Charla Santiago Numerico

THE COUPLED PROBLEM

THE CONTINUOUS FORMULATION

THE GALERKIN FORMULATION

A POSTERIORI ERROR ESTIMATOR

NUMERICAL EXAMPLES

A priori and a posteriori error analyses of atwo-fold saddle point approach for a nonlinear

Stokes-Darcy coupled problem

GABRIEL N. GATICA, RICARDO OYARZUA,FRANCISCO-JAVIER SAYAS.

WONAPDE 2010UNIVERSIDAD DE CONCEPCION – CHILE.

G. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas Stokes-Darcy coupled problem

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Contents

1 THE COUPLED PROBLEM

2 THE CONTINUOUS FORMULATION

3 THE GALERKIN FORMULATION

4 A POSTERIORI ERROR ESTIMATOR

5 NUMERICAL EXAMPLES

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Geometry of the problem

Incompressible viscous fluid in ΩS Porous medium in ΩD

(flowing back and forth across Σ) (saturated with the same fluid)

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Let fS ∈ L2(ΩS) and fD ∈ L20(ΩS).

Coupled problem: Find velocities (uS,uD) and pressures (pS, pD)

Stokes equations

σS = − pS I + ν∇uS in ΩS

−divσS = fS in ΩS

div uS = 0 in ΩS

uS = 0 on ΓS

Darcy equations

uD = −κ (·, |∇ pD|)∇ pD in ΩD

div uD = fD in ΩD

uD · n = 0 on ΓD

Coupling terms

uS · n = uD · n on Σ

σSn + pDn +ν

κ(uS · t)t = 0 on Σ

ν > 0: fluid viscosity, κ: friction constantG. N. Gatica, S. Meddahi ,R. Oyarzua, F.-J. Sayas Stokes-Darcy coupled problem

THE COUPLED PROBLEM

NUMERICAL 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∂

∂ρκ(x, ρ) ≤ k1, and

|∇xκ(x, ρ)| ≤ k1.

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

We note thatdiv uS = 0 ∈ ΩS ⇒ pS = −1

2trσS

Rewriting the Stokes equations

pS = − 12 trσS in ΩS

ν−1σdS = ∇uS in ΩS

−divσS = fS in ΩS

uS = 0 on ΓS

whereσd

S := σS −12

trσSI.

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Additional unknowns

ϕ := −uS ∈ H1/2(Σ), λ := pD ∈ H1/2(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

σSn + λn− νκ−1(ϕ · t)t = 0 on Σ

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Additional unknowns

ϕ := −uS ∈ H1/2(Σ), λ := pD ∈ H1/2(Σ)

tD := ∇pD in ΩD

tD = ∇pD in ΩD

div uD = fD in ΩD

uD · n = 0 on ΓD

ϕ · n + uD · n = 0 on Σ

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Additional unknowns

ϕ := −uS ∈ H1/2(Σ), λ := pD ∈ H1/2(Σ)

tD := ∇pD in ΩD

tD = ∇pD in ΩD

div uD = fD in ΩD

uD · n = 0 on ΓD

ϕ · n + uD · n = 0 on Σ

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Spaces

L2(Ω?) := [L2(Ω?)]2, ? ∈ S, D

H1/2(Σ) := [H1/2(Σ)]2

H(div ; ΩS) :=τ : ΩS → R2×2 : atτ ∈ H(div ; ΩS), ∀a ∈ R2

HΓD(div ; ΩD) := v ∈ H(div ,ΩD) : v · n = 0 on ΓD

Unknowns

(σS, tD) ∈ H(div ; ΩS)× L2(ΩD)

(uS,uD,ϕ) ∈ L2(ΩS)×HΓD(div ,ΩD)×H1/2(Σ)

(pD, λ) ∈ L2(ΩD)×H1/2(Σ)

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Spaces

L2(Ω?) := [L2(Ω?)]2, ? ∈ S, D

H1/2(Σ) := [H1/2(Σ)]2

H(div ; ΩS) :=τ : ΩS → R2×2 : atτ ∈ H(div ; ΩS), ∀a ∈ R2

HΓD(div ; ΩD) := v ∈ H(div ,ΩD) : v · n = 0 on ΓD

Unknowns

(σS, tD) ∈ H(div ; ΩS)× L2(ΩD)

(uS,uD,ϕ) ∈ L2(ΩS)×HΓD(div ,ΩD)×H1/2(Σ)

(pD, λ) ∈ L2(ΩD)×H1/2(Σ)

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Variational equations

ν−1(σdS, τ

dS)S + (div τ S,uS)S + 〈τ Sn,ϕ〉Σ = 0 ∀ τ S ∈ H(div ; ΩS)

(divσS,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 ∈ H1/2(Σ)

〈σSn,ψ〉Σ + 〈ψ · n, λ〉Σ −ν

κ〈ψ · t,ϕ · t〉Σ = 0 ∀ψ ∈ H1/2(Σ)

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Decomposition of σS

σS + c I with the new unknowns σS ∈ H0(div ; ΩS) and c ∈ R ,

H0(div ; ΩS) := τ S ∈ H(div ; ΩS) :∫

tr (τ S) = 0

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

ν−1 (σdS, τ

dS)S + (div τ S,uS)S + 〈τ S n,ϕ〉Σ = 0 ∀ τ S ∈ H(div ; ΩS)

ν−1 (σdS, τ

dS)S + (div τ S,uS)S + 〈τ S n,ϕ〉Σ = 0 ∀ τ S ∈ H0(div ; ΩS)

d 〈ϕ · n, 1〉Σ = 0 ∀ d ∈ R

〈σS n,ψ〉Σ + 〈ψ · n, λ〉Σ −ν

κ〈ϕ · t,ψ · t〉Σ = 0 ∀ψ in H1/2(Σ)

〈σS n,ψ〉Σ + 〈ψ·n, λ〉Σ−ν

κ〈ϕ·t,ψ·t〉Σ + c 〈ψ·n, 1〉Σ = 0 ∀ψ ∈ H1/2(Σ)

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

ν−1 (σdS, τ

dS)S + (div τ S,uS)S + 〈τ S n,ϕ〉Σ = 0 ∀ τ S ∈ H(div ; ΩS)

ν−1 (σdS, τ

dS)S + (div τ S,uS)S + 〈τ S n,ϕ〉Σ = 0 ∀ τ S ∈ H0(div ; ΩS)

d 〈ϕ · n, 1〉Σ = 0 ∀ d ∈ R

〈σS n,ψ〉Σ + 〈ψ · n, λ〉Σ −ν

κ〈ϕ · t,ψ · t〉Σ = 0 ∀ψ in H1/2(Σ)

〈σS n,ψ〉Σ + 〈ψ·n, λ〉Σ−ν

κ〈ϕ·t,ψ·t〉Σ + c 〈ψ·n, 1〉Σ = 0 ∀ψ ∈ H1/2(Σ)

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Global spaces

X := H(div ; ΩS)× L2(ΩD)

M := L2(ΩS)×HΓD(div ,ΩD)×H1/2(Σ)

Q := L20(ΩD)×H1/2(Σ)× R

Global unknowns

t := (σS, tD) ∈ X

u := (uS,uD,ϕ) ∈ M

p := (pD, λ, c) ∈ Q

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Global spaces

X := H(div ; ΩS)× L2(ΩD)

M := L2(ΩS)×HΓD(div ,ΩD)×H1/2(Σ)

Q := L20(ΩD)×H1/2(Σ)× R

Global unknowns

t := (σS, tD) ∈ X

u := (uS,uD,ϕ) ∈ M

p := (pD, λ, c) ∈ Q

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Continuous formulation

Find (t,u,p) := ((σS, 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

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Operators and functionals

[A(t), s] := ν−1(σdS, τ

dS)S + (κ(·, |tD|)tD, sD)D

[B1(s),v] := (div τ S,vS)S + (vD, sD)D + 〈τ Sn,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

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Equivalent augmented formulation

Find (t,u,p) := ((σS, 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] + (divσS,div τ S)S

= ν−1(σdS, τ

dS)S + (divσS,div τ S)S + (κ(·, |tD|)tD, sD)D

[F, s] := −(fS,div τ S)

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Lemma: Inf-sup condition for B

There exists β > 0 such that

supv∈M

[B(v),q]‖v‖M

≥ β ‖q‖Q ∀q ∈ Q.

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

kernel(B)

v := (vS,vD,ψ) ∈M : 〈n,ψ〉Σ = 0, vD ·n = −ψ ·n on 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

sups∈X

[B1(s),v]‖s‖X

≥ β1 ‖v‖M ∀v ∈ M.

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

kernel(B)

v := (vS,vD,ψ) ∈M : 〈n,ψ〉Σ = 0, vD ·n = −ψ ·n on 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

sups∈X

[B1(s),v]‖s‖X

≥ β1 ‖v‖M ∀v ∈ M.

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

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‖2X

and[A(t) − A(r), s] ≤ γ ‖t− r‖X ‖s‖X,

for all t, r, s ∈ X.

THE COUPLED PROBLEM

NUMERICAL 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′ .

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Discrete spaces (? ∈ S,D)

Hh(Ω?) ⊆ H(div ; Ω?) , Lh(Ω?) ⊆ L2(Ω?) , Λ?h(Σ) ⊆ H1/2(Σ)

Lh(Ω?) := [Lh(Ω?)]2 , ΛSh(Σ) := [ΛS

h(Σ)]2

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

0(ΩD) .

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Global discrete spaces

Xh := Hh,0(ΩS)× Lh(ΩD)

Mh := Lh(ΩD)×Hh,ΓD(ΩD)×ΛSh(Σ)

Qh := Lh,0(ΩD)× ΛDh (Σ)× R

Global discrete unknowns

th := (σS,h, tD,h) ∈ Xh

uh := (uS,h,uD,h,ϕh) ∈Mh

:= (pD,h, λh, ch) ∈ Qh

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Global discrete spaces

Xh := Hh,0(ΩS)× Lh(ΩD)

Mh := Lh(ΩD)×Hh,ΓD(ΩD)×ΛSh(Σ)

Qh := Lh,0(ΩD)× ΛDh (Σ)× R

Global discrete unknowns

th := (σS,h, tD,h) ∈ Xh

uh := (uS,h,uD,h,ϕh) ∈Mh

:= (pD,h, λh, ch) ∈ Qh

THE COUPLED PROBLEM

NUMERICAL 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;

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Equivalent augmented galerkin scheme

[A(th), s] + [B1(s),uh] = [F, s] ∀ s ∈ Xh,

[B(uh),q] = [G,q] ∀q ∈ Qh;

THE COUPLED PROBLEM

NUMERICAL 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 existsψ0 ∈ H1/2(Σ), such that

supvh∈Hh,ΓD (ΩD)\0

∫ΩD

qh div vh + 〈vh · n, ξh〉Σ

‖vh‖div ,ΩD

≥ βD

(‖qh‖0,ΩD + ‖ξh‖1/2,Σ

)∀ (qh, ξh) ∈ Lh,0(ΩD)× ΛD

h (Σ),

ψ0 ∈ ΛSh(Σ) ∀h and 〈ψ0 · n, 1〉Σ 6= 0.

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Discrete Hypothesis

(H.2) div Hh(ΩD) ⊆ Lh(ΩD).

(H.3) Hh(ΩD) ⊆ Lh(ΩD), and there exists βS, independent of h, suchthat

supτ h∈Hh(ΩS)\0

∫ΩS

vh div τh + 〈τh · n, ψh〉Σ

‖τh‖div ,ΩS

≥ βS

(‖vh‖0,ΩS + ‖ψh‖1/2,Σ

)∀ (vh, ψh) ∈ Lh(ΩS)× ΛS

h(Σ), where

Hh(ΩD) :=

vh ∈ Hh(ΩD) : div vh = 0.

(H.4) div Hh(ΩS) ⊆ Lh(ΩS).

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Theorem

Assume 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,

[B(uh),q] = [G,q] ∀q ∈ Qh

Moreover there exist C, C > 0, independent of h, such that

‖(th,uh,ph)‖ ≤ C

‖F|Xh

‖X′h

+ ‖G1|Mh‖M′

h+ ‖G|Qh

‖Q′h

‖(t−th,u−uh,p−ph)‖ ≤ C

sh∈Xh

‖t−sh‖X + infvh∈Mh

‖u−vh‖M+ infvh∈Qh

‖p−ph‖Q

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Particular choice of discrete spaces

Let T Sh and T D

h be respective triangulations of the domains ΩS andΩD.

Raviart–Thomas space of the lowest order (T ∈ T Sh ∪ T D

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 ∈ T ?h

Lh(Ω?) :=qh : Ω? → R : qh|T ∈ P0(T ) ∀T ∈ T ?

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Let T Sh and T D

h be respective triangulations of the domains ΩS andΩD.

Raviart–Thomas space of the lowest order (T ∈ T Sh ∪ T D

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 ∈ T ?h

Lh(Ω?) :=qh : Ω? → R : qh|T ∈ P0(T ) ∀T ∈ T ?

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Discrete spaces on the interface (ΛSh(Σ) = ΛD

h (Σ) = Λh(Σ) )

Let us assume that the number of edges of Σh is an even numberand 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, anddefine

Λh(Σ) := P1(Σ2h) ∩ C(Σ) .

G.N. GATICA, R. OYARZUA AND F-J. SAYAS, Analysis of fully-mixed finite element methods for the Stokes-Darcy

coupled problem. Preprint 09-08, Departamento de Ingenieria Matematica, Universidad de Concepcion, (2009).

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

A posteriori error indicator for ΩS

Θ2S,T := ‖fS + divσS,h‖20,T + h2

T ‖σdS,h‖20,T + h2

T ‖rotσdS,h‖20,T

e∈Eh(T )∩Eh(Σ)

∥∥∥(σS,h + chI)n + λhn− ν

κ(ϕh · t)t

∥∥∥2

∥∥∥ν−1σdS,ht +∇ϕht

∥∥∥2

0,e+ he‖ϕh + uS,h‖20,e

∑e∈E(T )∩(Eh(ΩS)∪Eh(ΓS))

he‖[σd

S,ht]‖20,e + he‖[uS,h]‖20,e

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

A posteriori error indicator for ΩD

Θ2D,T := ‖fD − div uD,h‖20,T + h2

T ‖rot (tD,h)‖20,T

+h2T ‖tD,h‖20,T + ‖κ(·, |tD,h|)tD,h + uD,h‖0,T

e∈E(T )∩(Eh(ΩD)∪Eh(ΓD))

he‖[pD,h]‖20,e + he ‖[tD,h · t]‖20,e

e∈E(T )∩Eh(Σ)

∥∥∥∥tD,h · t−dλh

∥∥∥∥2

+ he ‖ϕh · n + uD,h · n‖20,e

+he‖pD,h − λh‖20,e

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Theorem

There exist Crel, Ceff > 0, independent of h and h such that

Ceff Θ ≤ ‖σ − σh‖X + ‖u− uh‖M + ‖p− ph‖Q ≤ CrelΘ,

∑T∈T S

Θ2S,T +

∑T∈T D

Θ2D,T

THE COUPLED PROBLEM

NUMERICAL 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(σS) := ‖σS − σS,h‖div ,ΩS , e(λ) := ‖λ− λh‖1/2,Σ

e(ϕ) := ‖ϕ−ϕh‖1/2,Σ

eT :=e(tD)2 + e(uS)2 + e(pD)2 + e(uD)2 + e(σS)2 + e(λ)2 + e(ϕ)2

THE COUPLED PROBLEM

NUMERICAL 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 + y3 ,

pD(x, y) := sin(πx)3(y + 1)2 ,

κ(·, s) := 2 +1

1 + s.

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Errors and rates of convergence.

N e(tD) r(tD) e(uS) r(uS) e(pD) r(pD) e(uD) r(uD) e(σS) r(σS)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 1.0999860 0.250 1.106 0.057 1.019 0.026 1.193 0.522 1.117 0.302 1.02739172 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 1.1789860 0.125 0.217 1.187 0.258 1.06139172 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/Θ172 6.6958 — 4.2323 — 1.5821644 3.1075 1.1629 2.0629 1.0887 1.50642500 1.5690 1.0077 0.9789 1.0992 1.60289860 0.7374 1.1005 0.4929 1.0001 1.4961

39172 0.3647 1.0209 0.2461 1.0071 1.4819156164 0.1820 1.0054 0.1233 0.9990 1.4754

THE COUPLED PROBLEM

NUMERICAL 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] ,

uS(x, y) = curl(x2(x2 − 1)2(y − 1)2(y − 0.5)2

(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 ,

κ(·, s) := 2 +1

1 + s.

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Uniform refinement.N eT r(eT) Θ eT/Θ

110 39.0943 — 38.8240 1.0070396 53.6275 — 54.1512 0.9903

1508 65.9962 — 66.4284 0.99355892 45.7855 0.5366 46.0250 0.994823300 24.1982 0.9276 24.2902 0.996292676 12.9528 0.9053 12.9270 1.0020

Adaptive refinement.N eT r(eT) Θ e/Θ

110 39.0943 — 38.8240 1.0070289 54.9264 — 55.3642 0.9921479 66.9987 — 67.5925 0.9912657 52.4883 1.5449 53.0637 0.9892

1315 35.2800 1.1450 35.9303 0.98193759 21.5520 0.9385 21.8753 0.98524017 20.4178 1.6288 20.7288 0.98507875 15.6882 0.7829 15.9045 0.986410191 13.5102 1.1595 13.7352 0.983616558 10.4623 1.0535 10.5737 0.989528745 7.9688 0.9871 8.0054 0.995448715 6.1675 0.9715 6.1166 1.008370713 5.1460 0.9718 5.0318 1.0227

109264 4.2753 0.8520 4.0772 1.0486

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Errors vs degrees of freedom.

100 1000 10000 100000

adaptative

♦♦

♦♦♦ ♦

♦♦

♦uniform

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Grids with 110, 1315, 4017 and 28745 degrees of freedom.

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

I. BABUSKA AND G.N. GATICA, On the mixed finite element method with Lagrange multipliers. NumericalMethods for Partial Differential Equations, vol. 19, 2, pp. 192-210, (2003).

F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods. Springer Verlag, 1991.

G.N. GATICA, N. HEUER, AND S. MEDDAHI, On the numerical analysis of nonlinear twofold saddle pointproblems. IMA Journal of Numerical Analysis, vol. 23, 2, pp. 301-330, (2003).

G.N. GATICA, S. MEDDAHI, AND R. OYARZUA, A conforming mixed finite-element method for the coupling offluid flow with porous media flow. IMA Journal of Numerical Analysis, vol. 29, 1, pp. 86-108, (2009).

G.N. GATICA, R. OYARZUA AND F-J. SAYAS, Convergence of a family of Galerkin discretizations for theStokes-Darcy problem. Numerical Methods for Partial Differential Equations, to appear.

G.N. GATICA, R. OYARZUA AND F-J. SAYAS, Analysis of fully-mixed finite element methods for theStokes-Darcy coupled problem. Preprint 09-08, Departamento de Ingenieria Matematica, Universidad deConcepcion, (2009).

G.N. GATICA, W.L. WENLAND, Coupling of mixed finite elements and boundary elements for linear andnonlinear elliptic problems. Applicable Analysis, 63, 39-75, (1996).

G.N. GATICA AND F-J. SAYAS, Characterizing the inf-sup condition on product spaces. NumerischeMatematik, vol. 109,2,pp 209-231, (2008).

W.J. LAYTON, F. SCHIEWECK, AND I. YOTOV, Coupling fluid flow with porous media flow. SIAM Journal onNumerical Analysis, vol. 40, 6, pp. 2195-2218, (2003).

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

THE COUPLED PROBLEM

NUMERICAL EXAMPLES

Charla Santiago Numerico

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