Stochastic multiscale modeling of subsurface and surface flows. Part III: Multiscale mortar finite elements for coupled Stokes-Darcy flows

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1 Stochastic multiscale modeling of subsurface and surface flows. Part III: Multiscale mortar finite elements for coupled Stokes-Darcy flows Ivan otov Department of Mathematics, University of Pittsburgh KAUST WEP Workshop January 3 February, 2 Joint work with Vivette Girault, Paris VI, and Danail Vassilev, University of Pittsburgh Department of Mathematics, University of Pittsburgh

2 Outline Mathematical model for the coupled flow problem Interface conditions Existence and uniqueness for a global weak formulation Equivalence to a domain decomposition weak formulation Finite element discretizations Conforming Stokes elements and mixed finite elements for Darcy Mortar finite elements on all subdomain interfaces Discrete inf-sup condition Existence and uniqueness of a discrete solution Convergence analysis Non-overlapping domain decomposition - reduction to a mortar interface problem Coupling of Stokes-Darcy flow with transport Department of Mathematics, University of Pittsburgh 2

3 Applications of coupled Stokes and Darcy flows Interactions between surface water and groundwater flows Flows in fractured porous media Flows through vuggy rocks Flows through industrial filters Fuel cells Blood flows Glaucoma Department of Mathematics, University of Pittsburgh 3

4 Model problem Γ Γ Ω = fluid region (Stokes flow) n 2 n Γ Ω = saturated porous Γ medium (Darcy s law) Γ I Γ Γ 2 u j : Ω j R d, fluid velocity in Ω j, p j : Ω j R, fluid pressure in Ω j. Deformation tensor D and stress tensor T in Ω s : D(u s ) := 2 ( u s + ( u s ) T ), T(u s, p s ) := p s I + 2µD(u s ). Department of Mathematics, University of Pittsburgh 4

5 Flow equations Stokes flow on Ω s : div T(u s, p s ) 2µ div D(u s ) + p s = f s in Ω (conservation of momentum), div u s = in Ω s (conservation of mass), u s = on Γ s (no slip). Darcy flow on Ω d : µk u d + p d = f d in Ω d (Darcy s law), div u d = q d in Ω d (conservation of mass), u d n d = on Γ d (no flow), Solvability condition: Ω d q d dx = Department of Mathematics, University of Pittsburgh 5

6 Mass conservation across Γ sd : Continuity of normal stress on Γ d : Interface conditions u s n s + u d n d = on Γ sd. n s T n s p s 2µn s D(u s ) n s = p d on Γ sd. Slip with friction interface condition: (Beavers-Joseph (967),Saffman (97), Jones (973), Jäger and Mikelić (2)) n s T τ j 2µn s D(u s ) τ j = µα Kj u τ j, j =, d, on Γ sd, where K j = τ j K τ j. Department of Mathematics, University of Pittsburgh 6

7 Global variational formulation For u d, v d in L 2 (Ω d ) d and u s, v s in H (Ω s ) d, a(u, v) = µ K u d v d +2 µ Ω d D(u s ) : D(v s )+ Ω s d j= For v d H(div; Ω d ), v s H(div; Ω s ) and q L 2 (Ω), b(v, q) = q div v d Ω d q div v s. Ω s µα (u s τ j )(v s τ j ). Γ sd Kj = {v H(div; Ω) ; v s H (Ω d ) d, v Γs =, (v n) Γd = }, v, v = ( v 2 H(div;Ω) + v s 2 H (Ω s )) /2. Find (u, p) L 2 (Ω) such that v, a(u, v) + b(v, p) = r L 2 (Ω), b(u, r) = Ω Ω f v, r q d. Department of Mathematics, University of Pittsburgh 7

8 Existence and uniqueness of a weak solution Lemma: q L 2 (Ω), sup v b(v, q) v β q L 2 (Ω) Lemma: a(v, v) γ v 2 v V = {v : div = } Proof: By Korn s inequality, a(v, v) C( v s 2 H (Ω s ) + v d 2 L 2 (Ω d ) ). Coercivity now follows from v V, v s L 2 (Ω s ) C ( v s 2 H (Ω s ) + v d 2 /2. L 2 (Ω d )) Lemma: The variational problem has a unique solution. Department of Mathematics, University of Pittsburgh 8

9 Domain decomposition variational formulation Let Ω s = Ω s,i, Ω d = Ω d,i. Interface conditions Stokes-Stokes interfaces: Darcy-Darcy interfaces: [v s ] =, [T n] = on Γ ss [u d n] =, [p d ] = on Γ dd = {v L 2 (Ω) d : v Ωs,i H (Ω s,i ) d, v Ωd,i H(div, Ω d,i ), v n L r ( Ω d,i )(r > ), v Γs =, (v n) Γd = }, W = L 2 (Ω), Λ = {ξ : ξ Γij H /2 (Γ ij ) Γ ij Γ ss, ξ Γij H /2 (Γ ij ) Γ ij Γ dd Γ sd }. Department of Mathematics, University of Pittsburgh 9

10 Domain decomposition variational formulation, cont. a s,i (u s,i, v s,i ) = 2 µ D(u s,i ) : D(v s,i )+ Ω s,i a d,i (u d,i, v d,i ) = µ K u d,i v d,i, Ω d,i d j= Ω s,i Γ sd µα Kj (u s,i τ j )(v s,i τ j ) b i (v i, w i ) = w i div v i Ω i a(, ) = a s,i (, ) + a d,i (, ), b(, ) = b i (, ) b Γ (v, λ) = [v] λ + [v n] λ + [v n] λ Γ ss Γ dd Γ sd Find u, p W, λ Λ: v, a(u, v) + b(v, p) + b Γ (v, λ) = w W, b(u, w) = w q d, Ω ξ Λ, b Γ (u, ξ) =. Ω f v, Department of Mathematics, University of Pittsburgh

11 Equivalence Lemma: The two variational formulations are equivalent. Department of Mathematics, University of Pittsburgh

12 Partition T h i on Ω i ; T h i Finite element discretization and T h j need not match at Γ ij. Stokes elements h s,i W h in Ω s,i : MINI (Arnold-Brezzi-Fortin), Taylor- Hood, Bernardi-Raugel; contain at least polynomials of degree r s and r s resp. Pressure Velocity Mixed finite elements d,i h W d,i h in Ω d,i : RT, BDM, BDFM, BDDF; contain at least polynomials of degree r d velocity pressure Lagrange multplier T H ij h := h i, W h := W h i L 2 (Ω) - partition of Γ ij, possibly different from the traces of T h i and T h j Λ H ij : continuous or discontinuous piecewise polynomials of degree at least m s on Γ ss or m d on Γ dd and Γ sd Λ H := Λ H ij Nonconforming approximation: Λ h Λ Department of Mathematics, University of Pittsburgh 2

13 Mortar finite element method Find u h h, p h W h, λ H Λ H : v h h, a(u h, v h ) + b(v h, p h ) + b Γ (v h, λ H ) = w h W h, b(u h, w h ) = w h q d, Equivalently, letting ξ H Λ H, b Γ (u h, ξ H ) =. Ω Ω f v h, V h = {v h h : b Γ (v h, ξ H ) = ξ H Λ H }, Find u h V h, p h W h : v h V h, a(u h, v h ) + b(v h, p h ) = f v h, w h W h, b(u h, w h ) = Ω Ω w h q d. Department of Mathematics, University of Pittsburgh 3

14 Mortar compatibility conditions On Γ ij Γ dd Γ sd, i < j (assume that Ω i is a Darcy domain), ξ H Λ H, sup ϕ h i V h i n ϕ h i, ξh Γij ϕ h i L 2 (Γ ij ) β d ξ H L 2 (Γ ij ). On Γ ij Γ ss, i < j, ξ H Λ H, sup ϕ h i V h i Γ ij H /2 (Γ ij) d ϕ h i, ξh Γij ϕ h i H /2 (Γ ij ) β s ξ H H /2 (Γ ij ). These are much more general than the mortar conditions used in [Bernardi-Maday- Patera], [Ben Belgacem] for Laplace and Stokes, and in [Layton-Schieweck-.], [Riviere-.], [Galvis-Sarkis] for Stokes-Darcy. The above condition on Γ dd is similar to the one used in [Arbogast-Cowsar-Wheeler-.] and [Arbogast-Pencheva-Wheeler-.]. Department of Mathematics, University of Pittsburgh 4

15 Π h s v H (Ω s,i) C v H (Ω s,i) Π h d v H(div;Ωs,i) C v H (Ω s,i) Weakly continuous interpolants V h s = {v h h s : [v h ], ξ H Γss = ξ H Λ H } V h d = {v h h d : [v h n], ξ H Γdd = ξ H Λ H } There exisit Π h s : H (Ω s ) V h d such that (div(v Π h sv), w h ) Ωs,i = w h W h, Ω s,i Ω s, There exist Π h d : H (Ω d ) V h s such that (div(v Π h dv), w h ) Ωd,i = w h W h, Ω d,i Ω d, Department of Mathematics, University of Pittsburgh 5

16 Discrete inf-sup condition Lemma: w h W h, sup b(vh, wh ) v h V h v Proof: Given w h W h, construct v H (Ω): β w W div v = w h, v H (Ω) C w h L 2 (Ω). Then construct an interpolant Π h : H (Ω) V h such that b(π h v v, q h ) = q h W h, Π h v C v H (Ω). Department of Mathematics, University of Pittsburgh 6

17 Existence and uniqueness of a discrete solution Lemma: There exists a unique solution to: Find u h V h, p h W h : v h V h, a(u h, v h ) + b(v h, p h ) = (f, v h ), w h W h, b(u h, w h ) = (q d, w h ). Proof: Coercivity: The discrete Korn s inequality gives which, combined with a s,i (v h, v h ) α s,i v h 2 H (Ω s,i ), v h V h, v h 2 L 2 (Ω s,i ) v h 2 H (Ω s,i ), implies as,i(v h, v h ) α s v h 2 H (Ω s,i ). On Ω d, ad,i (v h, v h ) α d v h 2 H(div;Ω d,i) vh : (div v h, w h ) = w h W h. Coercivity and inf-sup condition imply existence and uniqueness of a solution. Department of Mathematics, University of Pittsburgh 7

18 Approximation properties of V h Construct an interpolant I h : H (Ω) V h, I h v = (I h s v, I h d v), v I h s v H (Ω s,i) C h r s v H rs+ (Ω s,i), v I h d v H(div;Ωs,i) C (h r d+ v H rd + (Ωd,i) + hr s v H rs+ (Ω s,i)). Department of Mathematics, University of Pittsburgh 8

19 Convergence analysis Theorem: Proof: u u h + p p h W C(h r s + h r d+ + H m s+/2 + H m d+/2 ) u u h + p p h W C( inf v h V h u vh + where R h a(u, = sup vh ) + b(vh, p) (f, vh ) v h V h v h Bound on the non-conforming error: Θ(v h ) := a(u, v h ) + b(v h, p) (f, v h ) = b Γ (v h, λ) inf p w h W ) + R h, w h W h [v h n], λ Γsd CH md+ v h [v h n], λ Γdd CH md+/2 v h [v h ], λ Γss CH ms+/2 v h Department of Mathematics, University of Pittsburgh 9

20 Example : Smooth solution Ω = Ω Ω 2, where Ω = [, ] [ 2, ] and Ω 2 = [, ] [, 2 ] Taylor-Hood on triangles in Ω ; lowest order Raviart-Thomas on rectangules in Ω 2. u = u 2 = [ sin( x G + ω)ey/g cos( x G + ω)ey/g ], p = ( G K µ G )cos( x G + ω)e/(2g) + y.5, p 2 = G K cos( x G + ω)ey/g, where µ =., K =, α =.5, G = µk, and ω =.5. α Department of Mathematics, University of Pittsburgh 2

21 Convergence rates for Example mesh u u,h,ω rate p p,h,ω rate 4x4 9.74e e-3 8x8 2.4e e x6 6.5e e x32.59e e x64 4.5e e-5.9 Table : Numerical errors and convergence rates in Ω. mesh u 2 u 2,h,Ω2 rate p p 2,h,Ω2 rate 4x4 3.8e-2 8.e-3 8x8 9.2e e-3.7 6x6 2.48e e x e e x64.72e e-5.99 Table 2: Numerical errors and convergence rates in Ω 2. Department of Mathematics, University of Pittsburgh 2

22 Example 2: discontinuous tangential velocity u = u 2 = [ (2 x)(.5 y)(y ξ) y3 3 + y2 2 (ξ +.5).5ξy.5 + sin(ωx) ], [ ω cos(ωx)y χ(y +.5) + sin(ωx) p = sin(ωx) + χ + µ(.5 ξ) + cos(πy), 2K p 2 = χ (y +.5) 2 sin(ωx)y K 2 K, ], where ω = 6 and the other parameters are defined as in Example. Department of Mathematics, University of Pittsburgh 22

23 Computed pressure and velocity for Example 2 U V Left: horizontal velocity; Right: vertical velocity Department of Mathematics, University of Pittsburgh 23

24 Convergence rates for Example 2 mesh u u,h,ω rate p p,h,ω rate 4x4 3.54e- 3.e-2 8x8 8.6e e x6 2.5e e x e e x64.4e e-4.99 Table 3: Numerical errors and convergence rates in Ω. mesh u 2 u 2,h,Ω2 rate p p 2,h,Ω2 rate 4x4 2.6e-.8e- 8x8 5.79e e x6.47e e x32 3.7e e x e e-4 2. Table 4: Numerical errors and convergence rates in Ω 2. Department of Mathematics, University of Pittsburgh 24

25 Solution of the algebraic flow system A B L B T L T u p λ = f f 2 ( R L L T ) ( x λ ) = ( f ) x - subdomain unknowns; λ - interface unknowns Matrix is symmetric but indefinite. Form the Shur complement system: L T R Lλ = L T R f () Iterative method for () (e.g. CG). Each iteration requires evaluating R = R... R n, i.e., solving subdomain problems. Advantages: subdomain solves in parallel; reuse existing Stokes and Darcy codes Department of Mathematics, University of Pittsburgh 25

26 Domain decomposition Subdomain problems with specified interface normal stress (R Lλ): find (u h,i (λ), p h,i (λ)) Vi h W h i such that a i (u h,i(λ), v i ) + b i (v i, p h,i(λ)) = λ v i n i, v i V i h, Γ I b i (u h,i(λ), w i ) =, w i Wh. i λ Complementary subdomain problems (R f): find (ū h, p h ) Vh i W h i such that BC a i (ū h, v i ) + b i (v i, p h ) = f i v i, v i V i BC h, BC q Ω i q 2 BC b i (ū h, w i ) = q i w i, w i Wh. i BC BC Ω i Interface problem: find λ h Λ h such that s h (λ h, µ) b I (u h(λ h ), µ) = b I (ū h, µ), µ Λ h, Recover global velocity and pressure: u h = u h (λ h) + ū h, p h = p h (λ h) + p h. Department of Mathematics, University of Pittsburgh 26

27 Subdomain problems Both types of local problems are well posed due to the local discrete inf-sup conditions. Neumann-Robin BC on Γ I for the Stokes region: n T n = λ, n T τ j µα Kj u τ j =. Dirichlet BC on Γ I for the Darcy region: p 2 = λ. Interface (Steklov Poincaré) operator: S h : Λ h Λ h, (S h λ, µ) = s h (λ, µ) λ, µ Λ h ; S h : λ [u h(λ) n] Interface problem: Find λ h Λ h such that S h λ h = g h. Department of Mathematics, University of Pittsburgh 27

28 Condition number estimate Lemma: For all λ M h = {µ M h : Γ I µ = }, C, h λ 2 Γ I s (λ, λ) C,2 λ 2 Γ I (S : H /2 (Γ I ) H /2 (Γ I )) C 2, λ 2 Γ I s 2 (λ, λ) C 2,2 h λ 2 Γ I (S 2 : H /2 (Γ I ) H /2 (Γ I )) Theorem: cond(s h ) = O(h ) Proof: (C, h + C 2, ) λ 2 Γ I s(λ, λ) (C,2 + C 2,2 h ) λ 2 Γ, Department of Mathematics, University of Pittsburgh 28

29 Algorithm on non-matching grids Apply the Conjugate Gradient method for S h λ h = g h. Computing the action of the operator (needed at each CG iteration): Given mortar data λ h Λ h, project onto subdomain grids: λ h Q hi λ h Solve local Stokes and Darcy problems in parallel with boundary data Q hi λ h Project fluxes onto the mortar space and compute the jump: u h,i n i Q T h i u h,i n i, S h λ h = Q T h u h, n + Q T h 2 u h,2 n 2 Department of Mathematics, University of Pittsburgh 29

30 Multiple subdomains Three types of interfaces: Stokes Stokes, Stokes Darcy, and Darcy Darcy. The algorithm above works for Stokes Darcy, and Darcy Darcy interfaces: only u n is continuous. On Stokes Stokes interfaces u τ also has to be continuous. Augment S h with the tangential stress as primary variable: S h : ( ) n T n n T τ j where [ ] is the jump operator. ( [u n ] [u τ j ] S h is a Neumann-to-Dirichlet operator. Different number of primary variables on different interfaces. ) or, equivalently, S h : n T [u ], Department of Mathematics, University of Pittsburgh 3

31 Condition number for multiple subdomains Γ S : Stokes-Stokes interfaces Γ D : Stokes-Darcy and Darcy-Darcy interfaces C (h λ 2 Γ S + λ 2 Γ D ) s(λ, λ) C 2 ( λ 2 Γ S + h λ 2 Γ D ) C h λ 2 Γ s(λ, λ) C 2 h λ 2 Γ Theorem: cond(s h ) = O(h 2 ) Department of Mathematics, University of Pittsburgh 3

32 Condition number studies: two subdomains /h eigmin eigmax iter Department of Mathematics, University of Pittsburgh 32

33 Condition number studies: multiple subdomains 4 subdomains /h eigmin eigmax iter Varying number of subdomains /H eigmin eigmax iter Department of Mathematics, University of Pittsburgh 33

34 Coupling Stokes-Darcy flow with transport concentration of a chemical c(x, t) porosity of the medium φ(x) diffusion - dispersion tensor D t (x, t) (symmetric and positive definite) source Q φc t + (uc D t c) = φq, (x, t) Ω (, T ) IC: c(x, ) = c (x), x Ω BC: { (uc Dt c) n = uc I n on Γ in (D t c) n = on Γ out where Γ in := {x Ω : u n < } and Γ out := {x Ω : u n }. Department of Mathematics, University of Pittsburgh 34

35 Stability and accuracy of a discontinuous Galerkin method z = D t c; (c h, z h ) : approximation of (c, z); Norm for (c h, z h ): (c h, z h ) 2 := φ /2 c h (T ) 2,Ω + + T { u h n, (c h )2 Γ + l T where [c h ] is the jump across the interior boundary edge γ l. Theorem: (Vassilev and.) D /2 t z h 2,Ω dt u h n l, [c h ] 2 γl } dt (c h, z h ) { φ /2 c 2,Ω + T c I u n /2 2,Γ in dt} /2 + T φ /2 Q,Ω dt Theorem: (Vassilev and.) (c c h, z z h ) C(h k + u u h ) C(h k + h r + h r 2+ ) Department of Mathematics, University of Pittsburgh 35

36 Convergence test U V D = 3 I D = mesh c c h L (L 2 ) rate z z h L 2 (L 2 ) rate c c h L (L 2 ) rate 4x4.99e+ 8.95e-3 2.7e+ 8x8 3.27e e e x6 8.48e e e x e e e x64 5.6e e e-3.74 Department of Mathematics, University of Pittsburgh 36

37 Contaminant transport example Permeability perm V Computed velocity Department of Mathematics, University of Pittsburgh 37

38 Transport: initial plume.2.2 Time = C Time = Department of Mathematics, University of Pittsburgh 38

39 Transport: initial plume.2.2 Time = 3 C Time = 3 Department of Mathematics, University of Pittsburgh 39

40 Transport: initial plume.2.2 Time = 5 C Time = 5 Department of Mathematics, University of Pittsburgh 4

41 Transport: initial plume.2.2 Time = 9 C Time = 9 Department of Mathematics, University of Pittsburgh 4

42 Transport: initial plume.2.2 Time = 6 C Time = 6 Department of Mathematics, University of Pittsburgh 42

43 Transport: inflow.2.2 Time = 2 C Time = 2 Department of Mathematics, University of Pittsburgh 43

44 Transport: inflow.2.2 Time = C Time = Department of Mathematics, University of Pittsburgh 44

45 Transport: inflow.2.2 Time = 7 C Time = 7 Department of Mathematics, University of Pittsburgh 45

46 Summary Well-posed mathematical flow model based on continuity of flux and normal stress, and the Beavers-Joseph-Saffman condition Error analysis for mortar multiscale finite element approximations with very general mortar conditions Non-overlapping domain decomposition: s.p.d. interface problem; local Stokes and Darcy solves required at every CG iteration Condition number of interface problem: O(h ) for two subdomains; O(h 2 ) for multiple subdomains Conforming and DG Stokes elements and mixed finite elements for flow are coupled with LDG for transport Department of Mathematics, University of Pittsburgh 46

47 Current and future work Balancing preconditioner for the interface problem A posteriori error estimation and adaptive mesh refinement Extensions to Navier-Stokes and Brinkman Polygonal elements using MFD and DG P.2 K Permeability Pressure and velocity Department of Mathematics, University of Pittsburgh 47