy Wathen Oxford University Joint work with: Howard Elman (University of Maryland, USA) David Silvester (University of Manchester, UK) Acknowledgements: EPSRC, British Energy

2 ICFD 25 p.2/21 Steady Incompressible Navier-Stokes: ν 2 u + u. u + p = f u = 0 mixed finite element approximation (or other approx eg. MAC scheme): u u h = u i φ i X h (H 1 ) d p p h = p k ψ k M h L 2 Galerkin F(u) B T u p f 0 B 0 = u: velocity coefficients, p: pressure coefficients

3 ICFD 25 p.3/21 Steady Incompressible Navier-Stokes: ν 2 u + u. u + p = f u = 0 F(u) B T u p f 0 Approximation B 0 = u: velocity coefficients, p: pressure coefficients B/B T : discrete divergence/gradient F(u) = νa + N(u): discrete advection diffusion operator A: discrete (vector) Laplacian, N: advection

4 ICFD 25 p.4/21 Linearisation: Slow flow Stokes νa B T B 0 u p = f 0

5 ICFD 25 p.4/21 Linearisation: Slow flow Stokes νa B T u p f 0 B 0 = Picard (simple fixed point) Oseen νa + N(u k ) B T u (k+1) f 0 B 0 p (k+1) =

6 ICFD 25 p.4/21 Linearisation: Slow flow Stokes νa B T u p f 0 B 0 = Picard (simple fixed point) Oseen νa + N(u k ) B T u (k+1) f 0 B 0 p (k+1) = Newton F(u k ) + M(u k ) B T δu (k+1) B 0 δp (k+1) = residual M(u) = F u (u).u: zeroth order term

7 ICFD 25 p.5/21 Fast solution of these linearised INDEFINITE systems: Direct (elimination) methods: dimensions 10 4, 10 5 Multigrid: effective smoothers Krylov subspace methods (Conjugate Gradients, MINRES, GMRES,...): PRECONDITIONING

8 An observation (Murphy, Golub, W) H B T preconditioned by H 0 0 S H B T 0 S B 0 has 3 distinct eigenvalues has 2 distinct eigenvalues where S = BH 1 B T (Schur Complement) MINRES /GMRES terminates in 3 / 2 iterations want approximations Ĥ, Ŝ 3 / 2 clusters fast convergence ICFD 25 p.6/21

9 Ĥ: ICFD 25 p.7/21

10 ICFD 25 p.7/21 Ĥ: for Stokes: H = νa is just discrete Laplacian use multigrid

11 ICFD 25 p.7/21 Ĥ: for Stokes: H = νa is just discrete Laplacian use multigrid for Oseen: H = F(u k ) = νa + N(u k ) is discrete advection-diffusion use multigrid

12 ICFD 25 p.7/21 Ĥ: for Stokes: H = νa is just discrete Laplacian use multigrid for Oseen: H = F(u k ) = νa + N(u k ) is discrete advection-diffusion use multigrid for Newton: H = F(u k ) + M(u k ) is discrete 2nd order operator use multigrid?

13 Ŝ? (Schur Complement Approximation) ICFD 25 p.8/21

14 ICFD 25 p.8/21 Ŝ? (Schur Complement Approximation) Stokes: directly use div-stabilty γ p sup u (p, u) u in matrix form: γ(p T Qp) 1/2 max u = max w=a 1/2 u p T Bu (u T Au) 1/2 p T BA 1/2 w (w T w) 1/2 = (p T BA 1 B T p) 1/2

15 ICFD 25 p.8/21 Ŝ? (Schur Complement Approximation) Stokes: directly use div-stabilty and boundedness γ p sup u (p, u) u Γ p in matrix form: γ(p T Qp) 1/2 max u = max w=a 1/2 u p T Bu (u T Au) 1/2 p T BA 1/2 w (w T w) 1/2 = (p T BA 1 B T p) 1/2 Γ(p T Qp) 1/2 shows Q, the pressure mass matrix, is (spectrally) equivalent to the Schur complement BA 1 B T use Ŝ = Q (or diag(q) or diag scaled Chebyshev for Q)

16 Preconditioner for Stokes: AMG 0 0 Q = Ĥ 0 0 Ŝ Theory (Silvester and W): Convergence (in right norm) independent of h Practice: number of MINRES iterations for 10 6 residual reduction (CPU time) A MG : 1 V-cycle Ŝ: 4 diag scaled Conj Grad iterations Driven Cavity flow Mixed Element Grid Q 1 P 0 Q 2 Q 1 Q 2 P 1 Q 2 P 0 direct (6) 31 (5) 29 (5) 25 (5) (.3) (8) 33 (10) 31 (7) 25 (6) (3) (21) 31 (21) 31 (19) 27 (16) (31) (76) 31 (74) 29 (69) 27 (59) (221) (313) 29 (309) 29 (305) 27 (267) (8961) ICFD 25 p.9/21

17 ICFD 25 p.10/21 Results from Rene Schneider (Leeds/Chemnitz): P 2 P 1 on 1 processor of a cluster of Sun Fire 6800 with UltraSPARC II Cu 900MHz processors CPU time degrees of freedom iterations for solution for setup e-2 1.5e e-1 5.0e e e+1 5.9e e e+2 1.0e e+3 4.2e e+3 1.7e e+4 6.8e+2

18 Ŝ? (Schur Complement Approximation) Oseen: non-symmetric S = BF 1 B T, F = νa + N, advection-diffusion Note: BB T QA p : discrete Laplacian on pressure space If F p is similarly an advection-diffusion operator on the pressure space, can expect FB T B T F p BB T BF 1 B T F p = SF p S 1 F p (BB T ) 1 F p A 1 p Q 1 Outcome: choose Ŝ 1 = F p A 1 p Q 1 (Kay & Loghin) with A 1 p MG cycle and Q 1 diag scaled Conj Grad ICFD 25 p.11/21

19 ICFD 25 p.12/21 FMG B Oseen preconditioner: T, 0 Ŝ Ŝ 1 = F p A 1 p Q 1 This is the Pressure Convection-Diffusion Preconditioner Theory (Krzyzanowski, Loghin, Elman, Silvester & W): Eigenvalues bounded independent of h ( GMRES convergence bounded independent of h?) Mild dependence on ν Practice: number of GMRES iterations for Oseen solve (zero initial vector), driven cavity flow Q 2 Q 1 Q 1 P 0 Grid ν =1/10 1/100 1/1000 1/10 1/100 1/

20 ICFD 25 p.13/21 3-D Driven Cavity flow, u top = (1/ 3, 2/ 3, 0) Maximum number of GMRES iterations for each Picard iteration (Oseen solve) Q 2 Q 1 element ν = 1/Re degees of freedom 1/20 1/40 1/80 1/ (results from David Kay)

21 ICFD 25 p.14/21 3-D Driven Cavity flow, u top = (1/ 3, 2/ 3, 0) Maximum number of GMRES iterations for each Picard iteration (Oseen solve) Q 2 Q 1 element, degees of freedom: element aspect ratio: maximum edge length/minimum edge length ν = 1/Re element aspect ratio 1/50 1/100 1/

22 Comment: alternative derivation (using Greens tensors) and analysis of ν-dependence: Kay, Loghin & W, Elman, Silvester & W,Loghin & W ICFD 25 p.15/21

23 ICFD 25 p.15/21 Comment: alternative derivation (using Greens tensors) and analysis of ν-dependence: Kay, Loghin & W, Elman, Silvester & W,Loghin & W Also alternative component preconditioning blocks can easily be used: Results from Vicki Howle (Sandia National Lab, USA) using (Smoothed Aggregation) Algebraic Multigrid ν = 1/Re Grid 1 1/10 1/20 1/

24 ICFD 25 p.16/21 Preconditioner for Newton: as M(u k ) is zeroth order, use F 1 (u k ) + M 1,1 (u k ) M 1,2 (u k ) 0 F 2 (u k ) + M 2,2 (u k ) 0 Ŝ B T with Ŝ 1 = F p A 1 p Q 1 (as before for Oseen) Theory: eigenvalues bounded and bounded away from 0 independently of h (Elman, Loghin & W) Practice: needs approximations to F i (u k ) + M i,i (u k ) as well as multigrid cycles for A p, Conj Grad for Q and construction of F p (multiply).

25 ICFD 25 p.17/21 Driven cavity: number of GMRES iterations for first Newton iteration Q 2 Q 1 Q 1 P 0 Grid ν =1/10 1/100 1/1000 1/10 1/100 1/ >

26 ICFD 25 p.18/21 Alternative algebraic preconditioner for Oseen/ Newton (Elman): instead of FB T B T F p start from so BFB T BB T F p (BB T ) 1 BFB T A 1 p Q 1 F p A 1 p Q 1 (BB T ) 1 BFB T (BB T ) 1 S 1 and (BB T ) 1 ( ) 1 use (Laplace) multigrid!

27 ICFD 25 p.18/21 Alternative algebraic preconditioner for Oseen/ Newton (Elman): instead of FB T B T F p start from BFB T BB T F p (BB T ) 1 BFB T A 1 p Q 1 F p A 1 p Q 1 In fact with the correct mesh scaling: Ŝ 1 = (BD 1 B T ) 1 BD 1 FD 1 B T (BD 1 B T ) 1 S 1 where D is diagonal of velocity mass matrix D. This is the Least-Squares Commutator Preconditioner

28 ICFD 25 p.19/21 Ŝ 1 = (BD 1 B T ) 1 BD 1 FD 1 B T (BD 1 B T ) 1 S 1 Note again only multiply by advection-diffusion operator F and multigrid cycles for Laplacian, but mild mesh-dependence of convergence for this algebraic preconditioner Practice: number of GMRES iterations for Oseen solve (zero initial vector), driven cavity flow, Q 2 Q 1 mixed element Oseen Newton Grid ν =1/10 1/100 1/1000 1/10 1/100 1/

29 ICFD 25 p.20/21 Important points: need only approximate solvers/preconditioners for Laplacian and advection-diffusion simple multigrid for such scalar problems can be applied any (stable or stabilised) discretisation need advection-diffusion operator for pressure space

30 ICFD 25 p.21/21 Main Reference Elman, H.C., Silvester, D.J. & Wathen, A.J., 2005, Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics, Oxford University Press, 2005 and associated matlab software: IFISS freely downloadable from