Algebraic Multigrid Methods for the Oseen Problem

Transcription

1 Algebraic Multigrid Methods for the Oseen Problem Markus Wabro Joint work with: Walter Zulehner, Linz This work has been supported by the Austrian Science Foundation Fonds zur Förderung der wissenschaftlichen Forschung (FWF) under the grant P14953 Robust Algebraic Multigrid Methods and their Parallelization p.1/27

2 Overview Introduction (Navier-Stokes / Oseen equations) p.2/27

3 Overview Introduction (Navier-Stokes / Oseen equations) Multigrid Methods for the solution of the Oseen equations p.2/27

4 Overview Introduction (Navier-Stokes / Oseen equations) Multigrid Methods for the solution of the Oseen equations Algebraic Multigrid for the Oseen equations Decoupled Approach Coupled Approach Prolongation, AMGe Smoothing p.2/27

5 Overview Introduction (Navier-Stokes / Oseen equations) Multigrid Methods for the solution of the Oseen equations Algebraic Multigrid for the Oseen equations Decoupled Approach Coupled Approach Prolongation, AMGe Smoothing Numerical results p.2/27

6 Complex 3D domain Ω, Aim p.3/27

7 Complex 3D domain Ω, Aim the incompressible Navier-Stokes equations t u ν u + (u )u + p = f, + boundary/initial conditions, div u = 0, p.3/27

8 Complex 3D domain Ω, Aim the incompressible Navier-Stokes equations t u ν u + (u )u + p = f, + boundary/initial conditions, Finite Element Discretization A(u) B BT u = C p div u = 0, f g p.3/27

9 Complex 3D domain Ω, Aim the incompressible Navier-Stokes equations t u ν u + (u )u + p = f, + boundary/initial conditions, Finite Element Discretization A(u) B BT u = C p div u = 0, f g We want an efficient Solver. p.3/27

10 Convective Term Linearization: Fixed point iteration Oseen equation ν u + (w )u + p = f p.4/27

11 Convective Term Linearization: Fixed point iteration Oseen equation ν u + (w )u + p = f Stabilization (SUPG): ((w )u, v)+β h (w u, w v), with β h = O(h). p.4/27

12 Convective Term Linearization: Fixed point iteration Oseen equation ν u + (w )u + p = f Stabilization (SUPG): ((w )u, v)+β h (w u, w v), with β h = O(h). Solution of linear problem? p.4/27

13 MGM for Saddle-Point-Problems p.5/27

14 MGM for Saddle-Point-Problems Iterative Decoupling Pressure correction methods: SIMPLE, Uzawa Precond. Krylov space methods (e.g. GMRES, BiCGstab) MG for the elliptic systems. p.5/27

15 MGM for Saddle-Point-Problems Iterative Decoupling Pressure correction methods: SIMPLE, Uzawa Precond. Krylov space methods (e.g. GMRES, BiCGstab) Coupled MG for the whole system. Needs appropriate MG components (smoother, inter-grid-transfer). MG for the elliptic systems. p.5/27

16 Decoupled Approach Examples p.6/27

17 Decoupled Approach Examples SIMPLE: Aũ = f B T p k, Ŝ p = Bũ Cp k g, u k+1 = ũ D 1 B T p, p k+1 = p k + p, with D A, Ŝ = C + BD 1 B T. p.6/27

18 Decoupled Approach Examples SIMPLE: Aũ = f B T p k, Ŝ p = Bũ Cp k g, u k+1 = ũ D 1 B T p, p k+1 = p k + p, with D A, Ŝ = C + BD 1 B T. Preconditioner of Silvester, Wathen et.al.: P 1 = A 1 0 I BT I 0, 0 I 0 I 0 Q 1 A p L 1 A A, Q pressure mass matrix, A p... pressure conv.-diff. operator, L pressure Laplacian ( C. Powell) p.6/27

19 Algebraic Multigrid Complex geometry discretization No further refinement possible. many unknowns. p.7/27

20 Algebraic Multigrid Complex geometry discretization No further refinement possible. many unknowns. Algebraic Multigrid. Starting from the finest levels generate the coarse levels (almost) only from matrix information. p.7/27

21 Algebraic Multigrid Complex geometry discretization No further refinement possible. many unknowns. Algebraic Multigrid. Starting from the finest levels generate the coarse levels (almost) only from matrix information. O.K. for elliptic problems, application to the decoupled approach is straightforward. What about coupled AMG? p.7/27

22 Coupled Approach Strategy p.8/27

23 Coupled Approach Strategy Separate pressure and velocity-component unknowns. p.8/27

24 Coupled Approach Strategy Separate pressure and velocity-component unknowns. Keep interaction of u and p the same. p.8/27

25 Coupled Approach Strategy Separate pressure and velocity-component unknowns. Keep interaction of u and p the same. Try to translate as much as possible from geometric MG for saddle point problems. p.8/27

26 Coupled Approach Strategy Separate pressure and velocity-component unknowns. Keep interaction of u and p the same. Try to translate as much as possible from geometric MG for saddle point problems. Try to use as much as possible from elliptic AMG. p.8/27

27 Coupled Approach Strategy Separate pressure and velocity-component unknowns. Keep interaction of u and p the same. Try to translate as much as possible from geometric MG for saddle point problems. Try to use as much as possible from elliptic AMG. AMG as black box? p.8/27

28 Coupled Approach Strategy Separate pressure and velocity-component unknowns. Keep interaction of u and p the same. Try to translate as much as possible from geometric MG for saddle point problems. Try to use as much as possible from elliptic AMG. AMG as black box? p.8/27

29 Notation Grid transfer: ( Prolongators Pi+1 i = Ĩi i+1 Ji+1 ) i Ĩ i i+1 = ( I i i+1 I i i+1 Restrictors P i+1 i Coarse level systems: K i = ( ) A i Bi T B i C i I i i+1, (= P i i+1t ). = P i i 1 K i 1P i 1 i ), p.9/27

30 Notation Grid transfer: ( Prolongators Pi+1 i = Ĩi i+1 Ji+1 ) i Ĩ i i+1 = ( I i i+1 I i i+1 Restrictors P i+1 i Coarse level systems: K i = ( ) A i Bi T B i C i I i i+1, (= P i i+1t ). = P i i 1 K i 1P i 1 i With two exceptions (h-dependence in conv. stab.)! ), p.9/27

31 Some Mixed Finite Elements P1-P1-stab: C = α elts K h 2 K ( p, q) 0,K p.10/27

32 Some Mixed Finite Elements P1-P1-stab: modified Taylor-Hood: C = α elts K h 2 K ( p, q) 0,K p.10/27

33 Some Mixed Finite Elements P1-P1-stab: modified Taylor-Hood: C = α elts K h 2 K ( p, q) 0,K Crouzeix-Raviart: p.10/27

34 PSfrag replacements Coarsening Strategy P1-P1-stab: same hierarchy Coarsening fine mesh 1st level u, p 2nd level u, p. N -th level u, p p.11/27

35 PSfrag replacements Coarsening Strategy P1-P1-stab: same hierarchy PSfrag replacements Coarsening fine mesh 1st level u, p 2nd level u, p. N -th level u, p mod. Taylor-Hood: shifted hierarchy (first try) Refinement Coarsening fine (u) mesh coarse (p) mesh 1st level u p 2nd level u p u p... N th level u p p.11/27

36 Stability P1-P1-stab (1) Geometric case: Franca, Stenberg (1991) using idea of Verfürth (1984): (div v, p) sup 0 v V h v 1 γ p 0 δ ( elts K Stability h 2 K p 2 0,K ) 1 2 p Q h p.12/27

37 Stability P1-P1-stab (1) Geometric case: Franca, Stenberg (1991) using idea of Verfürth (1984): (div v, p) sup 0 v V h v 1 (In the form γ p 0 δ sup (v,q) V Q ( elts K Stability h 2 K p 2 0,K ) 1 2 B((w,r),(v,q)) (v,q) X c (w, r) X, p Q h B((w, r), (v, q)) = a(w, v) b(v, r) b(w, q) c(r, q)) p.12/27

38 Stability P1-P1-stab (2) AMG case: Assume A i is of essentially positive type i.e. for all v i V i ( a kj )(v k v j ) 2 γ ( a kj )(v k v j ) 2. k,j k,j For all v i V i we can find a Π i+1 i v i V i+1 such that v i Ĩ i i+1π i+1 i v i 2 D i β 1 v i 2 A i. (Ii+1 i... e.g. Ruge, Stüben, 87, 01; Vaněk et al., 95) p.13/27

39 Stability P1-P1-stab (2) AMG case: Assume A i is of essentially positive type i.e. for all v i V i ( a kj )(v k v j ) 2 γ ( a kj )(v k v j ) 2. k,j k,j For all v i V i we can find a Π i+1 i v i V i+1 such that v i Ĩ i i+1π i+1 i v i 2 D i β 1 v i 2 A i. (Ii+1 i... e.g. Ruge, Stüben, 87, 01; Vaněk et al., 95) Then for all levels i sup vbt i p γ p Mi δ ( p T C i p ) 1 2 p Q i. 0 v V i v Ai p.13/27

40 Stability mod. Taylor-Hood (1) arbitrary grid-transfer problems. p.14/27

41 Stability mod. Taylor-Hood (1) arbitrary grid-transfer problems. Example: I i i+1 P1-P1 situation. locally equal I unstabilized p.14/27

42 Stability mod. Taylor-Hood (1) arbitrary grid-transfer problems. Example: Ii+1 i locally equal I unstabilized P1-P1 situation. More restrictive coarsening leads to poor approximation. p.14/27

43 Stability mod. Taylor-Hood (1) arbitrary grid-transfer problems. Example: Ii+1 i locally equal I unstabilized P1-P1 situation. More restrictive coarsening leads to poor approximation. Drop idea of same hierarchy. Guarantee that there are enough velocity unknowns. How? p.14/27

44 Stability mod. Taylor-Hood (2) First try (brute force): double shift. 1st level 1st level 2nd level 2nd level PSfrag replacements N th level N th level fine (u) mesh coarse (p) mesh fine (u) mesh coarse (p) mesh u p u p u p u p u u p p u p u p Refinement Refinement Coarsening Coarsening p.15/27

45 Problems Stability Arbitrary elements? p.16/27

46 AMGe Stability Arbitrary elements? One possibility AMGe (Jones, Vassilevski et.al.): element agglomeration minimal energy basis functions p.16/27

47 AMGe for Crouzeix-Raviart (1) Nonconf. geometric prolongation: p.17/27

48 AMGe for Crouzeix-Raviart (1) Nonconf. geometric prolongation: Conforming AMGe: A E C 0 C T x f = 0 λ z C = (0 I), z = x b f. p.17/27

49 AMGe for Crouzeix-Raviart (2) Nonconforming AMGe: A E C T C 0 x f = 0 λ z C, z s.t. x f = 1 on 1-edge, average 0 on 0-edges. p.18/27

50 AMGe for Crouzeix-Raviart (2) Nonconforming AMGe: A E C T C 0 x f = 0 λ z C, z s.t. x f = 1 on 1-edge, average 0 on 0-edges. (in num. practice: truncate small entries to reduce fill-in) p.18/27

51 Stability Crouzeix-Raviart Lemma: Assume existence of lin.op. Π k k 1 : U k 1 U k with and b(π k k 1v k 1, q k ) = b(v k 1, J k 1 k q k ) q k Q k, v k 1 U k 1 (1) Π k k 1v k 1 A C v k 1 A v k 1 U k 1. (2) Then inf-sup in U k 1 Q k 1 implies inf-sup in U k Q k. p.19/27

52 Stability Crouzeix-Raviart Lemma: Assume existence of lin.op. Π k k 1 : U k 1 U k with and b(π k k 1v k 1, q k ) = b(v k 1, J k 1 k q k ) q k Q k, v k 1 U k 1 (1) Π k k 1v k 1 A C v k 1 A v k 1 U k 1. (2) Then inf-sup in U k 1 Q k 1 implies inf-sup in U k Q k. (1), (2) can be shown using purely geometric arguments. p.19/27

53 Smoothing Use smoothers from GMG for saddle point systems, e.g. Braess-Smoother: Â(û j+1 u j ) = f Au j B T p j, Ŝ(p j+1 p j ) = Bû j+1 Cp j g, Â(u j+1 û j+1 ) = B T (p j+1 p j ), Ŝ C + BÂ 1 B T (AMG). p.20/27

54 Smoothing Use smoothers from GMG for saddle point systems, e.g. Braess-Smoother: Â(û j+1 u j ) = f Au j B T p j, Ŝ(p j+1 p j ) = Bû j+1 Cp j g, Â(u j+1 û j+1 ) = B T (p j+1 p j ), Ŝ C + BÂ 1 B T (AMG). Vanka-Smoother: Solve smaller local problems and combine solutions via (mult.) Schwarz method. Zulehner/Schöberl: theoretical basis for GMG. p.20/27

55 Software package AMuSE All numerical tests were performed using AMuSE Algebraic Multigrid for Stokes type Equations, developed by F. Kickinger ( ), C. Reisinger ( ), M. W. (since 1999). p.21/27

56 Numerical Results P1isoP2-P e-05 1e-06 1e-07 1e-08 1e-09 Coupled, MSM SIMPLE BiCGstab, Silvester-Wathen 1e e-05 1e-06 1e-07 1e-08 1e-09 1e-10 1e-11 Braess Braess, shifted SIMPLE 1e D valve, 10 5 vars, ν = D valves, vars, ν = 0.001, velocity at inlet 0.5, narrowest part p.22/27

57 Numerical Results 3D Valves p.23/27

58 Numerical Results P1-P1-stab e-05 1e-06 1e-07 1e-08 1e-09 Coupled SIMPLE 1e vars, mesh: tets, hexas, pyramids, prisms; ν = p.24/27

59 Numerical Results P1-P1-stab e-05 1e-06 1e-07 1e-08 1e-09 Coupled SIMPLE 1e vars, mesh: tets, hexas, pyramids, prisms; ν = p.24/27

60 Numerical Results Crouzeix-Raviart 2D valve, W-5-5 level faces elem rate (at 4th level: 0.8 reduction/minute) p.25/27

61 Smoothing Comparison 2D and/or less complex problems: Vanka-smoother leads to a fast and stable solver Complex 3D problems: Braess-smoother can compensate stability problems. The patches for the Vanka-smoother grow rapidly (for conforming elements) poor efficiency p.26/27

62 Summary We have a coupled AMG method for saddle point problems originating in the Navier Stokes equations. We can use existing components from AMG for elliptic problems, resp. GMG for saddle point problems. p.27/27

63 Summary We have a coupled AMG method for saddle point problems originating in the Navier Stokes equations. We can use existing components from AMG for elliptic problems, resp. GMG for saddle point problems. BUT It is absolutely no Black-Box method! We have no rigorous prove for most aspects (e.g. convergence)! p.27/27