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Charla Santiago Numerico

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Charla Santiago Numerico

  1. 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 atwo-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. 2. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESContents 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. 3. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESGeometry 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. 4. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESLet fS ∈ L2 (ΩS ) and fD ∈ L2 (ΩS ). 0Coupled 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 κν > 0: fluid viscosity, κ: friction constant G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem
  5. 5. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESAssumption on κThere 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. 6. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESWe note that 1 div uS = 0 ∈ ΩS ⇒ pS = − trσ S 2Rewriting 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 ΓSwhere 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. 7. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESAdditional unknowns ϕ := −uS ∈ H1/2 (Σ), λ := pD ∈ H 1/2 (Σ) tD := pD in ΩDRewriting the Darcy equations tD = pD in ΩD uD = − κ (·, |tD |)tD in ΩD div uD = fD in ΩD uD · n = 0 on ΓDRewriting 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
  8. 8. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESAdditional unknowns ϕ := −uS ∈ H1/2 (Σ), λ := pD ∈ H 1/2 (Σ) tD := pD in ΩDRewriting the Darcy equations tD = pD in ΩD uD = − κ (·, |tD |)tD in ΩD div uD = fD in ΩD uD · n = 0 on ΓDRewriting 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
  9. 9. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESAdditional unknowns ϕ := −uS ∈ H1/2 (Σ), λ := pD ∈ H 1/2 (Σ) tD := pD in ΩDRewriting the Darcy equations tD = pD in ΩD uD = − κ (·, |tD |)tD in ΩD div uD = fD in ΩD uD · n = 0 on ΓDRewriting 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. 10. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESSpaces 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. 11. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESSpaces 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. 12. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESVariational 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. 13. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESDecomposition of σ Sσ 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. 14. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESν −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. 15. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESν −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. 16. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESGlobal spaces X := H(div ; ΩS ) × L2 (ΩD ) M := L2 (ΩS ) × HΓD (div , ΩD ) × H1/2 (Σ) Q := L2 (ΩD ) × H 1/2 (Σ) × R 0Global 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. 17. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESGlobal spaces X := H(div ; ΩS ) × L2 (ΩD ) M := L2 (ΩS ) × HΓD (div , ΩD ) × H1/2 (Σ) Q := L2 (ΩD ) × H 1/2 (Σ) × R 0Global 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. 18. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESContinuous formulationFind (t, u, p) := ((σ S , tD ), (uS , uD , ϕ), (pD , λ, c)) ∈ X × M × Q suchthat, [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. 19. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESOperators 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. 20. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESEquivalent augmented formulationFind (t, u, p) := ((σ S , tD ), (uS , uD , ϕ), (pD , λ, c)) ∈ X × M × Q suchthat, ˜ [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. 21. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESLemma: Inf-sup condition for BThere 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. 22. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESkernel(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. 23. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESkernel(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. 24. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESLemma ˜The nonlinear operator A : X → X is strongly monotone andLipschitz continuous, that is, there exist α, γ > 0 such that ˜ ˜ [A(t) − A(r), t − r] ≥ α t − r 2 Xand ˜ ˜ [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. 25. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESTheoremFor 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. 26. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESDiscrete 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. 27. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESGlobal discrete spaces Xh := Hh,0 (ΩS ) × Lh (ΩD ) Mh := Lh (ΩD ) × Hh,ΓD (ΩD ) × ΛS (Σ) h Qh := Lh,0 (ΩD ) × ΛD (Σ) × R hGlobal 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. 28. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESGlobal discrete spaces Xh := Hh,0 (ΩS ) × Lh (ΩD ) Mh := Lh (ΩD ) × Hh,ΓD (ΩD ) × ΛS (Σ) h Qh := Lh,0 (ΩD ) × ΛD (Σ) × R hGlobal 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. 29. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESGalerkin 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. 30. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESEquivalent 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. 31. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESDiscrete 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. 32. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESDiscrete 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. 33. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESTheoremAssume that (H.0), (H.1), (H.2), (H.3) and (H.4) hold. Then thereexists 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. 34. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESParticular 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. 35. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESParticular 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. 36. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESParticular 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. 37. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESA 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. 38. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESA 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. 39. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESTheorem ˜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. 40. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESNotations 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 1/2eT := 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. 41. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESExample 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. 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 1.008156164 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. 43. T HE COUPLED PROBLEM T HE CONTINUOUS FORMULATION T HE GALERKIN FORMULATION A POSTERIORI ERROR ESTIMATOR N UMERICAL EXAMPLESExample 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. 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 1.0486G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem
  45. 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 NG. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas ´ Stokes-Darcy coupled problem
  46. 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. 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
  48. 48. 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
  49. 49. 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
  50. 50. 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
  51. 51. 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
  52. 52. 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
  53. 53. 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
  54. 54. 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
  55. 55. 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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