BETI for acoustic and electromagnetic scattering

Transcription

1 BETI for acoustic and electromagnetic scattering O. Steinbach, M. Windisch Institut für Numerische Mathematik Technische Universität Graz Oberwolfach 18. Februar 2010 FWF-Project: Data-sparse Boundary and Finite Element Domain Decomposition Methods in Electromagnetics 1 / 38

2 Outline Model problem The Tearing and Interconnecting approach Preconditioning Outlook 2 / 38

3 Outline Model problem The Tearing and Interconnecting approach Preconditioning Outlook 3 / 38

4 Problem definition Find u : Ω R so that holds. Remark: u(x) k 2 u(x) = 0 for x Ω, γ D u(x) = g(x) for x Γ D, γ N u(x) = u n (x) = h(x) for x Γ N, ) (ikeγ D u(x) + γ N u(x) = k(x) for x Γ R. k(x) piecewise constant Extension to exterior boundary value problems 4 / 38

5 Domain: Geometric domain decomposition Ω = i I Ω i with Ω i Ω j = if i j. Ω i is bounded for i > 0, Ω 0 may be unbounded Boundary: Ω := Γ D Γ N, Γ I := Γ ij, i,j I Γ S := Γ i = Γ I Ω. i I Γ 23 Γ D Ω 2 Γ 12 Ω 1 Γ 34 Γ 14 Ω 3 Ω 4 Γ N 5 / 38

6 Dirichlet to Neumann map Helmholtz u k 2 u = 0 in Ω, u = g on Γ, t = Su BEM: (symmetric analytic version) S k := D k + ( 1 2 I + K k 1 )Vk ( 1 2 I + K k) FEM: (discrete) S k := A ΓΓ A ΓI A 1 II A IΓ Remark: These definitions do not allow an exact discretization of the Steklov-Poincare operator. If k 2 is not a Dirichlet eigenfrequency then the Steklov-Poincare operator is well defined, bounded and fulfills a Garding inequality. If k 2 is not a Neumann eigenfrequency then the Steklov-Poincare operator is invertible. 6 / 38

7 Formulation of the domain decomposition approach Find u H 1 (Ω) such that u i k 2 u i = 0 in Ω i, γ N u i = h on Γ N, γ N u i = γ N u j on Γ ij where u i = u Ωi. By using the Steklov-Poincare operator, it is possible to reduce the problem to functions defined on the boundary: Find γ D u H 1/2 (Γ S ) such that S i γ D u Γi + S j γ D u Γj =0 on Γ ij S i γ D u = h on Γ N Γ I. 7 / 38

8 Formulation of the domain decomposition approach Find γ D u i H 1/2 (Γ i ) such that S i γ D u i + S j γ D u j = 0 on Γ ij, γ D u i =γ D u j on Γ ij, If we rewrite the interface conditions we get: γ N u = h on Γ N. γ N u i + γ N u j =0 on Γ ij, γ D u i γ D u j =0 on Γ ij This can be replaced by a Dirichlet-Robin type coupling condition [Després, 1990, Gander, 2002]: (γ N u i + ikpγ D u i ) + (γ N u j ikpγ D u j ) =0 on Γ ij, γ D u i γ D u j =0 with an elliptic operator P : H 1/2 (Γ ij ) H 1/2 (Γ ij ). on Γ ij 8 / 38

9 Outline Model problem The Tearing and Interconnecting approach Preconditioning Outlook 9 / 38

10 Basic idea of Tearing and Interconnecting Ω 2 Ω 1 Ω 5 Ω 3 Ω 4 All degrees of freedom are global degrees of freedom (red) [Farhat and Roux, 1991] 10 / 38

11 Basic idea of Tearing and Interconnecting Ω 2 Ω 1 Ω 5 Ω 3 Ω 4 All degrees of freedom are global degrees of freedom (red) Now we tear the domains more degrees of freedom (but local ones) [Farhat and Roux, 1991] 10 / 38

12 Basic idea of Tearing and Interconnecting Ω 2 Ω 1 Ω 5 Ω 3 Ω 4 All degrees of freedom are global degrees of freedom (red) Now we tear the domains more degrees of freedom (but local ones) Now we connect corresponding d.o.f. again by Lagrangian multipliers [Farhat and Roux, 1991] 10 / 38

13 Tearing and Interconnecting Find γ D u H 1/2 (Γ S ) so that S i γ D u Γi + S j γ D u Γj = 0 on Γ ij, S i γ D u =h(x) on Γ N Γ ij. After a summation and discretization we end up with the linear system S h u = p A T i S h,i A i u = i=1 p A T i f i. i=1 Now we can apply the tearing and interconnecting idea and get: S h,1 B1 T u 1 f S h,p Bp T u p =. f p. B 1... B p 0 λ 0 11 / 38

14 Dirichlet-to-Robin map After eliminating the primal degrees of freedom we get Remarks: p i=1 B 1 S 1 h,i BT i λ = rhs. S i may be not well defined S h,i may be not invertible Idea: Using Robin boundary conditions instead of Neumann boundary conditions for coupling [Benamou and Desprès, 1997]: γ N u i + ikpγ D u i = g on Γ R This leads to the formulation (S i + ikp)γ D u Γi + (S j ikp)γ D u Γj =0 on Γ ij S i γ D u = h on Γ N Γ i. 12 / 38

15 Dirichlet-to-Robin map As in the Dirichlet to Neumann case, this leads to p A T i S h,i A i u = i=1 p A T i f i. But the discrete splitting leads to S h,1 ± ikp h,1 B1 T u 1 f S h,p ± ikp h,p Bp T u p =. f p. B 1... B p 0 λ 0 The local mapping operators S j ± ikp j are injective and fulfill a Garding inequality, so the local matrices S h,j ± ikp h,j are invertible if h is small enough. i=1 13 / 38

16 Robin-System In general we have to solve local systems like (S + ip)u = (D + ip)u + ( 1 2 I + K )V 1 ( 1 I + K)u = rhs (1) 2 but this operator is not well defined if k 2 is a local Dirichlet eigenvalue. Instead we can solve the system ( D + ip 1 2 I + K ) ( ( ) u rhs v) 1 2 I K V =. 0 Properties: Equivalent to system (1) if k 2 is not a Dirichlet eigenvalue Garding inequality + surjective uniquely solvable Covers natural properties of the boundary value problem 14 / 38

17 Robin-System Global problem: Find γ D u H 1/2 (Γ S ) and γ N u j H 1/2 (Γ j ) such that p ( Dj + ip j ( 1 2 I + K ) ( ) ( ) ) j γd u Γj γd v ( 1 2 I + K), Γj j V j γ N u j γ N v j j=1 = p j=1 ( ) ( ) rhsj γd v, Γj 0 γ N v j for all γ D v H 1/2 (Γ S ) and γ N v j H 1/2 (Γ j ) Properties: Fulfills a Garding inequality Injective 15 / 38

18 Robin-System After Tearing and Interconnecting (with 1 2 I +K = K and D +ip = D) we get ( ) ( ) ( ) D h K h B T 10 γd u ( ) 1 f K h V h 1 γ N u ( ) ( ). D h K h B T ( ) p γd u =. ( ). p f p K h V h 0 ( p B1 0 ) (... Bp 0 ) γ N u p 0 0 λ 0 After eliminating the primal degrees of freedom we get Fλ = p i=1 ( Bi 0 ) ( D h K h K h V h ) 1 i ( B T i 0 ) λ = rhs. 16 / 38

19 Example Neumann boundary value problem: Domain: u(x) k 2 u(x) = 0 for x Ω, γ N u(x) = h(x) for x Γ. Ω 0 Ω 1 Frequencies: k = 2.0 and k = (first eigenfrequency of the unit cube). 17 / 38

20 Robin-System Frequency: 2.0 Frequency: N i M i GMRES rel. error u 1 u 1,h L2(Γ 1) N i M i GMRES rel. error u 1 u 1,h L2(Γ 1) / 38

21 Example Neumann boundary value problem: Domain: Unit cube u(x) k 2 u(x) = 0 for x Ω, γ N u(x) = h(x) for x Γ. Frequencies: k = / 38

22 Robin-System p N i M i GMRES u 1 u 1,h L2 (Γ 1 ) / 38

23 Outline Model problem The Tearing and Interconnecting approach Preconditioning Outlook 21 / 38

24 Local preconditioning To precondition the system ( Dh + ip A = h ( 1 2 M ) h + K h ) ( 1 2 M h + K h ) V h we use a block diagonal preconditioner based on on the idea of operators of opposite order [Steinbach and Wendland, 1998]: with C 1 A ( M 1 = 1,h V h M 1 1,h M 1 0,h D h M 1 0,h ) V h [i, j] = Vψ i, ψ j D h[i, j] = Vφ i, φ j ψ i, ψ j S 1 h (Γ), φ i, φ i S 0 h(γ). 22 / 38

25 Local preconditioning We solve now C 1 A A = rhs on the unit cube. Iteration numbers in dependency of k and h. N i M i Remark: Iteration numbers depend on the domain Ω 23 / 38

26 Global Preconditioning Idea from FETI-H method [Farhat et al., 2000] The residual r = d Fλ should be orthogonal to a given vector space in each step: Q t r = 0. (2) This is also a solution constraint: r = d Fλ = n B i u i. i=1 So r is the jump of the solution on the interface. 24 / 38

27 Global Preconditioning Introduce new iterate λ = λ + µ = λ + Qγ (3) where γ is chosen such that Q t r = 0. Inserting (3) in (2) leads to Q t FQγ = Q t (d Fλ). Solving this equation and inserting it in (3) leads to λ = Pλ + λ with P = I Q(Q t FQ) 1 Q t F, λ = Q(Q t FQ) 1 Q t d. 25 / 38

28 Global Preconditioning Replacing λ by λ in Fλ = d leads to FPλ + Fλ = d By multiplying this equation with P t we get P t FPλ = P t d since P t Fλ = 0. Finally we have to solve this and the coarse system Q t FQγ = Q t d Remark: P t FP = FP 26 / 38

29 Local planar waves Planar waves: Column-wise construction of Q: Q j ( ) = e ikθt i x( ) with Q j j-th column of Q θ j j-th unitary direction vector x( ) vector of nodal point coordinates evaluated at ( ) Local waves: The same approach on subdomain level Q = [Q 1 Q s Q Ns ] 27 / 38

30 Domain: Unit cube (0, 1) 3 Frequency: k = 2.0 Numerical example Iteration number Number of subdomains Number of directions per subdomain Refinement level 28 / 38

31 Domain: Unit cube (0, 1) 3 Frequency: k = 4.0 Numerical example Iteration number Number of subdomains Number of directions per subdomain Refinement level 29 / 38

32 Domain: Unit cube (0, 1) 3 Frequency: k = 8.0 Numerical example Iteration number Number of subdomains Number of directions per subdomain Refinement level 30 / 38

33 Outline Model problem The Tearing and Interconnecting approach Preconditioning Outlook 31 / 38

34 Helmholtz-Maxwell Helmholtz Maxwell Equation scalar vectorial Operator k 2 I curlcurl k 2 I S k,d k Garding inequality generalized Garding inequality Energy space H 1 (Ω) H(curl,Ω) Dirichlet trace space H 1/2 (Γ) H 1/2 (curl Γ, Γ) Neumann trace space H 1/2 (Γ) H 1/2 (div Γ, Γ) Discretization Ph 0 and P1 h Raviart-Thomas elements 32 / 38

35 Very first Maxwell example Local problem Number of iterations to solve ( 1 D h 2 M ) ( ) h + K h x ( 1 2 M = rhs h + K h ) V h y with given right hand side rhs on the unit cube (0, 1) 3 for various frequencies / 38

36 Very first Maxwell example Neumann boundary value problem: curlcurlu(x) k 2 u(x) = 0 for x Ω, γ N u(x) = h(x) for x Γ. Domain: Unit cube (0, 1) 3, divided in p 3 subcubes it Table: 8 domains it Table: 27 domains No preconditioning No Robin interfaces 34 / 38

37 Very first Maxwell example Neumann boundary value problem: curlcurlu(x) k 2 u(x) = 0 for x Ω, γ N u(x) = h(x) for x Γ. Domain: Unit cube (0, 1) 3, divided in p 3 subcubes er Table: 8 domains er Table: 27 domains No preconditioning No Robin interfaces 35 / 38

38 Outlook Analysis for the Maxwell case Numerical Analysis for the Maxwell case Preconditioning for Maxwell Implementing remaining parts for Maxwell Preprint 2009/05. O. Steinbach, M. Windisch Stable Boundary element domain decomposition methods for the Helmholtz problem Proceedings DD19. O. Steinbach, M. Windisch Robust BE Domain Decomposition Methods in Acoustics 36 / 38

39 Benamou, J.-D. and Desprès, B. (1997). A domain decomposition method for the helmholtz equation and related optimal control problems. J. Comput. Phys., 136(1): Després, B. (1990). Décomposition de domaine et problème de Helmholtz. C. R. Acad. Sci. Paris Sér. I Math., 311(6): Farhat, C., Macedo, A., and Lesoinne, M. (2000). A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems. Numer. Math., 85(2): Farhat, C. and Roux, F.-X. (1991). A method of finite element tearing and interconnecting and its parallel solution algorithm. 32(6): / 38

40 Gander, M. J. (2002). Optimized Schwarz methods for Helmholtz problems. In Domain decomposition methods in science and engineering (Lyon, 2000), Theory Eng. Appl. Comput. Methods, pages Internat. Center Numer. Methods Eng. (CIMNE), Barcelona. Steinbach, O. and Wendland, W. L. (1998). The construction of some efficient preconditioners in the boundary element method. Adv. Comput. Math., 9(1-2): Numerical treatment of boundary integral equations. 38 / 38