Iterative Methods for Stokes/Darcy Coupling

Transcription

1 Iterative Methods or Stokes/Darcy Coupling Marco Discacciati Ecole Polytechnique Fédérale de Lausanne Institut d Analyse et Calcul Scientiique Chair o Modelling and Scientiic Computing marco.discacciati@epl.ch Summary. We present iterative subdomain methods based on a domain decomposition approach to solve the coupled Stokes/Darcy problem using inite elements. The dependence o the convergence rate on the grid parameter h and on the physical data is discussed; some diiculties encountered when applying the algorithms are indicated together with possible improvement strategies. 1 Introduction and problem setting The simulation o incompressible lows in heterogenous media is an interesting topic with many applications: considering the particular case o ree luids which can iltrate through porous media, we recall or example the hydrological environmental applications and mass transer in biomechanics. The Stokes/Darcy coupled system provides a linear model to describe such phenomena. We consider a bounded domain Ω R d (d = 2, 3) ormed by two non-overlapping subdomains Ω and Ω p separated by a surace Γ = Ω Ω p. Ω is the region occupied by the luid whose motion is described by the Stokes equations which can be written in adimensional orm as: Re 1 u + p = u = 0 in Ω, (1) where u and p are the adimensional velocity and pressure, respectively, while Re is the Reynolds number deined as Re = L U /ν, ν > 0 being the luid kinematic viscosity and L, U a characteristic length and velocity, respectively. The iltration through the porous region Ω p is modeled using Darcy s equations, whose adimensional orm reads: u p = εre p p p u p = 0 in Ω p, (2)

2 564 Marco Discacciati where u p and p p are the adimensional luid velocity and pressure, respectively, Re p is the Reynolds number Re p = δ p U p /ν, U p being a characteristic velocity through the porous medium and δ p a characteristic pore size. Finally, ε = δ p /L p is the adimensional ratio between the micro and the macro scales in Ω p. Across the interace Γ the continuity o normal stresses and luxes is required; precisely, we impose: u n = u p n n T(u, p ) n = p p on Γ (3) τ j T(u, p ) n = αu τ j where T(u, p ) is the stress tensor, n is the unit normal outward vector to Ω and τ j n = 0 (j = 1,...,d 1); α is a dimensionless coeicient depending essentially on ν and the hydraulic conductivity o the porous medium. For an extensive discussion about these coupling conditions we reer to Discacciati et al. [2002], Jäger and Mikelić [1996], Layton et al. [2003], Payne and Straughan [1998]. The mathematical analysis o the coupled problem has been addressed in previous works concerning both the continuous case and the inite element approximation (see Discacciati and Quarteroni [2003], Layton et al. [2003]). In order to solve the coupled problem, an iterative substructuring method was proposed and analyzed in Discacciati and Quarteroni [2004]. Here, we test it on a model problem in order to investigate its eectiveness and robustness, with particular emphasis on the role that physical and grid parameters play on the convergence properties. We consider the computational domain Ω R 2, with Ω = (0, 1) (1, 2), Ω p = (0, 1) 2 and interace Γ = (0, 1) {1}. We rewrite Darcy s equation as (εre p p p ) = 0 in Ω p, and consider the ollowing analytic solution: u = (y 2 2y+1, x 2 x) T, p = 2(x+y 1)/Re +1/(3εRe p ) and p p = (x(1 x)(y 1) + y 3 /3 y 2 + y)/(εre p ) + 2x/Re. Concerning the inite element discretization, P 2 P 1 Taylor-Hood elements have been used or Stokes equations, while P 2 Lagrangian elements have been adopted or Darcy s problem. All the computational meshes are conorming on Γ. 2 Dirichlet-Neumann (DN) methods Considering the interace conditions (3) 1 and (3) 2, we can choose as scalar interace variable λ either λ = u n on Γ or λ = p p on Γ. These two choices deine two dierent DN-type methods, which can be outlined as ollows, respectively: Algorithm DN 1 0. choose λ = u n on Γ and an initial guess λ (0) on Γ ;

3 For k = 0, 1,... until convergence, Do Iterative Methods or Stokes/Darcy Coupling solve Darcy s equation with b.c. εre p p p (k+1) n = λ (k) on Γ ; 2. solve Stokes problem with b.c. n T(u (k+1), p (k+1) ) n = p p (k+1) and (3) 3 on Γ; 3. λ (k+1) = θu (k+1) n + (1 θ)λ (k) on Γ, θ (0, 1) ; End For Algorithm DN 2 0. choose λ = p p on Γ and an initial guess λ (0) on Γ ; For k = 0, 1,... until convergence, Do 1. solve Stokes problem with b.c. n T(u (k+1), p (k+1) ) n = λ (k) and (3) 3 on Γ; 2. solve Darcy s equation with b.c. εre p p p (k+1) n = u (k+1) n on Γ ; 3. λ (k+1) = θp (k+1) p + (1 θ)λ (k) on Γ, θ (0, 1) ; End For The two DN methods are equivalent to preconditioned Richardson methods to solve the symmetric Steklov-Poincaré equations associated to the coupled problem, and they allow to characterize optimal preconditioners or Krylov type methods (e.g. the Conjugate Gradient) or the corresponding interace problems (see Discacciati and Quarteroni [2004]). 2.1 Numerical results We consider Re = 1, εre p = 1 and tol = 10 5 ; in Table 1 we report the number o iterations or both the Richardson and the Preconditioned Conjugate Gradient (PCG) method. These convergence results are satisactory as they show the optimality o the preconditioners with respect to the grid parameter h. Number o DN 1 PCG DN 2 PCG mesh elements (θ = 0.7) λ = u n (θ = 0.7) λ = p p Table 1. Number o iterations on dierent grids with Re = 1 and εre p = 1 However, i the luid viscosity and the hydraulic conductivity decrease, small relaxation parameters θ must be adopted to guarantee convergence, in

4 566 Marco Discacciati accordance with the theoretical estimate o the upper bound θ max given in Discacciati and Quarteroni [2004]. Unortunately, in some cases θ should be so small that in practice it prevents the numerical scheme rom converging. To quote an example, i Re = 10 3 and εre p = 10 2, then θ should be unreasonably small (smaller than 10 4!) in DN 1 to prevent divergence. This diiculty should not be ascribed to the non-optimal choice o the relaxation parameter θ. In act, i we apply the PCG method which embeds the choice o the optimal acceleration parameter (see, e.g., Quarteroni et al. [2000] p. 150), the iterative algorithm converges, but the optimal properties o the preconditioners are lost, since the number o iterations depends on the mesh parameter h, as reported in Table 2. Mesh elements PCG iterations (λ = u n) Table 2. Number o iterations on dierent grids with Re = 10 3 and εre p = 10 2 On the basis o the numerical results we have obtained we can conclude that DN methods are eective only when the ratio Re /(εre p ) is suiciently small, while dealing with large values causes some diiculties. However, the latter are the very values o interest in real-lie applications and, thereore, a robust numerical method is required. 3 Dirichlet-Neumann or a time-dependent problem We introduce a ormal argument to better understand the results obtained in Sect. 2.1 and to set up a more eective numerical scheme. This approach will be treated rom a precise mathematical viewpoint in a orthcoming work Discacciati [2004]. The underlying idea is that our diiculties in solving the Stokes/Darcy problem may come rom the dierent structure o equations (1) 1 and (2) 1, which become even more dissimilar when Re 1 and εre p 1. In act, in that case, under the physically reasonable hypothesis that u and p p are suiciently small, (1) 1 reduces almost to C I + p =, while (2) 1 becomes u p +C p I = 0, where C and C p denote two positive constants 1. We rewrite (2) 1 as (εre p ) 1 u p + p p = 0 in Ω p, (4) and ormally comparing (4) to (1) 1, we are led to modiy the latter by adding a mass term like (εre p ) 1 u p as ollows:

5 Iterative Methods or Stokes/Darcy Coupling 567 β(εre p ) 1 u Re 1 u + p =, β R +, (5) possibly with a consequent modiication o the right hand side (see Remark 1) that we have denoted by. In this way we obtain a generalized Stokes momentum equation, and note that now (5) has the same behaviour o (4) in the cases o our interest, that is when Re 1 and εre p 1. We expect that the mass term β(εre p ) 1 u would help improving the positivity o the discrete Steklov-Poincaré operator which acts as a preconditioner in the DN 1 method. With this aim, we have carried out some numerical tests using the PCG algorithm with λ = u n as interace variable to solve the modiied problem (2), (5). The convergence results reported in Table 3 show that the numerical scheme has really improved. Re εre p β Iterations on the mesh with 688 el el el Table 3. Number o iterations to solve problem (2), (5) or dierent values o Re, εre p and β Remark 1. Equation (5) can be regarded as a discretization in time o the time-dependent Stokes momentum equation t u Re 1 u + p = in Ω. Precisely, i we consider β(εre p ) 1 u,n+1 Re 1 u,n+1 + p,n+1 = n+1 n 0 with n+1 = (x, t n+1 )+β(εre p ) 1 u,n, we have a backward Euler discretization in time with β(εre p ) 1 playing the role o the inverse o a time step. From the physical viewpoint, since the luid velocities in Ω are much higher than the ones through the porous medium (see Ene and Sanchez- Palencia [1975]), a time-dependent model better represents the phenomena occurring during the iltration process. 3.1 The tdn algorithm Let [0, T] be a characteristic time interval; using or the sake o simplicity the irst-order backward Euler scheme, denoting by t > 0 the time step and N = T/ t, the iterative method that we propose to solve the time-dependent coupled problem reads (the subscript n reers to the nth time level):

6 568 Marco Discacciati Algorithm tdn For n = 0,...,N 1 Do 0. choose an initial guess λ (0) n+1 or the normal velocity on Γ at the (n + 1)th time level; For k = 0, 1,... until convergence, Do 1. solve Darcy s equation with b.c. εre p p (k+1) p,n+1 n = λ(k) n+1 on Γ ; 2. solve Stokes problem ( t) 1 u (k+1),n+1 Re 1 u(k+1),n+1 + p(k+1),n+1 = ( t) 1 u,n + n+1 u (k+1),n+1 = 0 in Ω with b.c. n T(u (k+1),n+1, p(k+1),n+1 ) n = p(k+1) 3. λ (k+1) n+1 = θu (k+1),n+1 End For End For n + (1 θ)λ(k) n+1 p,n+1 and (3) 3 on Γ ; on Γ, θ (0, 1) ; 3.2 Numerical tests We consider the horizontal section o a channel 12 m long and 8 m wide which is partially occupied by a porous medium with discontinuous conductivity, as represented in Fig. 1 (let). A parabolic inlow proile is imposed on the let hand side boundary with maximal velocity equal to 0.1m/s. On the right an outlow condition is imposed. The time interval is t [0, 0.5] and the time step t = 10 3 s; or space discretization three dierent computational meshes have been adopted. In a irst case we have considered Re = and a discontinuous coeicient εre p = 10 3 in Ω p (1), εre p = 10 7 in Ω p (2). In Fig. 1 (right) we have represented the computed solution at time t = 0.05 s, while in Fig. 2 a zoom o the velocity ield through the porous medium is shown; it can be seen that the velocity is almost null in the less permeable areas o the porous medium. Finally, Table 4 (let) reports the number o iterations obtained or three computational grids at dierent time levels, showing that the number o iterations is low and independent o h. The same test has been perormed considering dierent values o the parameters: Re = , εre p = 10 1 in Ω p (1) and εre p = 10 5 in the less permeable part o the porous medium Ω p (2). The convergence results show that the number o iterations is essentially independent o these parameters, as it can be seen comparing the previous convergence results with those reported in Table 4 (right).

7 Iterative Methods or Stokes/Darcy Coupling 569 Ω Ω (2) p Outlow Inlow Γ 8m m Ω p (1) Outlow Fig. 1. Computational domain (let) and computed velocity ield at t = 0.05 s (right) Fig. 2. Zoom o the velocity ield through the porous medium Time Iterations on the mesh with level 232 el. 928 el el Time Iterations on the mesh with level 232 el. 928 el el Table 4. Number o iterations on dierent grids with Re = , εre p = 10 3 and 10 7 (let); with Re = , εre p = 10 1 and 10 5 (right) 4 Conclusions and perspectives Numerical results show that considering a time-dependent problem allows to set up a ar more eicient DN algorithm or problems with parameters in a range o physical interest. However, as we have shown, the value o t generally depends on εre p and Re, and in some cases we could be orced to consider very small time steps t 1. This could be quite annoying since one might be interested in considering long time scales, or example in modeling the iltration o pollutants in groundwater.

8 570 Marco Discacciati This limitation on t drives us to reconsider the steady coupled model. In act, should we ind an algorithm whose behaviour were as much as possible independent o the physical parameters, then not only we would be able to solve the steady problem itsel, but we could also use it in the ramework o the time-dependent model where t would be chosen under the sole requirements o stability and accuracy. A possible approach we are currently considering is a Robin-Robin type method ollowing the ideas presented in Lube et al. [2001] or Oseen equations; its analysis and numerical results will be presented in a uture work. Acknowledgement. I wish to thank Pro. A. Quarteroni, Pro. A. Valli and Dr. N. Neuss or their suggestions and the helpul discussions concerning the subject o this work. Reerences M. Discacciati. PhD thesis, Ecole Polytechnique Fédérale de Lausanne, Switzerland, In preparation. M. Discacciati, E. Miglio, and A. Quarteroni. Mathematical and numerical models or coupling surace and groundwater lows. Appl. Numer. Math., 43:57 74, M. Discacciati and A. Quarteroni. Analysis o a domain decomposition method or the coupling o Stokes and Darcy equations. In F. Brezzi, A. Bua, S. Corsaro, and A. Murli, editors, Numerical Mathematics and Advanced Applications, ENUMATH 2001, pages Springer. Milan, M. Discacciati and A. Quarteroni. Convergence analysis o a subdomain iterative method or the inite element approximation o the coupling o Stokes and Darcy equations. Comput. Visual. Sci., To be published, H. I. Ene and E. Sanchez-Palencia. Equations et phénomenès de surace pour l écoulement dans un modèle de milieu poreux. J. Mécanique, 14(1):73 108, W. Jäger and A. Mikelić. On the boundary conditions at the contact interace between a porous medium and a ree luid. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23: , W. L. Layton, F. Schieweck, and I. Yotov. Coupling luid low with porous media low. SIAM J. Num. Anal., 40: , G. Lube, L. Müller, and F. C. Otto. A nonoverlapping domain decomposition method or stabilized inite element approximations o the Oseen equations. J. Comput. Appl. Math., 132: , L. E. Payne and B. Straughan. Analysis o the boundary condition at the interace between a viscous luid and a porous medium and related modelling questions. J. Math. Pures Appl., 77: , A. Quarteroni, R. Sacco, and F. Saleri. Numerical Mathematics. Springer- Verlag, New York Berlin Heidelberg, 2000.