Charla Santiago Numerico

1. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES A priori and a posteriori error analyses of a two-fold saddle point approach for a nonlinear Stokes-Darcy coupled problem ´ G ABRIEL N. G ATICA , R ICARDO OYARZ UA , F RANCISCO -J AVIER S AYAS . WONAPDE 2010 ´ U NIVERSIDAD DE C ONCEPCI ON – C HILE . G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

2. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Contents 1 T HE COUPLED PROBLEM 2 T HE CONTINUOUS FORMULATION 3 T HE GALERKIN FORMULATION 4 A POSTERIORI ERROR ESTIMATOR 5 N UMERICAL EXAMPLES G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

3. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Geometry of the problem ν ΓS ΩS t Σ ν ΩD ΓD ν Incompressible viscous fluid in ΩS Porous medium in ΩD (flowing back and forth across Σ) (saturated with the same fluid) G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

4. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Let fS ∈ L2 (ΩS ) and fD ∈ L2 (ΩS ). 0 Coupled problem: Find velocities (uS , uD ) and pressures (pS , pD )  σ S = − pS I + ν uS in ΩS   − div σ S = fS in ΩS     Stokes equations   div uS = 0 in ΩS  uS = 0 on ΓS     uD = − κ (·, | pD |) pD in ΩD   Darcy equations div uD = fD in ΩD  uD · n = 0 on ΓD  uS · n = uD · n on Σ   Coupling terms ν  σ S n + pD n + (uS · t)t = 0 on Σ κ ν > 0: ﬂuid viscosity, κ: friction constant G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

5. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Assumption on κ There exist constants k0 , k1 > 0, such that for all (x, ρ) ∈ ΩD × R+ : k0 ≤ κ(x, ρ) ≤ k1 , ∂ k0 ≤ κ(x, ρ) + ρ κ(x, ρ) ≤ k1 , and ∂ρ | x κ(x, ρ)| ≤ k1 . G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

6. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES We note that 1 div uS = 0 ∈ ΩS ⇒ pS = − trσ S 2 Rewriting the Stokes equations pS = − 1 trσ S 2 in ΩS ν −1 σd S = uS in ΩS − div σ S = fS in ΩS uS = 0 on ΓS where 1 σ d := σ S − tr σ S I. S 2 G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

7. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Additional unknowns ϕ := −uS ∈ H1/2 (Σ), λ := pD ∈ H 1/2 (Σ) tD := pD in ΩD Rewriting the Darcy equations tD = pD in ΩD uD = − κ (·, |tD |)tD in ΩD div uD = fD in ΩD uD · n = 0 on ΓD Rewriting the coupling terms ϕ · n + uD · n = 0 on Σ σ S n + λn − νκ−1 (ϕ · t)t = 0 on Σ G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

10. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Spaces L2 (Ω ) := [L2 (Ω )]2 , ∈ {S, D} H1/2 (Σ) := [H 1/2 (Σ)]2 H(div ; ΩS ) := τ : ΩS → R2×2 : at τ ∈ H(div ; ΩS ), ∀a ∈ R2 HΓD (div ; ΩD ) := {v ∈ H(div , ΩD ) : v · n = 0 on ΓD } Unknowns (σ S , tD ) ∈ H(div ; ΩS ) × L2 (ΩD ) (uS , uD , ϕ) ∈ L2 (ΩS ) × HΓD (div , ΩD ) × H1/2 (Σ) (pD , λ) ∈ L2 (ΩD ) × H 1/2 (Σ) G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

11. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Spaces L2 (Ω ) := [L2 (Ω )]2 , ∈ {S, D} H1/2 (Σ) := [H 1/2 (Σ)]2 H(div ; ΩS ) := τ : ΩS → R2×2 : at τ ∈ H(div ; ΩS ), ∀a ∈ R2 HΓD (div ; ΩD ) := {v ∈ H(div , ΩD ) : v · n = 0 on ΓD } Unknowns (σ S , tD ) ∈ H(div ; ΩS ) × L2 (ΩD ) (uS , uD , ϕ) ∈ L2 (ΩS ) × HΓD (div , ΩD ) × H1/2 (Σ) (pD , λ) ∈ L2 (ΩD ) × H 1/2 (Σ) G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

12. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Variational equations ν −1 (σ d , τ d )S + (div τ S , uS )S + τ S n, ϕ S S Σ = 0 ∀ τ S ∈ H(div ; ΩS ) (div σ S , vS )S = −(fS , vS )S ∀ vS ∈ L2 (ΩS ) (κ (·, |tD |)tD , sD )D + (uD , sD )D = 0 ∀ sD ∈ L2 (ΩD ) (tD , vD )D + (div vD , pD )D + vD · n, λ Σ = 0 ∀ vD ∈ H(div; ΩD ) (div uD , qD )D = (qD , fD )D ∀ qD ∈ L2 (ΩD ) ϕ · n, ξ Σ + uD · n, ξ Σ = 0 ∀ ξ ∈ H 1/2 (Σ) ν σ S n, ψ Σ + ψ · n, λ Σ − ψ · t, ϕ · t Σ = 0 ∀ ψ ∈ H1/2 (Σ) κ G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

13. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Decomposition of σ S σ S + c I with the new unknowns σ S ∈ H0 (div ; ΩS ) and c ∈ R, where H0 (div ; ΩS ) := {τ S ∈ H(div ; ΩS ) : tr (τ S ) = 0} ΩS G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

14. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES ν −1 (σ d , τ d )S + (div τ S , uS )S + τ S n, ϕ S S Σ = 0 ∀ τ S ∈ H(div ; ΩS ) ν −1 (σ d , τ d )S + (div τ S , uS )S + τ S n, ϕ S S Σ = 0 ∀ τ S ∈ H0 (div ; ΩS ) d ϕ · n, 1 Σ = 0 ∀d ∈ R ν σ S n, ψ Σ + ψ · n, λ Σ − ϕ · t, ψ · t Σ = 0 ∀ψ in H1/2 (Σ) κ ν σ S n, ψ Σ + ψ·n, λ Σ− ϕ·t, ψ·t Σ + c ψ·n, 1 Σ = 0 ∀ ψ ∈ H1/2 (Σ) κ G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

15. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES ν −1 (σ d , τ d )S + (div τ S , uS )S + τ S n, ϕ S S Σ = 0 ∀ τ S ∈ H(div ; ΩS ) ν −1 (σ d , τ d )S + (div τ S , uS )S + τ S n, ϕ S S Σ = 0 ∀ τ S ∈ H0 (div ; ΩS ) d ϕ · n, 1 Σ = 0 ∀d ∈ R ν σ S n, ψ Σ + ψ · n, λ Σ − ϕ · t, ψ · t Σ = 0 ∀ψ in H1/2 (Σ) κ ν σ S n, ψ Σ + ψ·n, λ Σ− ϕ·t, ψ·t Σ + c ψ·n, 1 Σ = 0 ∀ ψ ∈ H1/2 (Σ) κ G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

16. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Global spaces X := H(div ; ΩS ) × L2 (ΩD ) M := L2 (ΩS ) × HΓD (div , ΩD ) × H1/2 (Σ) Q := L2 (ΩD ) × H 1/2 (Σ) × R 0 Global unknowns t := (σ S , tD ) ∈ X u := (uS , uD , ϕ) ∈ M p := (pD , λ, c) ∈ Q G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

17. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Global spaces X := H(div ; ΩS ) × L2 (ΩD ) M := L2 (ΩS ) × HΓD (div , ΩD ) × H1/2 (Σ) Q := L2 (ΩD ) × H 1/2 (Σ) × R 0 Global unknowns t := (σ S , tD ) ∈ X u := (uS , uD , ϕ) ∈ M p := (pD , λ, c) ∈ Q G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

18. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Continuous formulation Find (t, u, p) := ((σ S , tD ), (uS , uD , ϕ), (pD , λ, c)) ∈ X × M × Q such that, [A(t), s] + [B1 (s), u] = [F, s], ∀s ∈ X [B1 (t), v] − [S(u), v] + [B(v), p] = [G1 , v] ∀v ∈ M [B(u), q] = [G, q] ∀q ∈ Q G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

19. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Operators and functionals [A(t), s] := ν −1 (σ d , τ d )S + (κ(·, |tD |)tD , sD )D S S [B1 (s), v] := (div τ S , vS )S + (vD , sD )D + τ S n, ψ Σ [B(v), q] := (div vD , qD )D + vD · n, ξ Σ + ψ · n, ξ Σ + d n, ψ Σ [S(u), v] := νκ −1 ψ · t, ϕ · t Σ [F, s] := 0, [G1 , v] := (fS , vS )S [G, q] := (fD , qD )D G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

20. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Equivalent augmented formulation Find (t, u, p) := ((σ S , tD ), (uS , uD , ϕ), (pD , λ, c)) ∈ X × M × Q such that, ˜ [A(t), s] + [B1 (s), u] ˜ = [F, s] ∀s ∈ X [B1 (t), v] − [S(u), v] + [B(v), p] = [G1 , v] ∀v ∈ M [B(u), q] = [G, q] ∀q ∈ Q ˜ [A(t), s] := [A(t), s] + (div σ S , div τ S )S = ν −1 (σ d , τ d )S + (div σ S , div τ S )S + (κ(·, |tD |)tD , sD )D S S ˜ [F, s] := −(fS , div τ S ) G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

21. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Lemma: Inf-sup condition for B There exists β > 0 such that [B(v), q] sup ≥ β q Q ∀ q ∈ Q. v∈M v M v=0 G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

22. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES kernel(B) ˜ M := v := (vS , vD , ψ) ∈ M : n, ψ = 0, vD · n = −ψ · n on Σ Σ and div vD = 0 in ΩD . Lemma: Inf-sup condition for B1 ˜ Let M := kernel(B), that is ˜ M := v ∈ M : [B(v), q] = 0 ∀ q ∈ Q . Then, there exists β1 > 0 such that [B1 (s), v] ˜ sup ≥ β1 v M ∀ v ∈ M. s∈X s X s=0 G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

23. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES kernel(B) ˜ M := v := (vS , vD , ψ) ∈ M : n, ψ = 0, vD · n = −ψ · n on Σ Σ and div vD = 0 in ΩD . Lemma: Inf-sup condition for B1 ˜ Let M := kernel(B), that is ˜ M := v ∈ M : [B(v), q] = 0 ∀ q ∈ Q . Then, there exists β1 > 0 such that [B1 (s), v] ˜ sup ≥ β1 v M ∀ v ∈ M. s∈X s X s=0 G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

24. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Lemma ˜ The nonlinear operator A : X → X is strongly monotone and Lipschitz continuous, that is, there exist α, γ > 0 such that ˜ ˜ [A(t) − A(r), t − r] ≥ α t − r 2 X and ˜ ˜ [A(t) − A(r), s] ≤ γ t − r X s X, for all t, r, s ∈ X. G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

25. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Theorem For each (F, G1 , G) ∈ X × M × Q there exists a unique (t, u, p) ∈ X × M × Q such that [A(t), s] + [B1 (s), u] = [F, s], ∀ s ∈ X, [B1 (t), v] − [S(u), v] + [B(v), p] = [G1 , v] ∀ v ∈ M, [B(u), q] = [G, q] ∀ q ∈ Q, Moreover, there exists a constant C > 0, independent of the solution, such that (t, u, p) X×M×Q ≤ C{ F X + G1 M + G Q }. G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

26. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Discrete spaces ( ∈ {S, D}) Hh (Ω ) ⊆ H(div ; Ω ) , Lh (Ω ) ⊆ L2 (Ω ) , Λh (Σ) ⊆ H 1/2 (Σ) Lh (Ω ) := [Lh (Ω )]2 , ΛS (Σ) := [ΛS (Σ)]2 h h Hh (ΩS ) := { τ : ΩS → R2×2 : ct τ ∈ Hh (ΩS ) ∀ c ∈ R2 }, Hh,ΓD := v ∈ Hh (ΩD ) : v · n = 0 on ΓD Hh,0 (ΩS ) := Hh (ΩS ) ∩ H0 (div ; ΩS ), Lh,0 (ΩD ) := Lh (ΩD ) ∩ L2 (ΩD ) . 0 G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

27. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Global discrete spaces Xh := Hh,0 (ΩS ) × Lh (ΩD ) Mh := Lh (ΩD ) × Hh,ΓD (ΩD ) × ΛS (Σ) h Qh := Lh,0 (ΩD ) × ΛD (Σ) × R h Global discrete unknowns th := (σ S,h , tD,h ) ∈ Xh uh := (uS,h , uD,h , ϕh ) ∈ Mh ph := (pD,h , λh , ch ) ∈ Qh G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

28. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Global discrete spaces Xh := Hh,0 (ΩS ) × Lh (ΩD ) Mh := Lh (ΩD ) × Hh,ΓD (ΩD ) × ΛS (Σ) h Qh := Lh,0 (ΩD ) × ΛD (Σ) × R h Global discrete unknowns th := (σ S,h , tD,h ) ∈ Xh uh := (uS,h , uD,h , ϕh ) ∈ Mh ph := (pD,h , λh , ch ) ∈ Qh G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

29. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Galerkin scheme [A(th ), s] + [B1 (s), uh ] = [F, s] ∀ s ∈ Xh , [B1 (th ), v] − [S(uh ), v] + [B(v), ph ] = [G1 , v] ∀ v ∈ Mh , [B(uh ), q] = [G, q] ∀ q ∈ Qh ; G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

30. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Equivalent augmented galerkin scheme ˜ [A(th ), s] + [B1 (s), uh ] ˜ = [F, s] ∀ s ∈ Xh , [B1 (th ), v] − [S(uh ), v] + [B(v), ph ] = [G1 , v] ∀ v ∈ Mh , [B(uh ), q] = [G, q] ∀ q ∈ Qh ; G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

31. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Discrete Hypothesis (H.0) [P0 (ΩS )]2 ⊆ Hh (ΩS ) and P0 (ΩD ) ⊆ Lh (ΩD ). (H.1) There exist βD > 0, independent of h and there exists ψ 0 ∈ H1/2 (Σ), such that qh div vh + vh · n, ξh Σ ΩD sup ≥ βD qh 0,ΩD + ξh 1/2,Σ vh ∈ Hh,ΓD (ΩD )0 vh div ,ΩD ∀ (qh , ξh ) ∈ Lh,0 (ΩD ) × ΛD (Σ), h ψ 0 ∈ ΛS (Σ) ∀ h and h ψ 0 · n, 1 Σ = 0. G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

32. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Discrete Hypothesis (H.2) div Hh (ΩD ) ⊆ Lh (ΩD ). ˜ (H.3) Hh (ΩD ) ⊆ Lh (ΩD ), and there exists βS , independent of h, such that vh div τ h + τ h · n, ψh Σ ΩS sup ≥ βS vh 0,ΩS + ψh 1/2,Σ τ h ∈Hh (ΩS )0 τh div ,ΩS ∀ (vh , ψh ) ∈ Lh (ΩS ) × ΛS (Σ), where h ˜ Hh (ΩD ) := vh ∈ Hh (ΩD ) : div vh = 0 . (H.4) div Hh (ΩS ) ⊆ Lh (ΩS ). G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

33. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Theorem Assume that (H.0), (H.1), (H.2), (H.3) and (H.4) hold. Then there exists a unique (th , uh , ph ) ∈ Xh × Mh × Qh such that [A(th ), s] + [B1 (s), uh ] = [F, s] ∀ s ∈ Xh , [B1 (th ), v] − [S(uh ), v] + [B(v), ph ] = [G1 , v] ∀ v ∈ Mh , [B(uh ), q] = [G, q] ∀ q ∈ Qh ˜ Moreover there exist C, C > 0, independent of h, such that (th , uh , ph ) ≤ C F|Xh Xh + G1 |Mh Mh + G|Qh Qh . ˜ (t−th , u−uh , p−ph ) ≤ C inf t−sh X + inf u−vh M + inf p−ph Q sh ∈Xh vh ∈Mh vh ∈Qh G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

34. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Particular choice of discrete spaces Let Th and Th be respective triangulations of the domains ΩS and S D ΩD . S D Raviart–Thomas space of the lowest order (T ∈ Th ∪ Th ) RT0 (T ) := span (1, 0), (0, 1), (x1 , x2 ) . Discrete spaces in Ω ( ∈ {S, D}) Hh (Ω ) := vh ∈ H(div ; Ω ) : vh |T ∈ RT0 (T ) ∀ T ∈ Th , Lh (Ω ) := qh : Ω → R : qh |T ∈ P0 (T ) ∀ T ∈ Th . G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

35. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Particular choice of discrete spaces Let Th and Th be respective triangulations of the domains ΩS and S D ΩD . S D Raviart–Thomas space of the lowest order (T ∈ Th ∪ Th ) RT0 (T ) := span (1, 0), (0, 1), (x1 , x2 ) . Discrete spaces in Ω ( ∈ {S, D}) Hh (Ω ) := vh ∈ H(div ; Ω ) : vh |T ∈ RT0 (T ) ∀ T ∈ Th , Lh (Ω ) := qh : Ω → R : qh |T ∈ P0 (T ) ∀ T ∈ Th . G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

36. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Particular choice of discrete spaces Discrete spaces on the interface (ΛS (Σ) = ΛD (Σ) = Λh (Σ) ) h h Let us assume that the number of edges of Σh is an even number and there exists c > 0, independent of h, such that max |e1 |, |e2 | ≤ c min |e1 |, |e2 | , for each pair e1 , e2 ∈ Σh such that e1 ∪ e2 ∈ Σ2h . Then, we let Σ2h be the partition of Σ arising by joining pairs of adjacent elements, and deﬁne Λh (Σ) := P1 (Σ2h ) ∩ C(Σ) . ´ G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Analysis of fully-mixed ﬁnite element methods for the Stokes-Darcy coupled problem. Preprint 09-08, Departamento de Ingenieria Matematica, Universidad de Concepcion, (2009). G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

37. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES A posteriori error indicator for ΩS Θ2 := S,T fS + div σ S,h 2 0,T + h2 σ d T S,h 2 0,T + h2 rot σ d T S,h 2 0,T ν 2 + he (σ S,h + ch I)n + λh n − (ϕ · t)t κ h 0,e e∈Eh (T )∩Eh (Σ) 2 + he ν −1 σ d t + S,h ϕh t + he ϕh + uS,h 2 0,e 0,e + he [σ d t] S,h 2 0,e + he [uS,h ] 2 0,e e∈E(T )∩(Eh (ΩS )∪Eh (ΓS )) G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

38. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES A posteriori error indicator for ΩD Θ2 := D,T fD − div uD,h 2 0,T + h2 rot (tD,h ) T 2 0,T + h2 tD,h T 2 0,T + κ(·, |tD,h |)tD,h + uD,h 0,T 2 + he [pD,h ] 2 0,e + he [tD,h · t] 0,e e∈E(T )∩(Eh (ΩD )∪Eh (ΓD )) 2 dλh 2 + he tD,h · t − + he ϕh · n + uD,h · n 0,e dt 0,e e∈E(T )∩Eh (Σ) + he pD,h − λh 2 0,e G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

39. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Theorem ˜ There exist Crel , Cef f > 0, independent of h and h such that Cef f Θ ≤ σ − σ h X + u − uh M + p − ph Q ≤ Crel Θ, where  1/2   Θ = Θ2 + S,T Θ2 D,T .  S D  T ∈Th T ∈Th G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

40. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Notations e(tD ) := tD − tD,h 0,ΩD e(uS ) := uS − uS,h 0,ΩS , e(pD ) := pD − pD,h 0,ΩD , e(uD ) := uD − uD,h div ,ΩD e(σ S ) := σ S − σ S,h div ,ΩS , e(λ) := λ − λh 1/2,Σ e(ϕ) := ϕ − ϕh 1/2,Σ 1/2 eT := e(tD )2 + e(uS )2 + e(pD )2 + e(uD )2 + e(σ S )2 + e(λ)2 + e(ϕ)2 G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

41. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Example 1. µ = 1, κ = 1. ΩS := (−1, 1) × (0, 1) , ΩD := (−1, 1) × (−1, 0) , uS (x, y) := curl (x2 − 1)2 (y − 1)2 , pS (x, y) := x3 + y 3 , pD (x, y) := sin(πx)3 (y + 1)2 , 1 κ(·, s) := 2 + . 1+s G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

42. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Errors and rates of convergence. N e(tD ) r(tD ) e(uS ) r(uS ) e(pD ) r(pD ) e(uD ) r(uD ) e(σ S ) r(σ S ) 172 2.072 — 0.449 — 0.718 — 4.714 — 2.837 — 644 0.988 1.123 0.233 0.997 0.180 2.099 2.176 1.171 1.287 1.197 2500 0.535 0.904 0.115 1.045 0.060 1.622 1.123 0.976 0.619 1.099 9860 0.250 1.106 0.057 1.019 0.026 1.193 0.522 1.117 0.302 1.027 39172 0.124 1.019 0.028 1.007 0.013 1.032 0.258 1.021 0.151 1.008 156164 0.062 1.004 0.014 1.003 0.006 1.008 0.129 1.005 0.075 1.003 N ˜ h e(λ) r(λ) e(ϕ) r(ϕ) 172 0.998 1.174 — 2.859 — 644 0.499 0.892 0.417 1.187 1.332 2500 0.250 0.490 0.884 0.534 1.178 9860 0.125 0.217 1.187 0.258 1.061 39172 0.062 0.105 1.051 0.128 1.018 156164 0.031 0.052 1.014 0.064 1.006 N eT r(eT ) Θ r(Θ) eT /Θ 172 6.6958 — 4.2323 — 1.5821 644 3.1075 1.1629 2.0629 1.0887 1.5064 2500 1.5690 1.0077 0.9789 1.0992 1.6028 9860 0.7374 1.1005 0.4929 1.0001 1.4961 39172 0.3647 1.0209 0.2461 1.0071 1.4819 156164 0.1820 1.0054 0.1233 0.9990 1.4754 G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

43. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Example 2, µ = 1, κ = 1. ΩS := (−1, 1) × (0, 1)[0, 1] × [0.5, 1] , ΩD := (−1, 1) × (−1, 0) [0, 1] × [−1, −0.5] , x2 (x2 − 1)2 (y − 1)2 (y − 0.5)2 uS (x, y) = curl , (x2 + (y − 0.5)2 + 0.01)2 pS (x, y) = sin(2πx)3 (y + 1)2 (y + 0.5)2 , pD (x, y) = sin(2πx)3 (y + 1)2 (y + 0.5)2 , 1 κ(·, s) := 2 + . 1+s G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

44. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Uniform reﬁnement. N eT r(eT ) Θ eT /Θ 110 39.0943 — 38.8240 1.0070 396 53.6275 — 54.1512 0.9903 1508 65.9962 — 66.4284 0.9935 5892 45.7855 0.5366 46.0250 0.9948 23300 24.1982 0.9276 24.2902 0.9962 92676 12.9528 0.9053 12.9270 1.0020 Adaptive reﬁnement. N eT r(eT ) Θ e/Θ 110 39.0943 — 38.8240 1.0070 289 54.9264 — 55.3642 0.9921 479 66.9987 — 67.5925 0.9912 657 52.4883 1.5449 53.0637 0.9892 1315 35.2800 1.1450 35.9303 0.9819 3759 21.5520 0.9385 21.8753 0.9852 4017 20.4178 1.6288 20.7288 0.9850 7875 15.6882 0.7829 15.9045 0.9864 10191 13.5102 1.1595 13.7352 0.9836 16558 10.4623 1.0535 10.5737 0.9895 28745 7.9688 0.9871 8.0054 0.9954 48715 6.1675 0.9715 6.1166 1.0083 70713 5.1460 0.9718 5.0318 1.0227 109264 4.2753 0.8520 4.0772 1.0486 G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

45. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Errors vs degrees of freedom. 100 adaptative ♦ ♦ + uniform + ♦ + ♦ + + ♦ ♦ + ♦ ♦ e ♦ ♦ + 10 ♦ ♦ ♦ ♦ ♦ 100 1000 10000 100000 N G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

46. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES Grids with 110, 1315, 4017 and 28745 degrees of freedom. G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

47. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLES ˇ I. B ABU SKA AND G.N. G ATICA, On the mixed finite element method with Lagrange multipliers. Numerical Methods for Partial Differential Equations, vol. 19, 2, pp. 192-210, (2003). F. B REZZI AND M. F ORTIN, Mixed and Hybrid Finite Element Methods. Springer Verlag, 1991. G.N. G ATICA , N. H EUER , AND S. M EDDAHI, On the numerical analysis of nonlinear twofold saddle point problems. IMA Journal of Numerical Analysis, vol. 23, 2, pp. 301-330, (2003). ´ G.N. G ATICA , S. M EDDAHI , AND R. OYARZ UA, A conforming mixed finite-element method for the coupling of fluid flow with porous media flow. IMA Journal of Numerical Analysis, vol. 29, 1, pp. 86-108, (2009). ´ G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Convergence of a family of Galerkin discretizations for the Stokes-Darcy problem. Numerical Methods for Partial Differential Equations, to appear. ´ G.N. G ATICA , R. OYARZ UA AND F-J. S AYAS, Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem. Preprint 09-08, Departamento de Ingenieria Matematica, Universidad de Concepcion, (2009). G.N. G ATICA , W.L. W ENLAND, Coupling of mixed finite elements and boundary elements for linear and nonlinear elliptic problems. Applicable Analysis, 63, 39-75, (1996). G.N. G ATICA AND F-J. S AYAS, Characterizing the inf-sup condition on product spaces. Numerische Matematik, vol. 109,2,pp 209-231, (2008). W.J. L AYTON , F. S CHIEWECK , AND I. YOTOV, Coupling fluid flow with porous media flow. SIAM Journal on Numerical Analysis, vol. 40, 6, pp. 2195-2218, (2003). G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem

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