Total Overlapping Schwarz Preconditioners for Elliptic PDEs ( a ) ( b )

Transcription

1 1 Seminaire, PARIS XIII, 05/12/08 Total Overlapping Schwarz Preconditioners for Elliptic PDEs ( a ) ( b ) F. Ben Belgacem (LMAC-UTCompiègne) N. Gmati (LAMSIN-ENITunis) F. Jelassi (LMAC-UTCompiègne) a Main ideas originated in the Ph.D Thesis of F. Jelassi, b The Helmholtz equation is considered as elliptic.

2 2 Schwarz Method (1870) : A Brief Description Find u such that u = 0 in ] 1, 1[, u( 1) = u(+1) = 0, 1 α β + 1 Find u 2m+1 such that (u 2m+1 ) = 0, ] 1, β[, (u 2m+1 )( 1) = 0, (u 2m+1 )(β) = (u 2m )(β) Find u 2m+2 such that (u 2m+1 ) = 0, ]α, +1[, (u 2m+2 )(+1) = 0, (u 2m+2 )(α) = (u 2m+1 )(α)

3 3 Convergence This is a Mathematical Proof) CL Dirichlet x = 1 CL Dirichlet x = CL Dirichlet x = +1 CL Dirichlet x = CL Dirichlet x = 1 CL Dirichlet x = Linear convergence

4 4 Motivations Construction of Sub-structuring Preconditioners MSM, ASM, FETI,... (A Huge Literature). Coupling Methods, Models, or Programs. Matching Meshes,... (Great Number of papers) Parallelism (A lot of work)

5 5 Bibliography H. A. Schwarz, 1870 maximum principle K. Miller, 1965 efficient computational tool A. M. Matsokin, S. V. Nepomnyaschikh, 1985 P. E. Bjørstad, O. B. Widlund, 1986 P.-L. Lions, 1988 variationnal approach!!! M. Dryja, O. B. Widlund, 1989, 1990 (Additive Schwarz ) B. Smith, P. Bjørstad, W. Gropp, 1996 C. Farhat, F. X. Roux, 1991 (FETI) A. Quarteroni, Valli, 1999 A. Toselli, O. Widlund, 2005 BenDali, Boubendir, Fares, Collino, 2003

6 The Total Overlapping Schwarz Method (TOSM) 6

7 7 Bibliography (2) : post 2000 Liu, Jin, Adaptive absorbing BC/FEM (2001), Preconditioners for FEM/Boundary Integral (2002) Maury, Ismaïl, The Fat boundary Method (2001) Pironeau, Lions, Brezzi, Apoung-Kamga, Hecht Chimera Method, Numerical Zooming (2005) Glowinski, He, Lozinsky, Rappaz, Wagner, Multi-Domain (Multi-Scale) Method, FEM with patches (2004)

8 8 A Cracked Domain (I) γ + ω γ Σ Ω γ γ Find ϕ such that ϕ = f in Ω γ, ϕ = 0 on Γ, n ϕ = 0 in γ + γ

9 9 A Cracked Domain (II) Variational Principle (= ) Find ϕ such that ϕ ψ dx = fψ dx, Ω γ Ω γ ψ. Finite Element Approximation (= ) Find ϕ h such that A h ϕ h = f h. The stiffness matrix A h sees the cracks (ϕ h suffers from jumps) (= ) The condition number degrades rapidly with the number and of the geometrical form of the cracks. (= ) Necessity of Domain Decomposition devices in the computations.

10 10 The Mile Stone of (TOSM) : Decompose the Problem One sub-problem (= ) Find η in Ω γ such that η = f in Ω γ, [η] = [χ] on γ, [ n η] = 0 on γ, η = 0 on Γ. The other (= ) Find χ in ω γ such that χ = f in ω γ, χ = η on Σ n χ = 0 on γ + γ. We have that η = ϕ and χ = ϕ ωγ Look at (η χ) to be convinced.

11 11 Uncouple the Transmission (= ) (TOSM) Assume (χ m, η m ) is known, η m+1 is solution of η m+1 = f in Ω γ, [η m+1 ] = [χ m ] on γ, [ n η m+1 ] = 0 on γ, η m+1 = 0 on Γ, and χ m+1 satisfies χ m+1 = f in ω γ, χ m+1 = η m+1 on Σ n χ m+1 = 0 on γ + γ.

12 12 If η m+1 wants not to see the crack at all γ = any regular curve embracing the crack γ and located inside ω γ. f = is the trivial extension of f inside γ. Replace η m+1 by η m+1 solution of and χ m+1 satisfies η m+1 = f in Ω \ γ, [ η m+1 ] = χ m on γ, [ n η m+1 ] = n χ m on γ, η m+1 = 0 on Γ. χ m+1 = f in ω γ, χ m+1 = η m+1 on Σ n χ m+1 = 0 on γ + γ.

13 13 Advantages Mesh for η that do not see the crack (= ) The Stiffness Matrix is related to the safe domain Ω (= ) Better conditioned and Easier to solve. Deconnection between multiple cracks (= ) Solve local subproblems on χ at the vicinity of the individual cracks (small subproblems). Uncoupled two-scale computations : we may use Standard FEM for η and XFEM( a ) for χ (Blind to each other). a extended FEM

14 14 Higher Advantages for Exterior Problems (I) Ω γ = R d γ (= ) η m+1 the solution of η m+1 = f in R d γ, [η m+1 ] = [χ m ] on γ, [ n η m+1 ] = 0 on γ, η m+1 = 0 at, can be obtained thanks to an Integral Representation (G : Green function). η m+1 (x) = fg(x, ) dy + ([χ m ]) n G(x, ) dγ = T(χ m ). R d γ

15 15 Higher Advantages for exterior problems (II) (= ) Solve iteratively χ m+1 = f in ω γ, χ m+1 = (η m+1 ) = T γ (χ m ) on Σ n χ m+1 = 0 on γ + γ. (= ) Uncoupling FEM/Boundary Integral. T γ is a Regular Kernel Operator Allows Programs dedicated to Bounded Domains to handle Exterior Domains without considering Singular Kernels.

16 16 In what is it different of Schwarz Method (for cracks)? Assume (χ m, η m ) is known, η m+1 is solution of and χ m+1 satisfies η m+1 = f in Ω γ, η m+1 = χ m on γ + γ, η m+1 = 0 on Γ, χ m+1 = f in ω γ, χ m+1 = η m+1 on Σ n χ m+1 = 0 on γ + γ. Bounded domains : Both approaches, when written on γ, are economically equivalent if they have the same convergence rate (not the case). Unbounded domains : The Total Overlapping approach is pretty more economical.

17 17 Perforated Domains γ Ω γ Σ Ω γ ω γ Find ϕ such that ϕ = f in Ω γ, ϕ = 0 on Γ, n ϕ = 0 in γ

18 18 Decompose the Problem Extend f by zero in Ω γ = f One sub-problem (= ) Find η Ω γ Ω γ such that η = f in Ω γ Ω γ, [η] = χ on γ, [ n η] = 0 on γ, η = 0 on Γ. The other (= ) Find χ in ω γ such that We have that χ = f in ω γ, χ = η on Σ, n χ = 0 on γ. η Ωγ = ϕ Ωγ, η Ω γ = 0 and χ = ϕ ωγ

19 19 Toal Overlapping Schwarz Method Find η m+1 Ω γ Ω γ such that Find χ m+1 in ω γ such that η m+1 = f in Ω γ Ω γ, [η m+1 ] = χ m on γ, [ n η m+1 ] = 0 on γ, η m+1 = 0 on Γ, χ m+1 = f in ω γ, χ m+1 = η m+1 on Σ, n χ m+1 = 0 on γ. The Fat Boundary Method (Bertrand Maury 2001) (=) A Penalized (TO)-Schwarz Method ( =) Wrong Transmission Cdts.

20 20 Differences with the Classical Schwarz Method Find η m+1 Ω γ such that Find χ m+1 in ω γ such that η m+1 = f in Ω γ, η m+1 = χ m on γ, η m+1 = 0 on Γ, χ m+1 = f in ω γ, χ m+1 = η m+1 on Σ, n χ m+1 = 0 on γ. Classical Schwarz method (= ) Compute η m+1 in the perforated domain. (TO)-Schwarz method (= ) Compute η m+1 in the filled domain ( ).

21 21 Convergence? Classical Schwarz γ,σ, Γ : Circles with radius 1,, and 1 <. The Schwarz iterates are given by (χ m, η m )(r, θ) = (χ m p, ηp m )(r) cos(pθ), p=0 η m p = a m r p, r χ m p = b m r p + d m r p, 1 r Linear Convergence (= ) µ p = 2 2p 1 + 2p

22 22 Convergence? (TOSM) (χ m, η m )(r, θ) = (χ m p, ηp m )(r) cos(pθ), p=0 ηp m = a i rp, r 1 a e r p, r 1 χ m p = b m r p + d m r p Linear Convergence (= ) µ p = 2p (= ) Twice faster than the Cl. Sch p 2,5 2,5 2, ,5 1 1,5 1 1,5 1 0,5 0,5 0, ,5 1 1,5 2 2,5 3 3, ,5 1 1,5 2 2,5 3 3,5-0,

23 23 Convergence of (TOSM) Denote by e m = (ϕ χ m ), τ m = (ϕ η m ). Théo. 1 (fj, 2005) (TOSM) converges with a linear rate e m L2 (ω γ ) + τ m L2 (Ω γ ) be am m. a > 0 and increases when the distance between Σ and γ grows.

24 24 Numerical Examples: Exterior Problems MELINA fortran 90 (D. Martin, IRMAR) Implementation of specific procedures (F. Jelassi, LMAC)

25 25 Rankine s Oval (1) Potential flow : Source-sink dipole + Uniform flow f(z) = 5π 4 z + log ( z 0.2 z ) Rankine oval: Streamlines in ω γ

26 26 Rankine Oval (2) Ecart entre deux iteres successifs (l,w) = (0.6,0.2) (l,w) = (1,0.5) (l,w) = (2,1) (l,w) = (3,1.5) Ecart entre deux iteres successifs (l,w) = (0.6,0.2) (l,w) = (1,0.5) (l,w) = (2,1) (l,w) = (3,1.5) Nombre d iterations Nombre d iterations Consecutive iterates gap : Potential (left) and Stream function (right).

27 27 A Wing Profile (1)

28 28 A Wing Profile (2) Wing profile : NACA0012. Angle of Attack = π 12. Erreurs relatives 10-1 Carre Ellipse Spline Erreurs relatives 10-1 Carre Ellipse Spline Nombre d iteration Nombre d iteration Consecutive iterates gap: l 2 (left) et l (right).

29 29 Electrical Dipole : Non-connected Σ (1) The local problems are dis-connecte. They are solved individually

30 30 Electrical Dipole : Non-connected Σ (2) Circular Conductors radius = 0.5. Distance = 4. Electric Charge = πσ of opposite signs. Non-Connected Calculus. Erreur relative R = 0.7 R = 1 R = 1.5 Erreur relative R = 0.7 R = 1 R = Iterations Iterations Gaps (u sc u co ) : l 2 (left) et l (right).

31 31 Electric Quadrupole : Non-Connected Σ Quadrupole. Symmetric et Dis-symmetric Cases.

32 32 Electric Quadrupole in 3D : Non-Connected Electrostatic Conductors (Spheres of radius 1 U). Uniformly charged on their skins and Distant of 4 U from each other Erreurs relatives Cas symetrique Cas symetrique Cas non-symetrique Cas non-symetrique Nombre d iteration Gap between Consecutive Iterates with l 2 -norm (empty symbols) and l (full symbols).

33 33 Flat Quadrupole (1) (Cracked Domain) Equipotentials of the non-symmetric flat quadrupole. Charges concentration are higher along the sides where equipotentials are more dense

34 34 Flat Quadrupole (2) 10 0 Residual Schwarz GMRES Iterations (TO)-Schwarz method and Preconditioned GMRES residual curves for the cracked domain: the potential for the non symmetric quadrupole.

35 35 Electric Dipole within Heterogeneous Media GMRES Computations (left) and (TO)-Schwarz Method (right).

36 36 To overcome this Weakness (= ) Krylov Sub-Spaces Solvers Take back the coupled problem (= ) Find η λ Ω γ Ω γ such that η λ = f in Ω γ Ω γ, [η λ ] = χ λ on γ, [ n η λ ] = 0 on γ, η λ = 0 on Γ. Find χ λ in ω γ such that χ λ = f in ω γ, χ λ = λ( = η λ ) on Σ, n χ λ = 0 on γ. The condensed problem to solve λ (η λ ) Σ = 0 on Σ.

37 37 Static Condensation (I) The Constant Part of the Condensed Equation Define χ such that χ = f in ω γ, χ = 0 on Σ n χ = 0 on γ + γ. Define η that satisfies η = f in Ω γ, [η] = [χ] on γ, [ n η] = 0 on γ, η = 0 on Γ. Set G = (η) Σ. No real coupling between both sub-problems.

38 38 Static Condensation (II) The linear Part Define χ λ such that χ λ = 0 in ω γ, χ λ = λ on Σ n χ λ = 0 on γ + γ, Define η λ that satisfies η λ = 0 in Ω γ, [η λ ] = [χ λ ] on γ, [ n η λ ] = 0 on γ, η λ = 0 on Γ. Set B Σ (λ) = (η λ ) Σ

39 39 (TOS) Preconditioner The suitable λ(= χ) solves the equation χ λ = η λ on Σ, (I + B Σ )(λ) = G ( ) λ (η λ ) Σ = (η) Σ. Richardson method (= Jacobi Algorithm) λ m+1 + B Σ (λ m ) = G. ( ) (TOSM) with χ m = χ λ m + χ. B Σ is a small perturbation of I because all the singular values of it are smaller than one ( =) (TOSM) converges. B Σ is a highly Regular Kernel operator (= ) the singular values of B Σ decrease exponentially fast toward zero. (= ) Krylov Subspaces Method will dramatically accelerate the convergence.

40 40 Convergence of Krylov Methods Denote by r m = G (I + B Σ )(λ m ) = (χ λ η λ ) m. Théo. 2 (fbb-ng-fj, 2008) Krylov Methods converge with a quadratic rate in two dimensions that is r m L2 (Σ) be am2 m. The convergence rate in three dimensions is three halves since we have that r m L2 (Σ) be am m m. In both estimates, a > 0 and increases when the distance between Σ and γ grows.

41 41 Circular Quadrupole (1) (= ) Perforated Domain) The non-symmetric circular quadrupole..

42 42 Circular Quadrupole (2) Equipotentials of the symmetric circular quadrupole (in ω γ ).

43 43 Circular Quadrupole (3) Residual Schwarz GMRES Residual Schwarz GMRES Iterations Iterations Convergence curves for the quadrupole with circular conductors. The symmetric is in the left diagram and non-symmetric one in the right diagram. Full symbols are for connected ω γ while empty symbols are for non-connected ω γ. Circular symbols correspond to the TOSM and squared symbols are for the precondit. GMRES.

44 44 Back to the Flat Quadrupole (2b) Residual Schwarz GMRES Iterations (TO)-Schwarz method and Preconditioned GMRES residual curves for the cracked domain: the potential for the non symmetric quadrupole.

45 Wave Diffraction 45

46 46 Helmholtz Problem Ω : (Gray) Bounded Obstacle = Ω : medium surrounding Ω. Incident wave Scattered wave Find u such that u + κ 2 u = 0 in Ω, n u = g on γ, ( r iκ)u = e iκr O( 1 ) at, r2 Sommerfeld Condition at = Existence and Uniqueness

47 47 Exact Non-Reflecting Boundary Condition (FEM/BEM) Find u such that u + κ 2 u = 0 in ω γ, n u = g on γ, ( n + β)u = T β K (u) on Σ. The operator T β K is defined by (K β = ( n + β)g, Regular Green Function ) T β K (u)(x) = ( (u) n K β (x, y) + (g)k β (x, y) ) dγ y, on Σ γ Variational Formulation = Find u (A β + T β )u = G in ω γ. Exterior and Bounded problems are equivalent for β C

48 48 (TOSM) Find v m+1 such that Find u m+1 such that I(β) 0 is compulsory. v m+1 + κ 2 v m+1 = 0 in Ω Ω, [v m+1 ] = [u m ] on γ, [ n v m+1 ] = g on γ, ( r iκ)v m+1 = e iκr O( 1 ) at, r2 u m+1 + κ 2 u m+1 = 0 in ω γ, n u m+1 = g on Γ, ( n + β)u m+1 = ( n + β)v m+1 on Σ.

49 49 (TOSM) = (keep u m and eliminate v m ) Find u such that u m+1 + κ 2 u m+1 = 0 in ω γ, n u m+1 = g on Γ, ( n + β)u m+1 = T β K (um ) on Σ. Uncoupled variational Formulation (= ) u m known, find u m+1 such that A β u m+1 + T β u m = G. (1) I(β) 0 = A β is invertible. Exact non-reflection at = IneExact Non-Reflection at Σ + Iterations. (TOSM) = Liu and Jin Method (IEEE Trans. Antennas Propagat. 2001) = Preconditioned Richardson Method applied to (1).

50 50 Analytical Investigation Γ, Σ : Circles with radius 1, et R 1 R. Sound Hard Obstacle (a Neumann Condition) g(θ) = p Z g p exp(ipθ). Iterates of (TOSM) are provided by u m (r, θ) = p Z ϕ m (r)exp(ipθ), Attenuation Cœfficients of (TOSM) 1 τ p (β) = 1 H(1) p (κ) J p (κ) (κj p βj p )(κr) (κh p (1) βh p (1) )(κr).

51 51 Attenuation Cœfficients τ p (β) (1) Choice of β = iκ, Wave Length : λ = 2π κ, Thickness of the Computational Domain = eλ, 0,4 0,5 τ 0,3 κ = 30 κ = 10 κ = 1 τ 0,4 κ = 30 κ = 10 κ = 1 0,3 0,2 0,2 0,1 0, p p Attenuation Cœfficients τ p for e = 1 and e = 0.25.

52 52 Attenuation Cœfficients τ p (β) (2) τ 1 0,8 κ = 30 κ = 10 κ = 1 partie imaginaire 1 0,5 0 κ = 1 κ = 10 κ = 30 0,6-0,5 0,4 0, p -1,5-1 -0,5 0 0,5 1 1,5 partie reelle Attenuation Cœfficients for e = 0.01 (left). Eigenvalue of the Operator of Iteration (right). Relaxation of Richardson with a Small Parameter = It works.

53 53 (TOS) Preconditioner for Krylov Methods (1) Find u such that (I + (A β ) 1 T β ) u = (A β ) 1 G. by GMRES or Bi-Conjugate Gradient. Théo. 3 Assume that the fictitious boundary Σ is piecewise analytic. Then, for any real number a > 0, there exists c = c(a) > 0, such that a(m log m) r m L2 (Σ) ce m. Rem. 1 There is no reason why we have Less sharp convergence than for the Poisson Problem (= ) Hope to improve our Mathematical proofs.

54 Numerical Examples (Melina) 54

55 55 Non-Convex Guide (1) Slit Length = 0.4. Wave length = 1 or 0.1. Thickness of the computational domain = The truncating Σ and the boundary of the scatterer Γ have similar shapes. To the right is a zoom on the mesh we used.

56 56 Non-Convex Guide (2) 8 8 Im (eigenv) 6 Im (eigenv) 6 4 λ = 1 4 λ = 1 λ = λ = Re (eigenv) Re (eigenv) The distribution of the eigenvalues of non preconditioned system (to the left). The effect of the (TOS) preconditioner (to the right).

57 57 Non-Convex Guide (3) Residuals λ = 1 λ = 0.1 Residuals λ = 1 λ = Iterations Iterations Convergence curves of the GMRES with no preconditioner (to the left) With the Schwarz preconditioner (to the right). After capturing the first non-converging modes the Precond. GMRES exhibits a quadratic convergence.

58 58 Non-Convex 3D Guide Residuals λ = 1 λ = 0.5 λ = Iterations The shape of the scatterer (left panel) and (TOS) precondtioned GMRES convergence curves for an incident planar wave with wavelengths λ = 1, 0.5 and 0.2. The thickness of the computational domain e = 0.05.

59 59 Conclusion (TOSM) improves the Schwarz. Perspectives (1) More Computations (3D). (2) Many possible Extensions. (3) More Mathematics.