Dual-Primal Isogeometric Tearing and Interconnecting Solvers for Continuous and Discontinuous Galerkin IgA Equations

Transcription

1 Dual-Primal Isogeometric Tearing and Interconnecting Solvers for Continuous and Discontinuous Galerkin IgA Equations Christoph Hofer and Ulrich Langer Doctoral Program Computational Mathematics Numerical Analysis and Symbolic Computation Johannes Kepler University, Linz Hofer et. al JKU Linz 1 / 23

2 Introduction IETI-DP dg-ieti-dp Numerical examples Conclusion Hofer et. al JKU Linz 2 / 23

3 Introduction : Overview Introduction IETI-DP dg-ieti-dp Numerical examples Conclusion Hofer et. al JKU Linz 2 / 23

4 Introduction : Motivation What is the Dual Primal IsogEometric Tearing and Interconnecting (IETI-DP) method? adaption of Dual Primal Finite Element Tearing and Interconnecting (FETI-DP) for IgA non overlapping domain decomposition method Why we need it? fast solvers parallelizeable solvers robust solvers w.r.t. jumping coefficients Hofer et. al JKU Linz 3 / 23

5 Introduction : State of the Art Pechstein, C. (2012). Finite and boundary element tearing and interconnecting solvers for multiscale problems (Vol. 90). Springer Science & Business Media. Beirao da Veiga, L., Cho, D., Pavarino, L. F., Scacchi, S. (2013). BDDC preconditioners for isogeometric analysis. Mathematical Models and Methods in Applied Sciences, 23(06), Dryja, M., Galvis, J., Sarkis, M. (2013). A FETI-DP preconditioner for a composite finite element and discontinuous Galerkin method. SIAM Journal on Numerical Analysis, 51(1), Dryja, M., Sarkis, M. (2014). 3-D FETI-DP preconditioners for composite finite element-discontinuous Galerkin methods. In Domain Decomposition Methods in Science and Engineering XXI (pp ). Springer International Publishing. Hofer et. al JKU Linz 4 / 23

6 Introduction : Isogeometric Discretization The computational domain Ω can be represented as Ω := N Ω (k), where Ω (k) = G (k) (ˆΩ), ˆΩ = (0, 1) d, k=0 with Γ (k,l) := Ω (k) Ω (l) and G (k) (ξ) = i I P (k) i N i,p (ξ). Hofer et. al JKU Linz 5 / 23

7 Introduction : Examples Abbildung : B-Splines with p = 2 and knot vector Ξ = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5} 1 1 Book: Isogeometric Analysis, Towards Integration of CAD and FEA, J. A. Cotrell and T. J. R. Hughes and Y. Bazilevs Hofer et. al JKU Linz 6 / 23

8 Introduction : Remarks dim 2: N i,p is obtained via a tensorproduct using Ňi,p(x) := N i,p G 1 (x) as basis for discretization Ň i,p (x) is in general not interpolatory dim = 1: Ň i,p is interpolatory at G((0, 1)) dim = 2: Ň i,p is interpolatory at the corner points, and it is possible to associate certain N i,p to G((0, 1) 2 ) Hofer et. al JKU Linz 7 / 23

9 Introduction : Problem formulation - cg setting Find u h V D,h : a(u h, v h ) = F, v h v h V D,h, where V D,h is a conforming discrete subspaces of V D, e.g. a(u, v) = α u v dx, F, v = fv dx + g N v ds Ω Ω Γ N V D = {u H 1 : γ 0 u = 0 on Γ D }, V D,h = k span{n i,p G (k) 1 } H 1 (Ω). The variational equation is equivalent to Ku = f. Hofer et. al JKU Linz 8 / 23

10 IETI-DP : Overview Introduction IETI-DP dg-ieti-dp Numerical examples Conclusion Hofer et. al JKU Linz 8 / 23

11 IETI-DP : Hofer et. al JKU Linz 9 / 23

12 IETI-DP : Lagrange Multipliers Hofer et. al JKU Linz 10 / 23

13 IETI-DP : IETI-DP Given K (k) and f (k), we can reformulate [ Ke B T ] [ ] ue Ku = f = B 0 λ where K e = diag(k (k) ) and f e = [f (k) ]. [ ] f e, 0 Since K e is not invertible, we need additional primal variables incorporated in K e (partial assembly) K e, B, f e : continuous vertex values continuous edge/face averages Hofer et. al JKU Linz 11 / 23

14 IETI-DP : IETI-DP Given K (k) and f (k), we can reformulate [ Ke B T ] [ ] ue Ku = f = B 0 λ where K e = diag(k (k) ) and f e = [f (k) ]. [ ] f e, 0 Since K e is not invertible, we need additional primal variables incorporated in K e (partial assembly) K e, B, f e : continuous vertex values continuous edge/face averages Hofer et. al JKU Linz 11 / 23

15 IETI-DP : Introducing Primal Variables Hofer et. al JKU Linz 12 / 23

16 IETI-DP : Introducing Primal Variables Find (u, λ) RÑ U : ] [u ] [ Ke BT = B 0 λ K e is SPD, hence, we can define: ] [ f e. 0 F := 1 B K e B T d := 1 B K e f e The saddle point system is equivalent to solving: find λ U : F λ = d. Hofer et. al JKU Linz 13 / 23

17 IETI-DP : Theorem Let H k be the diameter and h k the local mesh size of Ω (k). Let the Scaled Dirichlet preconditioner M 1 sd be defined as: M 1 sd = B DS e B T D Then, under suitible assumption on the mesh, we have ( κ(m 1 sd F ker( B T ) C max 1 + log ) k where the constant C is independent of H k, h k. ( Hk h k )) 2, Hofer, C., Langer U.: DP-IETI Solvers for large-scale systems of multipatch cg IgA equations. RICAM-Report Beirao da Veiga, L., Cho, D., Pavarino, L. F., Scacchi, S. (2013). BDDC preconditioners for isogeometric analysis. Mathematical Models and Methods in Applied Sciences, 23(06), Hofer et. al JKU Linz 14 / 23

18 dg-ieti-dp : Overview Introduction IETI-DP dg-ieti-dp Numerical examples Conclusion Hofer et. al JKU Linz 14 / 23

19 dg-ieti-dp : Problem formulation - dg setting Find u V h := k span{n i,p G (k) 1 }, such that a dg (u, v) = F, v v V h, where N N a dg (u, v) := a k (u, v), F, v := fv k dx, k=1 k=1 Ω k a k (u, v) = α k u k v k dx Ω k + ( α k uk l E Γ (k,l) 2 n (v l v k ) + v ) k n (u l u k ) ds k + µα k l E Γ (k,l) h (u l u k )(v l v k )ds, k Hofer et. al JKU Linz 15 / 23

20 dg-ieti-dp : Illustration of multipatch dg Hofer et. al JKU Linz 16 / 23

21 dg-ieti-dp : Decoupling of interface dofs Hofer et. al JKU Linz 17 / 23

22 dg-ieti-dp : Lagrange multipliers for decoupled dofs Hofer et. al JKU Linz 18 / 23

23 dg-ieti-dp : Remarks the dg-ieti-dp method can be seen as a cg-ieti-dp method on an extended grid principle can be extended to 3D: V + E + F scaled Dirichlet preconditioner: numerical examples also show quasi-optimal behavior w.r.t. H/h = max{h i /h i } and robustness w.r.t. jumps in α. κ(m 1 sd F ) C(1 + log(h/h))2 Hofer, C., Langer, U. (2016). Dual-primal isogeometric tearing and interconnecting solvers for multipatch dg-iga equations, Computer Methods in Applied Mechanics and Engineering. for finite elements: proven for 2D and 3D Dryja, M., Galvis, J., Sarkis, M. (2013). A FETI-DP preconditioner for a composite finite element and discontinuous Galerkin method Dryja, M., Sarkis, M. (2014). 3-D FETI-DP preconditioners for composite finite element-discontinuous Galerkin methods Hofer et. al JKU Linz 19 / 23

24 Numerical examples : Overview Introduction IETI-DP dg-ieti-dp Numerical examples Conclusion Hofer et. al JKU Linz 19 / 23

25 Numerical examples : Numerical example (a) Initial mesh of 2D computational domain (b) Initial mesh of 3D computational domain (c) Jumping coefficient pattern Hofer et. al JKU Linz 20 / 23

26 Numerical examples : 2D/3D domain, α {10 4, 10 4 }, p = 2 2D 3D V V #dofs H/h κ It. #dofs H/h κ It V + E V + E + F Hofer et. al JKU Linz 21 / 23

27 Numerical examples : Gaps & Overlaps with dg-ieti-dp, α = 1 d g h 1 2D 3D V + E V + E + F H/h #dofs κ It. #dofs κ It gap/overlap 0 if h 0 C. Hofer, U. Langer, and I. Toulopoulos (2015): Discontinuous Galerkin isogeometric analysis of elliptic problems on segmentations with gaps. RICAM-Report C. Hofer, and I. Toulopoulos (2015): Discontinuous Galerkin Isogeometric Analysis of Elliptic Problems on Segmentations with Non-Matching Interfaces. RICAM-Report C. Hofer, U. Langer, and I. Toulopoulos (2016): Discontinuous Galerkin Isogeometric Analysis on non matching segmentations: Error estimates and efficient solvers (under preparation) Hofer et. al JKU Linz 22 / 23

28 Conclusion : Overview Introduction IETI-DP dg-ieti-dp Numerical examples Conclusion Hofer et. al JKU Linz 22 / 23

29 Conclusion : Conclusion and further work quasi-optimal condition number estimate for IETI-DP adapted and implemented the IETI-DP algorithm for dg-iga formulation dg-ieti-dp for non-matching interface parametrizations (gaps & overlaps between patches) MPI is used to parallelize the method. proof for condition number bound of dg-ieti-dp is ongoing work. using inexact solvers for local problems (e.g. multigrid) dg-ieti-dp for space-time discretization Hofer et. al JKU Linz 23 / 23

30 Conclusion : Conclusion and further work quasi-optimal condition number estimate for IETI-DP adapted and implemented the IETI-DP algorithm for dg-iga formulation dg-ieti-dp for non-matching interface parametrizations (gaps & overlaps between patches) MPI is used to parallelize the method. proof for condition number bound of dg-ieti-dp is ongoing work. using inexact solvers for local problems (e.g. multigrid) dg-ieti-dp for space-time discretization Hofer et. al JKU Linz 23 / 23

31 Conclusion : Conclusion and further work Gap +Overlap: footprint test case 10 0 r=0.5 r=0.5 r=0.51 u u h dg r=1.52 r=1.50 r=2.1 r=1.50 r=2.1 r=2.0 r=2.07 r=2.0 r=2.03 d o+g =h 1 d o+g =h 2 d o+g =h 2.5 d o+g =h h u u h dg Ch min(s,β) ( u W l,2 + κ 2 ), where we assume that d g h λ, and s = min(p + 1, l) 1 and β = λ 1 2 Hofer et. al JKU Linz 23 / 23

32 Conclusion : p dependence: 2D + 3D & different multiplicity keeping multiplicity & increasing smoothness ( ) C p 1 elements increasing multiplicity & keeping smoothness ( ) C 1 elements 3.2 dependence on p (2D) 45 dependence on p (3D) coeff. scaling (increase mult.) coeff. scaling (increase mult.) 3 stiff. scaling (increase mult.) coeff. scaling (keep mult.) 40 stiff. scaling (increase mult.) coeff. scaling (keep mult.) 2.8 stiff. scaling (keep mult.) 35 stiff. scaling (keep mult.) condition number - κ condition number κ degree - p degree - p Hofer et. al JKU Linz 23 / 23

33 Conclusion : Conclusion and further work dim = 2 dim = 3 V coeff. V + E coeff. #dofs h i /h j H/h κ It. #dofs h i /h j H/h κ It V + E V + E + F Hofer et. al JKU Linz 23 / 23