A parallel iterative approach for the Stokes Darcy coupling

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1 A parallel iterative approach for the Stokes Darcy coupling Marco Discacciati*, Alfio Quarteroni**, Alberto Valli*** * Institute of Analysis and Scientific Computing, EPFL ** Institute of Analysis and Scientific Computing, EPFL, and MOX, Polytechnic of Milan *** Department of Mathematics, University of Trento A parallel iterative approach for the Stokes Darcy coupling p.1/20

2 Stokes Darcy coupling The coupling of the Stokes system and the Darcy equation has been recently considered as a model for the filtration of fluids through porous media (percolation of water from a basin through the ground). A parallel iterative approach for the Stokes Darcy coupling p.2/20

3 Stokes Darcy coupling The coupling of the Stokes system and the Darcy equation has been recently considered as a model for the filtration of fluids through porous media (percolation of water from a basin through the ground). Let us start describing the coupled problem. We denote by Ω S the fluid region, and by Ω D the ground region. Moreover, Γ := Ω S Ω D will be the interface between Ω S and Ω D. A parallel iterative approach for the Stokes Darcy coupling p.2/20

4 Stokes Darcy coupling (cont.) The Stokes system is given by { ν u + p = f in Ω S (1) divu = 0 in Ω S, where u is the velocity field, p is the pressure, ν > 0 is the kinematic viscosity and f is a given force field. A parallel iterative approach for the Stokes Darcy coupling p.3/20

5 Stokes Darcy coupling (cont.) The Stokes system is given by { ν u + p = f in Ω S (1) divu = 0 in Ω S, where u is the velocity field, p is the pressure, ν > 0 is the kinematic viscosity and f is a given force field. The Darcy equation is given by ( ) K (2) div N ϕ = 0 in Ω D, where ϕ is the piezometric head, K is the hydraulic conductivity tensor and N > 0 is the volumetric porosity. A parallel iterative approach for the Stokes Darcy coupling p.3/20

6 Stokes Darcy coupling (cont.) For simplicity, as boundary conditions let us assume that: (3) u = 0 on Ω S \ Γ ϕ = 0 on Ω D \ Γ. A parallel iterative approach for the Stokes Darcy coupling p.4/20

7 Stokes Darcy coupling (cont.) For simplicity, as boundary conditions let us assume that: (3) u = 0 on Ω S \ Γ ϕ = 0 on Ω D \ Γ. The interface conditions [matching of the normal velocity and of the normal stress] are: (4) u n = K N ϕ n on Γ T(u,p) n = gϕn on Γ where g is the gravity acceleration, and the fluid stress tensor is given by T ij (u,p) := ν(d i u j + D j u i ) pδ ij. Here n denotes the unit normal vector on Γ, pointing from Ω S into Ω D. A parallel iterative approach for the Stokes Darcy coupling p.4/20

8 Well-posedness and iterative solution algorithms It can be proved that the coupled problem (1) (4) has a unique solution [Discacciati and Quarteroni, ENUMATH 2001; Layton, Schieweck and Yotov, SINUM 2003]. A parallel iterative approach for the Stokes Darcy coupling p.5/20

9 Well-posedness and iterative solution algorithms It can be proved that the coupled problem (1) (4) has a unique solution [Discacciati and Quarteroni, ENUMATH 2001; Layton, Schieweck and Yotov, SINUM 2003]. We are interested in devising an efficient solution algorithm that uses as building blocks a Stokes solver and an elliptic solver. To do this, it is natural to introduce iterative algorithms. A parallel iterative approach for the Stokes Darcy coupling p.5/20

10 Well-posedness and iterative solution algorithms It can be proved that the coupled problem (1) (4) has a unique solution [Discacciati and Quarteroni, ENUMATH 2001; Layton, Schieweck and Yotov, SINUM 2003]. We are interested in devising an efficient solution algorithm that uses as building blocks a Stokes solver and an elliptic solver. To do this, it is natural to introduce iterative algorithms. Let us present some of the iterative algorithms that have been proposed for the solution of (1) (4). A parallel iterative approach for the Stokes Darcy coupling p.5/20

11 Dirichlet/Neumann algorithm Dirichlet/Neumann: given ϕ 0 on Γ, for m 0 A parallel iterative approach for the Stokes Darcy coupling p.6/20

12 Dirichlet/Neumann algorithm Dirichlet/Neumann: given ϕ 0 on Γ, for m 0 solve the Stokes problem with T(u m,p m ) n = gϕ m n on Γ A parallel iterative approach for the Stokes Darcy coupling p.6/20

13 Dirichlet/Neumann algorithm Dirichlet/Neumann: given ϕ 0 on Γ, for m 0 solve the Stokes problem with T(u m,p m ) n = gϕ m n on Γ solve the Darcy problem with K N ϕm+1/2 n = u m n on Γ A parallel iterative approach for the Stokes Darcy coupling p.6/20

14 Dirichlet/Neumann algorithm Dirichlet/Neumann: given ϕ 0 on Γ, for m 0 solve the Stokes problem with T(u m,p m ) n = gϕ m n on Γ solve the Darcy problem with pose on Γ K N ϕm+1/2 n = u m n on Γ ϕ m+1 := θϕ m+1/2 + (1 θ)ϕ m, where θ > 0 is an acceleration parameter. A parallel iterative approach for the Stokes Darcy coupling p.6/20

15 Dirichlet/Neumann algorithm (cont.) [The name Dirichlet/Neumann is somehow arbitrary: indeed, we are solving two Neumann problems. However, for the velocity field (u ΩS, ϕ ΩD ) the step in Ω S is a Neumann step, whereas the step in Ω D is a Dirichlet step (for the normal component...).] A parallel iterative approach for the Stokes Darcy coupling p.7/20

16 Dirichlet/Neumann algorithm (cont.) [The name Dirichlet/Neumann is somehow arbitrary: indeed, we are solving two Neumann problems. However, for the velocity field (u ΩS, ϕ ΩD ) the step in Ω S is a Neumann step, whereas the step in Ω D is a Dirichlet step (for the normal component...).] The algorithm is convergent (for a suitable choice of θ) [Discacciati and Quarteroni, Comput. Visual. Sci. 2004]. A parallel iterative approach for the Stokes Darcy coupling p.7/20

17 Dirichlet/Neumann algorithm (cont.) [The name Dirichlet/Neumann is somehow arbitrary: indeed, we are solving two Neumann problems. However, for the velocity field (u ΩS, ϕ ΩD ) the step in Ω S is a Neumann step, whereas the step in Ω D is a Dirichlet step (for the normal component...).] The algorithm is convergent (for a suitable choice of θ) [Discacciati and Quarteroni, Comput. Visual. Sci. 2004]. However: for a finite element approximation, the convergence is independent of the mesh parameter h, but depends heavily on the viscosity ν and the conductivity K A parallel iterative approach for the Stokes Darcy coupling p.7/20

18 Robin/Robin algorithm 1 Sequential Robin/Robin: given η 0 on Γ, for m 0 A parallel iterative approach for the Stokes Darcy coupling p.8/20

19 Robin/Robin algorithm 1 Sequential Robin/Robin: given η 0 on Γ, for m 0 solve the Darcy problem with γ D K N ϕm+1 n + gϕ m+1 = η m on Γ A parallel iterative approach for the Stokes Darcy coupling p.8/20

20 Robin/Robin algorithm 1 Sequential Robin/Robin: given η 0 on Γ, for m 0 solve the Darcy problem with γ D K N ϕm+1 n + gϕ m+1 = η m on Γ solve the Stokes problem with n T(u m+1,p m+1 ) n + γ S u m+1 n = gϕ m+1 γ S K N ϕ m+1 n on Γ τ T(u m+1,p m+1 ) n = 0 on Γ A parallel iterative approach for the Stokes Darcy coupling p.8/20

21 Robin/Robin algorithm 1 Sequential Robin/Robin: given η 0 on Γ, for m 0 solve the Darcy problem with γ D K N ϕm+1 n + gϕ m+1 = η m on Γ solve the Stokes problem with pose on Γ n T(u m+1,p m+1 ) n + γ S u m+1 n = gϕ m+1 γ S K N ϕ m+1 n on Γ τ T(u m+1,p m+1 ) n = 0 on Γ η m+1 := n T(u m+1,p m+1 ) n + γ D u m+1 n. A parallel iterative approach for the Stokes Darcy coupling p.8/20

22 Robin/Robin algorithm 1 (cont.) Here γ S > 0 and γ D > 0 are suitable acceleration parameters, and τ is a unit tangent vector on Γ. A parallel iterative approach for the Stokes Darcy coupling p.9/20

23 Robin/Robin algorithm 1 (cont.) Here γ S > 0 and γ D > 0 are suitable acceleration parameters, and τ is a unit tangent vector on Γ. The algorithm is proved to be convergent (at least for γ S = γ D ) [Discacciati, PhD Thesis, EPFL 2004]. A parallel iterative approach for the Stokes Darcy coupling p.9/20

24 Robin/Robin algorithm 1 (cont.) Here γ S > 0 and γ D > 0 are suitable acceleration parameters, and τ is a unit tangent vector on Γ. The algorithm is proved to be convergent (at least for γ S = γ D ) [Discacciati, PhD Thesis, EPFL 2004]. Moreover: A parallel iterative approach for the Stokes Darcy coupling p.9/20

25 Robin/Robin algorithm 1 (cont.) Here γ S > 0 and γ D > 0 are suitable acceleration parameters, and τ is a unit tangent vector on Γ. The algorithm is proved to be convergent (at least for γ S = γ D ) [Discacciati, PhD Thesis, EPFL 2004]. Moreover: it can be interpreted as an alternating direction algorithm (which is useful for tuning the parameters γ S and γ D A parallel iterative approach for the Stokes Darcy coupling p.9/20

26 Robin/Robin algorithm 1 (cont.) Here γ S > 0 and γ D > 0 are suitable acceleration parameters, and τ is a unit tangent vector on Γ. The algorithm is proved to be convergent (at least for γ S = γ D ) [Discacciati, PhD Thesis, EPFL 2004]. Moreover: it can be interpreted as an alternating direction algorithm (which is useful for tuning the parameters γ S and γ D for a finite element approximation, the convergence is independent of the mesh parameter h, and, for suitable choices of γ S and γ D, in the numerical computations it looks also independent of the viscosity ν and the conductivity K A parallel iterative approach for the Stokes Darcy coupling p.9/20

27 Robin/Robin algorithm 2 The preceding Robin/Robin algorithm is sequential. Our aim now is to find an efficient parallel algorithm. A parallel iterative approach for the Stokes Darcy coupling p.10/20

28 Robin/Robin algorithm 2 The preceding Robin/Robin algorithm is sequential. Our aim now is to find an efficient parallel algorithm. The leading idea comes from the Neumann/Neumann algorithm (but it cannot be applied straightforwardly: lack of regularity!). A parallel iterative approach for the Stokes Darcy coupling p.10/20

29 Robin/Robin algorithm 2 The preceding Robin/Robin algorithm is sequential. Our aim now is to find an efficient parallel algorithm. The leading idea comes from the Neumann/Neumann algorithm (but it cannot be applied straightforwardly: lack of regularity!). Recalling that the Neumann/Neumann algorithm for the Laplace operator is indeed given by a couple of Dirichlet solvers, followed by a couple of (homogeneous) Neumann solvers, which play the role of correctors, we propose a double parallel Robin/Robin algorithm. A parallel iterative approach for the Stokes Darcy coupling p.10/20

30 Robin/Robin algorithm 2 (cont.) Parallel Robin/Robin: given η 0 on Γ, for m 0 A parallel iterative approach for the Stokes Darcy coupling p.11/20

31 Robin/Robin algorithm 2 (cont.) Parallel Robin/Robin: given η 0 on Γ, for m 0 solve in parallel the Stokes problem and the Darcy problem with (5) n T(u m+1,p m+1 ) n γ 1 u m+1 n = η m = gϕ m+1 + γ 1 K N ϕ m+1 n on Γ τ T(u m+1,p m+1 ) n = 0 on Γ A parallel iterative approach for the Stokes Darcy coupling p.11/20

32 Robin/Robin algorithm 2 (cont.) Parallel Robin/Robin: given η 0 on Γ, for m 0 solve in parallel the Stokes problem and the Darcy problem with (5) n T(u m+1,p m+1 ) n γ 1 u m+1 n = η m = gϕ m+1 + γ 1 K N ϕ m+1 n on Γ τ T(u m+1,p m+1 ) n = 0 on Γ A parallel iterative approach for the Stokes Darcy coupling p.11/20

33 Robin/Robin algorithm 2 (cont.) Parallel Robin/Robin: given η 0 on Γ, for m 0 solve in parallel the Stokes problem and the Darcy problem with (5) n T(u m+1,p m+1 ) n γ 1 u m+1 n = η m = gϕ m+1 + γ 1 K N ϕ m+1 n on Γ τ T(u m+1,p m+1 ) n = 0 on Γ pose (6) σ m+1 := u m+1 n + K N ϕm+1 n A parallel iterative approach for the Stokes Darcy coupling p.11/20

34 Robin/Robin algorithm 2 (cont.) solve in parallel the homogeneous (f = 0) Stokes problem and the Darcy problem with (7) n T(ω m+1,π m+1 ) n + γ 2 ω m+1 n = γ 2 σ m+1 = gχ m+1 γ 2 K N χ m+1 n on Γ τ T(ω m+1,π m+1 ) n = 0 on Γ A parallel iterative approach for the Stokes Darcy coupling p.12/20

35 Robin/Robin algorithm 2 (cont.) solve in parallel the homogeneous (f = 0) Stokes problem and the Darcy problem with (7) + n T(ω m+1,π m+1 ) n + γ 2 ω m+1 n = γ 2 σ m+1 = + gχ m+1 γ 2 K N χ m+1 n on Γ τ T(ω m+1,π m+1 ) n = 0 on Γ A parallel iterative approach for the Stokes Darcy coupling p.12/20

36 Robin/Robin algorithm 2 (cont.) solve in parallel the homogeneous (f = 0) Stokes problem and the Darcy problem with (7) n T(ω m+1,π m+1 ) n + γ 2 ω m+1 n = γ 2 σ m+1 = gχ m+1 γ 2 K N χ m+1 n on Γ τ T(ω m+1,π m+1 ) n = 0 on Γ pose on Γ (8) η m+1 := η m θ ( n T(ω m+1,π m+1 ) n + gχ m+1), where γ 1 > 0, γ 2 > 0 and θ > 0 are acceleration parameters. A parallel iterative approach for the Stokes Darcy coupling p.12/20

37 Consistency of the Robin/Robin algorithm 2 Let us show that the algorithm is consistent, namely, at convergence we have the solution. A parallel iterative approach for the Stokes Darcy coupling p.13/20

38 Consistency of the Robin/Robin algorithm 2 Let us show that the algorithm is consistent, namely, at convergence we have the solution. At convergence, from (8) it holds T(ω,π ) n = gχ n, therefore normal stresses of the corrections are matching. A parallel iterative approach for the Stokes Darcy coupling p.13/20

39 Consistency of the Robin/Robin algorithm 2 Let us show that the algorithm is consistent, namely, at convergence we have the solution. At convergence, from (8) it holds T(ω,π ) n = gχ n, therefore normal stresses of the corrections are matching. On the other hand, from (7) normal velocities of the corrections are jumping ω n + K N χ n = 2g γ 2 χ. A parallel iterative approach for the Stokes Darcy coupling p.13/20

40 Consistency of the Robin/Robin algorithm 2 (cont.) But this jump gives an additional positive term in the energy of the problem, namely, one obtains 2ν D i ωj 2 g + Ω S ij Ω D N K χ χ + Γ 2g 2 γ 2 χ 2 = 0. A parallel iterative approach for the Stokes Darcy coupling p.14/20

41 Consistency of the Robin/Robin algorithm 2 (cont.) But this jump gives an additional positive term in the energy of the problem, namely, one obtains 2ν D i ωj 2 g + Ω S ij Ω D N K χ χ + Γ 2g 2 γ 2 χ 2 = 0. Hence, ω = 0 and χ = 0, and consequently σ = 0. A parallel iterative approach for the Stokes Darcy coupling p.14/20

42 Consistency of the Robin/Robin algorithm 2 (cont.) Finally: from (6), σ is the jump of the normal velocities, therefore they are matching: u n = K N ϕ n A parallel iterative approach for the Stokes Darcy coupling p.15/20

43 Consistency of the Robin/Robin algorithm 2 (cont.) Finally: from (6), σ is the jump of the normal velocities, therefore they are matching: u n = K N ϕ n from the Robin interface condition (5), if the normal velocities are matching also the normal stresses are matching: T(u,p ) n = gϕ n A parallel iterative approach for the Stokes Darcy coupling p.15/20

44 Consistency of the Robin/Robin algorithm 2 (cont.) Finally: from (6), σ is the jump of the normal velocities, therefore they are matching: u n = K N ϕ n from the Robin interface condition (5), if the normal velocities are matching also the normal stresses are matching: T(u,p ) n = gϕ n We have found the right solution! A parallel iterative approach for the Stokes Darcy coupling p.15/20

45 Convergence of the Robin/Robin algorithm 2 The parallel Robin/Robin algorithm can be rewritten as a preconditioned Richardson scheme: η m+1 = η m + θ(k S + K D )[ψ (H S + H D )η m ], for a suitable right hand side ψ. A parallel iterative approach for the Stokes Darcy coupling p.16/20

46 Convergence of the Robin/Robin algorithm 2 The parallel Robin/Robin algorithm can be rewritten as a preconditioned Richardson scheme: η m+1 = η m + θ(k S + K D )[ψ (H S + H D )η m ], for a suitable right hand side ψ. The operators K S, K D, H S and H D are the main building blocks of the algorithm, and are defined as follows. A parallel iterative approach for the Stokes Darcy coupling p.16/20

47 Convergence of the Robin/Robin algorithm 2 (cont.) K S σ := n T(ω,π) n, where ω is a solution with the Robin datum n T(ω,π) n + γ 2 ω n = γ 2 σ on Γ τ T(ω,π) n = 0 on Γ. K S is a Robin-to-Neumann operator A parallel iterative approach for the Stokes Darcy coupling p.17/20

48 Convergence of the Robin/Robin algorithm 2 (cont.) K S σ := n T(ω,π) n, where ω is a solution with the Robin datum n T(ω,π) n + γ 2 ω n = γ 2 σ on Γ τ T(ω,π) n = 0 on Γ. K S is a Robin-to-Neumann operator K D σ := gχ, where χ is a solution with the Robin datum gχ γ 2 K N χ n = γ 2 σ on Γ. K D is a Robin-to-Dirichlet operator A parallel iterative approach for the Stokes Darcy coupling p.17/20

49 Convergence of the Robin/Robin algorithm 2 (cont.) H S η := u n, where u is a solution (for f = 0) with the Robin datum n T(u,p) n γ 1 u n = η on Γ τ T(u,p) n = 0 on Γ. H S is a Robin-to-Dirichlet operator A parallel iterative approach for the Stokes Darcy coupling p.18/20

50 Convergence of the Robin/Robin algorithm 2 (cont.) H S η := u n, where u is a solution (for f = 0) with the Robin datum n T(u,p) n γ 1 u n = η on Γ τ T(u,p) n = 0 on Γ. H S is a Robin-to-Dirichlet operator H D η := K N ϕ n, where ϕ is a solution with the Robin datum K gϕ + γ 1 N ϕ n = η on Γ. H D is a Robin-to-Neumann operator A parallel iterative approach for the Stokes Darcy coupling p.18/20

51 Convergence of the Robin/Robin algorithm 2 (cont.) These operators have the following properties (in the space L 2 (Γ)): A parallel iterative approach for the Stokes Darcy coupling p.19/20

52 Convergence of the Robin/Robin algorithm 2 (cont.) These operators have the following properties (in the space L 2 (Γ)): K S is symmetric, continuous and positive definite A parallel iterative approach for the Stokes Darcy coupling p.19/20

53 Convergence of the Robin/Robin algorithm 2 (cont.) These operators have the following properties (in the space L 2 (Γ)): K S is symmetric, continuous and positive definite K D is symmetric, continuous and non-negative A parallel iterative approach for the Stokes Darcy coupling p.19/20

54 Convergence of the Robin/Robin algorithm 2 (cont.) These operators have the following properties (in the space L 2 (Γ)): K S is symmetric, continuous and positive definite K D is symmetric, continuous and non-negative H S is symmetric, continuous and non-negative (for γ 1 small enough) A parallel iterative approach for the Stokes Darcy coupling p.19/20

55 Convergence of the Robin/Robin algorithm 2 (cont.) These operators have the following properties (in the space L 2 (Γ)): K S is symmetric, continuous and positive definite K D is symmetric, continuous and non-negative H S is symmetric, continuous and non-negative (for γ 1 small enough) H D is symmetric, continuous and positive definite A parallel iterative approach for the Stokes Darcy coupling p.19/20

56 Convergence of the Robin/Robin algorithm 2 (cont.) These operators have the following properties (in the space L 2 (Γ)): K S is symmetric, continuous and positive definite K D is symmetric, continuous and non-negative H S is symmetric, continuous and non-negative (for γ 1 small enough) H D is symmetric, continuous and positive definite Therefore, both the operator (H S + H D ) and the preconditioner (K S + K D ) 1 are symmetric, continuous and positive definite: convergence is achieved! (for a suitable choice of the parameter θ) A parallel iterative approach for the Stokes Darcy coupling p.19/20

57 Comments It is well-known that the Dirichlet-to-Neumann operator is symmetric, continuous and positive definite from the energy trace space H 1/2 (Γ) into ts dual; here we have seen that: the Robin-to-Neumann operator is symmetric, continuous and positive definite in L 2 (Γ), whereas: the Robin-to-Dirichlet operator is symmetric, continuous and non-negative in L 2 (Γ) A parallel iterative approach for the Stokes Darcy coupling p.20/20

58 Comments It is well-known that the Dirichlet-to-Neumann operator is symmetric, continuous and positive definite from the energy trace space H 1/2 (Γ) into ts dual; here we have seen that: the Robin-to-Neumann operator is symmetric, continuous and positive definite in L 2 (Γ), whereas: the Robin-to-Dirichlet operator is symmetric, continuous and non-negative in L 2 (Γ) the above results also hold for the finite element numerical approximation, uniformly with respect to the mesh parameter h A parallel iterative approach for the Stokes Darcy coupling p.20/20

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