Méthodes de raffinement espace-temps

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1 Méthodes de raffinement espace-temps Caroline Japhet Université Paris 13 & University of Maryland Laurence Halpern Université Paris 13 Jérémie Szeftel Ecole Normale Supérieure, Paris Pascal Omnes CEA, Saclay Journée GNR MOMAS / GDR Calcul, 5 Mai 2010 Journée MOMAS, 5 Mai / 36

2 Coupling process For a given problem, split the domain : domain decomposition Subdomains with curved interfaces and widely differing lengths Coupling heterogeneous models Different time and space steps in different subdomains Application to nuclear waste disposal (GdR MoMaS) Journée MOMAS, 5 Mai / 36

3 Far field 3D simulations of underground nuclear waste Research Group MoMaS With Jérôme Jaffré, Michel Kern and Jean Roberts (INRIA) Actual dimensions : 40km 40km 500m A blow-up in the vertical direction (30 times) The repository is located in the red part of the bottom layer. Journée MOMAS, 5 Mai / 36

4 Hydrogeological data Hydrogeologic Effective Permeability Dispersivity layers Thickness Porosity diffusion [m/s] Coefficients [m] [%] coefficient [m] [m 2 /s] Regional Local Tithonian Variable , 0.6 Kimmeridgian when it outcrops -9 Variable , 0.6 Kimmeridgian 10 under cover Oxfordian L2a-L2b , 0.6 Oxfordian Hp1-Hp , 30 Oxfordian C3a-C3b , 0.6 Callovo-Oxfordian Cox K v =10-14 K h = , 0.6 Journée MOMAS, 5 Mai / 36

5 Near-field Computation Research Group MoMaS With Pascal Omnes and Paul-Marie Berthe (CEA Saclay, DEN/DM2S/SFME/MTMS) Journée MOMAS, 5 Mai / 36

6 DDM for evolution problems The goals Different time and space steps in different domains, adapted to the physics, Different models in different subdomains, Easy to use, fast and cheap. The tools Work on the PDE level, globally in time, Use time windows for long time simulations, Use physical transmission conditions, transmit with optimized transmission condition, Robin transmission conditions Order 2 transmission conditions Then discretize separately. Journée MOMAS, 5 Mai / 36

7 DDM for evolution problems The goals Different time and space steps in different domains, adapted to the physics, Different models in different subdomains, Easy to use, fast and cheap. The tools Work on the PDE level, globally in time, Use time windows for long time simulations, Use physical transmission conditions, transmit with optimized transmission condition, Robin transmission conditions Order 2 transmission conditions Then discretize separately. Optimized Schwarz Waveform Relaxation Journée MOMAS, 5 Mai / 36

8 Heterogeneous advection-diffusion equation (Radionuclide transport) Lu = tu + (a(x)u ν(x) u) + cu = f in Ω [0, T ] u = 0 on Ω [0, T ], u(., 0) = u 0 in Ω a(x) Ωi = a i (x) and ν(x) Ωi = ν i (x), ν(x) > 0 the interface is placed on the jump of a, ν u 1 = u 2 y T t (ν 1 n1 a 1 n 1 )u 1 = ( ν 2 n2 + a 2 n 2 )u 2 Γ Lu 1 = f 1 Lu 2 = f 2 Ω 1 Ω 2 x 0 Journée MOMAS, 5 Mai / 36

9 Optimized Schwarz Waveform Relaxation Method (M. Gander, L. Halpern, F. Nataf (1998), V. Martin (2003), M. Gander, L. Halpern, M. Kern (2007), E. Blayo, L. Halpern, C. J. (2007), D. Bennequin, M. Gander, L. Halpern (2009), L. Halpern, C. J., J. Szeftel (2010) ) (ν 1 n1 a 1 n 1 + S 1,2 )u k 1 = ( ν 2 n2 + a 2 n 2 + S 1,2 )u k 1 2 y t T l+1 (ν 2 n2 a 2 n 2 + S 2,1 )u k 2 = ( ν 1 n1 + a 1 n 1 + S 2,1 )u k 1 1 T l Lu k 1 = f 1 Lu k 2 = f 2 Ω 1 Ω 2 x Choose S 1,2 and S 2,1 in order to optimize the convergence factor Journée MOMAS, 5 Mai / 36

10 Optimized Schwarz Waveform Relaxation Method S 1,2 = p 1,2 + q 1,2 ( t + τ 2 (r 1,2 s 1,2 τ 2 )) S 2,1 = p 2,1 + q 2,1 ( t + τ 1 (r 2,1 s 2,1 τ 1 )) (ν 1 n1 a 1 n 1 + S 1,2 )u k 1 = ( ν 2 n2 + a 2 n 2 + S 1,2 )u k 1 2 y t T l+1 (ν 2 n2 a 2 n 2 + S 2,1 )u k 2 = ( ν 1 n1 + a 1 n 1 + S 2,1 )u k 1 1 T l Lu k 1 = f 1 Lu k 2 = f 2 Ω 1 Ω 2 x where p 1,2, p 2,1, q 1,2, q 2,1 are taken such that They optimize the convergence factor, The transmission conditions imply the coupling conditions at convergence, The subdomain problems are well-posed. For the simulations : r i,j = a j τ j and s i,j = ν j Journée MOMAS, 5 Mai / 36

11 Optimized Schwarz Waveform Relaxation Method S 1,2 = p 1,2 + q 1,2 ( t + τ 2 (r 1,2 s 1,2 τ 2 )) S 2,1 = p 2,1 + q 2,1 ( t + τ 1 (r 2,1 s 2,1 τ 1 )) where p 1,2, p 2,1, q 1,2, q 2,1 optimize the convergence factor (ν 1 n1 a 1 n 1 + S 1,2 )u k 1 = ( ν 2 n2 + a 2 n 2 + S 1,2 )u k 1 2 y t T l+1 (ν 2 n2 a 2 n 2 + S 2,1 )u k 2 = ( ν 1 n1 + a 1 n 1 + S 2,1 )u k 1 1 T l Lu k 1 = f 1 Lu k 2 = f 2 Ω 1 Ω 2 x Journée MOMAS, 5 Mai / 36

12 Optimized Schwarz Waveform Relaxation Method S 1,2 = p 1,2 + q 1,2 ( t + τ 2 (r 1,2 s 1,2 τ 2 )) S 2,1 = p 2,1 + q 2,1 ( t + τ 1 (r 2,1 s 2,1 τ 1 )) where p 1,2, p 2,1, q 1,2, q 2,1 optimize the convergence factor (ν 1 n1 a 1 n 1 + S 1,2 )u k 1 = ( ν 2 n2 + a 2 n 2 + S 1,2 )u k 1 2 y t T l+1 (ν 2 n2 a 2 n 2 + S 2,1 )u k 2 = ( ν 1 n1 + a 1 n 1 + S 2,1 )u k 1 1 T l Lu k 1 = f 1 Lu k 2 = f 2 Ω 1 Ω 2 x How to discretize these conditions with nonmatching space-time grids? Journée MOMAS, 5 Mai / 36

13 Optimized Schwarz Waveform Relaxation Method S 1,2 = p 1,2 + q 1,2 ( t + τ 2 (r 1,2 s 1,2 τ 2 )) S 2,1 = p 2,1 + q 2,1 ( t + τ 1 (r 2,1 s 2,1 τ 1 )) where p 1,2, p 2,1, q 1,2, q 2,1 optimize the convergence factor (ν 1 n1 a 1 n 1 + S 1,2 )u k 1 = ( ν 2 n2 + a 2 n 2 + S 1,2 )u k 1 2 y t T l+1 (ν 2 n2 a 2 n 2 + S 2,1 )u k 2 = ( ν 1 n1 + a 1 n 1 + S 2,1 )u k 1 1 T l Lu k 1 = f 1 Lu k 2 = f 2 Ω 1 Ω 2 x How to discretize these conditions with nonmatching space-time grids? Time discontinuous Galerkin method Journée MOMAS, 5 Mai / 36

14 Subdomain solver : Time Discontinuous Galerkin (K. Eriksson, C. Johnson, V. Thomée (1985), L. Halpern, C. J. (2008)) Problem in a subdomain Ω i, in one time window I = (T l, T l+1 ) 8 < tu + (au ν u) + cu = f in Ω i I, u(, T l ) = u 0 in Ω i, : (ν n a n + p + q( t + τ (r s τ )) u = g on Γ I Let H σ σ (Ω i ) = {v H σ (Ω i ), v Γ H σ (Γ)}, σ > 1 2, and ((u, v)) = (u, v) L 2 (Ω i ) + q(u, v) L 2 (Γ) Weak formulation : Find u such that with Z a(u, v) = Ω i 1 (( t u, v)) + a(u, v) = l(v), v H 1 1 (Ω i ) Z Z 2 ((a u)v (a v)u) dx + ν u v dx + (c + 1 Ω i Ω i 2 a)uv dx Z a n + `(p Γ 2 + q 2 τ r)uv + q 2 ( τ (ru)v τ (rv)u) + qs τ u τ v dσ, l(v) = (f, v) L 2 (Ω i ) + (g, v) L 2 (Γ) Journée MOMAS, 5 Mai / 36

15 Discontinuous Galerkin time stepping method Let T be a partition of I = N n=0i n, with I n = (t n, t n+1 ], and t n = t n+1 t n. Let ϕ(t ± n ) = lim t tn±0 ϕ(t). We define P q(v ) = {ϕ : ϕ(t) = qx ϕ i t i, ϕ i V } P q(v, T ) = {ϕ : I V, ϕ In P q(v ), 0 n K }. i=0 The discontinous Galerkin method defines recursively on I n, an approximate solution U in P q(h1 1 (Ω i ), T ) such that 8 U(0, ) = u 0, Z >< ϕ P q(h1 1 (Ω i ), T ) : ( (( du, ϕ)) + a(u, ϕ) ) dt I n dt Z >: +(( U(, t n + ), ϕ(, t n + ) )) = (( U(, t n), ϕ(, t n + ) )) + l(ϕ)dt I n Journée MOMAS, 5 Mai / 36

16 Discontinuous Galerkin time stepping method (C. Makridakis, R. Nochetto (2006)) Let 0 < ξ 1,..., ξ q+1 = 1, the Gauss-Radau points such that Z 1 0 q+1 X f (t)dt w j f (ξ j ) is exact in P 2q, and the interpolation operator I n on [t n, t n+1 ] at points (t n, t n + ξ 1 t n,..., t n + ξ q+1 t n) with I nu(t n) = U(t n ). Let I : P q(h 1 1 (Ω i ), T ) P q+1 (H 1 1 (Ω i ), T ) be the operator whose restriction to each subinterval is I n. It satisfies IU(t + n ) = U(t n ). j=1 U IU ξ 1 U IU t 0 t n t n+1 t N+1 t 0 t n t n+1 t N+1 Journée MOMAS, 5 Mai / 36

17 Discontinuous Galerkin time stepping method The discrete equation can be rewritten as Z ( (( diu Z, V )) + a(u, V )) dt = L(V ) dt I n dt I n or in the strong formulation t(iu) + (au ν U) + cu = Pf in Ω i I `ν n a n U + p U + q( t(iu) + τ (ru s τ U)) = Pg on Γ I where P is the L 2 projection in time on P q(v, T ) Journée MOMAS, 5 Mai / 36

18 Time Nonconforming Domain Decomposition Method Let T i be the partition of the time window (T l, T l+1 ) in subdomain Ω i, with N i + 1 intervals I i n. We define interpolation operators I i and projection operators P i. Let m 1 small in order to make very few iterations in each time window. Let Vi m be a discrete approximation of u i in Ω i (T l 1, T l ) at step m of the method. Then, the next time window s solution U i is obtained after m DG-OSWR iterations, starting with a given initial guess (g i,j ) in P d (L 2 (Γ i,j ), T i ) t (I i Ui k ) + (aui k ν i Ui k ) + c i Ui k = P i f in Ω i (0, T ), `νi ni a i n i Uk i + S i,j Ui k = with U k i (, T l ) = V m i (, T l ), P i `(ν j ni a j n i ) U k 1 j + S e i,j U k 1 j on Γi,j (0, T ), S i,j U = p i,j U + q i,j ( t (I i U) + τ (r i,j U s i,j τ U)) es i,j U = p i,j U + q i,j ( t (I j U) + τ (r i,j U s i,j τ U)) Journée MOMAS, 5 Mai / 36

19 Theorem (L. Halpern, C. J., J. Szeftel (DD18,2009)) (Well-posedness and convergence) The coupled discret problem in time has a unique solution and the discret Schwarz algorithm is convergent for q i,j = 0, p j,i p i,j a i n i = 0, p i,j > a i n i 2, ν i ν j, a i a j, and rectangular subdomains p i,j = p j,i = p > 0, q i,j = q j,i = q > 0, ν i = ν j, s i,j = s > 0, a i = 0, r i,j = 0, and strips (Error estimates) If a = 0, p i,j a i n i 2 = p j,i a j n j 2 = p > 0, q i,j = 0, t = max n t n, then IX i=1 u U i 2 L (0,T,L 2 (Ω i )) C t 2(q+1) q+1 t u 2 L 2 (0,T ;H 2 (Ω)). Theorem (L. Halpern, C. J., J. Szeftel (DD19,2010)) (Well-posedness) For a i, ν i, r i,j, s i,j, p i,j and q i,j sufficiently smooth with q i,j 0, s i,j > 0 a.e., the subdomain problem has a unique solution. (Convergence) The continuous Schwarz algorithm converges for q i,j = q j,i = 0, p i,j p j,i, ν i ν j, a i a j and general decomposition if q i,j = q j,i = q > 0, p i,j p j,i, ν i ν j, a i a j and decomposition into bands. The proof relies on energy estimates and Gronwall Lemma. Generalization to general geometries and piecewise smooth coefficients of results in D. Bennequin, M. Gander, L. Halpern (2009) Journée MOMAS, 5 Mai / 36

20 Space discretization with nonconforming space grids (M. Gander, C. J., Y. Maday, F. Nataf (2004)) For stationary elliptic problems and Robin transmission conditions Finite element space : V = Q K i=1 X i h Ω 1 Ω 2 Basis functions of X h i Basis functions of W h i Z Z (ν 1 n1 a 1 n 1 + S 1,2 )u 1 ψ 1 = ( ν 2 n2 + a 2 n 2 + S 1,2 )u 2 ψ 1, ψ 1 Wh 1 ZΓ ZΓ (ν 2 n2 a 2 n 2 + S 2,1 )u 2 ψ 2 = ( ν 1 n1 + a 1 n 1 + S 2,1 )u 1 ψ 2, ψ 2 Wh 2 Γ Γ Extension to time advection-diffusion problems and order 2 transmission conditions Journée MOMAS, 5 Mai / 36

21 An efficient way to perform the projections between space-time grids (Martin J. Gander, C. J. (2009)) Generalization in 2d and 3d of the algorithm in M. Gander, L. Halpern, F. Nataf (2003) y T l+1 t Ω i Ω j x Let Vk i (resp. V j l ) the shape Z functions of Pq(R, T i) (resp. P q(r, T j )) How to compute M k,l = VkV i j l? I T l Algorithm with linear complexity and without an additional grid gander I i r I j k Journée MOMAS, 5 Mai / 36

22 Numerical results. Extension to a transport equation with discontinuous porosity ω Lu = ω u t + (a(x)u ν(x) u) + cu = f Initial condition u 0 (x) = 1 2 e( 100((x 0.7)2 +(y 1.5) 2 )) a 1 = ( sin( π 2 (y 1))cos(π(x 1 2 )), 3cos( π 2 (y 1))sin(π(x 1 2 ))), ν 1 = 0.003, ω 1 = 0.1, a 2 = a 1, ν 2 = 0.01, ω 2 = 1, final time T = 1.5 Boundary conditions : homogeneous Dirichlet (S,W), Neumann (N,E) Velocity field DG OSWR Solution, At time t=t= y 1 y x x 0 Journée MOMAS, 5 Mai / 36

23 Time nonconforming DG-OSWR solution (after 5 iterations) Domain 1 (left), 180 time steps, Domain 2 (right), 100 time steps On the right : error with the one domain solution Error at time t=t=1.5 DG OSWR solution at time t=t=1.5 5 x y y x x 1 0 Journée MOMAS, 5 Mai / 36

24 Space-time nonconforming DG-OSWR solution (after 5 iterations) Domain 1 (left), mesh size h 1 = 1/66, 180 time steps Domain 2 (right), mesh size h 2 = 1/44, 100 time steps Journée MOMAS, 5 Mai / 36

25 Relative error versus the time step One domain : t = domain 1 : t 1 = 1 120, domain 2 : t 2 = 1 26 then divide by 2 the time steps t 1 and t 2 The converged solution is such that the residual is smaller than Slope 2 L (0,T,L 2 (Ω 1 )) Error L (0,T,L 2 (Ω 2 )) Error Slope 3 L 2 (Ω 1 ) Error at time t=t L 2 (Ω 2 ) Error at time t=t 10 5 Error Time step Journée MOMAS, 5 Mai / 36

26 Relative error versus the mesh size and time step One domain : t = 1, h = domain 1 : t 1 = 1 60, h 1 = 0.056, domain 2 : t 2 = 1 20, h 2 = 0.11 then divide by 2 the time steps and mesh sizes 10 0 Slope 2 Relative L 2 error, Domain 1 Relative L 2 error, Domain Slope 2 L 2 (Ω 1 ) error at t=t L 2 (Ω 2 ) error at t=t Slope 1 H 1 (Ω 1 ) error at t=t H 1 (Ω 2 ) error at t=t Error 10 2 Error Time step Mesh size Journée MOMAS, 5 Mai / 36

27 Application to Porous Media With Pascal Omnes (CEA) Time domain : (0, T ), T = s 10 4 years time steps : 10 9 s in the clay (host rock) s in the repository mesh size : h = 0.15 m in the repository h = 2 m in the clay Isotropic case with ν = m 2 /s, ω = 0.06 in the clay ν = m 2 /s, ω = 1 in the repository computational domain (in meters) : (0, 10) (0, 100) for the clay (4.5, 5.5) (49.5, 50.5) for the repository Initial condition : u 0 = 0 in the clay u 0 = 1 in the repository Boundary conditions : homogeneous Neumann at x = 0 and x = 10, homogeneous Dirichlet at y = 0 and y = 100. Journée MOMAS, 5 Mai / 36

28 Computational domain y x Journée MOMAS, 5 Mai / 36

29 Computational domain (zoom) y 50 y x x Journée MOMAS, 5 Mai / 36

30 Darcy flow a = 0 a = K h with K = k Id, and k = 10 8 in the repository and k = in the clay Boundary conditions : homogeneous Neumann at x = 0 and x = 10 and Dirichlet conditions with h = 100 at y = 0 and h = 0 at y = y x Journée MOMAS, 5 Mai / 36

31 Time nonconforming DG-OSWR solution With 10 time windows and 4 iterations per window (Left : DG-OSWR solution. Right : error with the one domain solution) x y 50 y x x 0 Journée MOMAS, 5 Mai / 36

32 Time nonconforming DG-OSWR solution (repository) With 10 time windows and 4 iterations per window (Left : DG-OSWR solution. Right : error with the one domain solution) DG OSWR solution, At time t=t= Error between monodomain and DG SWR solutions, At time t=t=1 x y y x x Journée MOMAS, 5 Mai / 36

33 Relative error versus the time step One domain : t = domain 1 : t 1 = 1 128, domain 2 : t 2 = 1 28 then divide by 2 the time steps t 1 and t 2 The converged solution is such that the residual is smaller than k 2 Monodomain, Fine Grid Monodomain, Coarse grid DG OSWR, Repository DG OSWR, Host Rock Slope 3 DG OSWR, Repository DG OSWR, Host Rock Monodomain Error 10 3 Error Number of time refinement Time step Journée MOMAS, 5 Mai / 36

34 Choice of the parameters : an example in 1d (The Order 2 conditions reduce to S i,j = p i,j + q i,j t) L. Halpern, C. J., P. Omnes (ECCOMAS CFD, 2010) Optimization of the convergence factor : Fourier analysis + domains with highly variable length (classical analysis : two half-spaces case). Ω 1 = (0, ), Ω 2 = (0.4965, ) (repository), Ω 3 = (0.5035, 1), final time T = We take u 0 = 1 in the repository, u 0 = 0 in the clay, f = 0, c = 0, a = 1, ν 2 = 1, ω 2 = 1, ν 1 = ν 3 = 0.06, ω 1 = ω 3 = 0.06, t 2 = T 500, t 1 = t 3 = T 100, h 2 = and h 20 1 = h 3 = Error Robin, 2 domains Robin 2 sided, 2 domains Order 1, 2 domains Robin, 3 domains Robin 2 sided, 3 domains Order 1, 3 domains Number of iterations Journée MOMAS, 5 Mai / 36

35 Convergence in the 2d conforming case Constant parameters p 1,2 = p 2,1 = p, q 1,2 = q 2,1 = q (mean value of the (p, q) obtained by optimization of the convergence factor) (p,q), 3 domains optimization (p,q): numerical value (2 domains minimization) (p,q): numerical value (3 domains minimization) p Error q Number of iterations Journée MOMAS, 5 Mai / 36

36 Work in progress Extension to the MoMaS approach (Mixte Finite Elements), 3d benchmark (with Jérôme Jaffré, Michel Kern and Jean Roberts) Extension to finite volume methods and general decomposition (with corners) for Order 2 transmission conditions, 3d simulations (with P-M. Berthe and P. Omnes) Analysis of the fully discrete method Analysis of the influence of the decomposition in time windows Numerical and mathematical analysis of the convergence factor (with M.J. Gander) Journée MOMAS, 5 Mai / 36

37 Nonoverlapping algorithms t u ν u + f (u) = 0. f C 2, f (0) = 0. F. Caetano, M. Gander, L.H, J. Szeftel, DD19. Well-posedness and convergence for linear Robin TC (common existence time for all iterates by energy estimates) to appear in NTMC 09. New : nonlinear TC (DD19) Robin : B i (u) = u n i + p(u)u. Order 2 : B i (u) = u n i + p(u)u + q(u)( t u ν y u). p(u) = p ( 0, ν, f (u)), p and q asymptotic formulas. q(u) = q ( 0, ν, f (u)) Journée MOMAS, 5 Mai / 36

38 A simple example f (u) = 10(exp(u) 1), two subdomains, T = Nonlinear O2 Nonlinear Robin O2 p heat Robin p heat 10 5 Error Number of iterations Convergence history. Journée MOMAS, 5 Mai / 36

39 Overlapping algorithms TRAN Minh Binh. New study of well-posedness and convergence, Classical Schwarz algorithm for the semilinear heat equation, with explosion permitted : f is in C 1 (R) and f (x) C f x p 1, p 1 Extension to systems of stochastic differential equations. Journée MOMAS, 5 Mai / 36