A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary...

34
A posteriori error estimators for higher-order methods applied to 1D reaction-diffusion problems Torsten Linß [email protected] FernUniversit ¨ at in Hagen, Fakult ¨ at f ¨ ur Mathematik und Informatik T. Linß, Dresden 16-18 Nov 2011 – p. 1/24

Transcript of A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary...

Page 1: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

A posteriori error estimatorsfor higher-order methods appliedto 1D reaction-diffusion problems

Torsten Linß[email protected]

FernUniversitat in Hagen, Fakultat fur Mathematik und Informatik

T. Linß, Dresden 16-18 Nov 2011 – p. 1/24

Page 2: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

Introduction

Model problem

Lu := −ε2u′′ + cu = f in (0, 1), u(0) = u(1) = 0.

0 < ε ≪ 1, c ≥ γ2 on [0, 1], γ > 0

boundary/interiour layers

Goal: maximum-norm a posteriori error estimators

‖(u − uh)‖∞ ≤ η(data, uh)

of interpolation type

η(data, uh) = Crhκi ‖Dκuh‖[xi−1,xi] + osc

T. Linß, Dresden 16-18 Nov 2011 – p. 2/24

Page 3: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

Outline

Variational formulationGreen’s functions

FEMdiscretisationa posteriori error analysis

C1-collocation [with H. Zarin, G. Radojev]discretisationa posteriori error analysis

T. Linß, Dresden 16-18 Nov 2011 – p. 3/24

Page 4: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

Variational formulation

Boundary value problem:

Lu := −ε2u′′ + cu = f in (0, 1), u(0) = u(1) = 0

Variational formulation: Find u ∈ V := H10 (0, 1) such that

a(u, v) = f(v) ∀ v ∈ V

with

a(u, v) := ε2(u′, v′) + (cu, v), f(v) := (f, v) :=

∫ 1

0fv

T. Linß, Dresden 16-18 Nov 2011 – p. 4/24

Page 5: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

Green’s function

G : [0, 1] × [0, 1] → with

v(x) = a(v,G(x, ·)

)∀ v ∈ V and x ∈ (0, 1)

“⇐⇒”

v(x) =

∫ 1

0

(Lv

)(ξ)G(x, ξ) dξ

Characterisation:

(LG(·, ξ)) (x) = δ(x − ξ) x ∈ (0, 1), G(0, ξ) = G(1, ξ) = 0

(L∗G(x, ·)) (ξ) = δ(ξ − x), ξ ∈ (0, 1), G(x, 0) = G(x, 1) = 0

. . . L = L∗ =⇒ G(x, ξ) = G(ξ, x)

T. Linß, Dresden 16-18 Nov 2011 – p. 5/24

Page 6: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

Green’s function

G(x, ·), ε2 = 10−3

0

2

4

6

8

10

12

14

16

0.2 0.4 0.6 0.8 1

T. Linß, Dresden 16-18 Nov 2011 – p. 6/24

Page 7: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

Green’s function

Pointwise bounds:

0 ≤ G(x, ξ) ≤ e−γ|x−ξ|/ε

2εγ

and

Gξ(x, ξ) ≥ 0 for ξ < x,

Gξ(x, ξ) ≤ 0 for x < ξ.

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Page 8: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

Green’s function

L1-bounds:∫ 1

0c(ξ)G(x, ξ) dξ ≤ 1,

∫ 1

0

∣∣Gξ(x, ξ)∣∣ dξ = 2G(x, x) ≤ 1

εγ

and

ε2

∫ 1

0

∣∣Gξξ(x, ξ)∣∣ dξ ≤ 2.

T. Linß, Dresden 16-18 Nov 2011 – p. 8/24

Page 9: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

FEM discretisation

Mesh: ∆ : 0 = x0 < x1 < · · · < xN = 1,Ji := [xi−1, xi], hi := xi − xi−1.

Splines: piecewise Πr, Cm, bc’s

Smr (∆) :=

{v ∈ Cm[0, 1] : v|Ji

∈ Πr for i = 1, . . . , N}

Smr,0(∆) :=

{v ∈ Sm

r (∆) : v(0) = v(1) = 0}

Standard FEM: Find u∆ ∈ Vr = S0r,0(∆) such that

a(u∆, v

)= f

(v)

∀ v ∈ Vr.

→ quadrature is essential

T. Linß, Dresden 16-18 Nov 2011 – p. 9/24

Page 10: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

FEM discretisation

Lagrange Interpolation: sub mesh on [0, 1] → Ji

0 ≤ t0 < t1 < · · · < tr ≤ 1.

ILr : v ∈ C0[0, 1] 7→ IL

r v ∈ S0r (∆) with

ILr v(xi + tjhi) = v(xi + tjhi), i = 1, . . . , N, j = 0, . . . , r.

Then (w, v) ≈(w, v

)∆

:=(ILr w, v

),

FEM with quadrature: Find u∆ ∈ Vr such that

a∆(u∆, v) := ε2(u′

h, v′)

+(cu∆, v

)∆

= (f, v)∆ ∀v ∈ Vr,

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Page 11: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

FEM, a posteriori analysis

Error in point x ∈ (0, 1), Γ := G(x, ·):

(u − u∆) (x) = a(u − u∆,Γ) = (f,Γ) − a(u∆,Γ)

= (f,Γ) − (f, IMr Γ)∆ − a(u∆,Γ) + a∆(u∆, IM

r Γ)

Special interpolant: IMr : v ∈ C0[0, 1] 7→ IM

r v ∈ Vr with

IMr v(xi) = v(xi), i = 0, . . . , N,

and∫

Ji

(IMr v − v

)(ξ)π(ξ) dξ = 0 ∀π ∈ Πr−2, i = 1, . . . , N.

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Page 12: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

FEM, a posteriori analysis

Then

(u − u∆) (x) = (f − cu∆︸ ︷︷ ︸=: q

,Γ) − (f − cu∆, IMr Γ)∆︸ ︷︷ ︸

=(ILr q,IM

r Γ)

Error representation:

(u − u∆) (xk) =(q − IL

r q,Γ)−

(ILr q,Γ − IM

r Γ)

T. Linß, Dresden 16-18 Nov 2011 – p. 12/24

Page 13: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

FEM, a posteriori analysis

Then

(u − u∆) (x) = (f − cu∆︸ ︷︷ ︸=: q

,Γ) − (f − cu∆, IMr Γ)∆︸ ︷︷ ︸

=(ILr q,IM

r Γ)

Error representation:

(u − u∆) (xk) =(q − IL

r q,Γ)−

(ILr q,Γ − IM

r Γ)

First term: interpolation error/data oscillations

∣∣∣(q − IL

r q,Γ)∣∣∣ ≤

∥∥∥∥q − IL

r q

c

∥∥∥∥∞

⇒ sampling

T. Linß, Dresden 16-18 Nov 2011 – p. 12/24

Page 14: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

FEM, a posteriori analysis

Then

(u − u∆) (x) = (f − cu∆︸ ︷︷ ︸=: q

,Γ) − (f − cu∆, IMr Γ)∆︸ ︷︷ ︸

=(ILr q,IM

r Γ)

Error representation:

(u − u∆) (xk) =(q − IL

r q,Γ)−

(ILr q,Γ − IM

r Γ)

First term: interpolation error/data oscillations

∣∣∣(q − IL

r q,Γ)∣∣∣ ≤

∥∥∥∥q − IL

r q

c

∥∥∥∥∞

⇒ sampling

T. Linß, Dresden 16-18 Nov 2011 – p. 12/24

Page 15: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

FEM, a posteriori analysis

Taylor:(ILr q

)(ξ) =

r∑

j=0

(ILr q

)(j)

i−1/2

j!

(ξ − xi−1/2

)j

T. Linß, Dresden 16-18 Nov 2011 – p. 13/24

Page 16: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

FEM, a posteriori analysis

Taylor:(ILr q

)(ξ) =

r∑

j=0

(ILr q

)(j)

i−1/2

j!

(ξ − xi−1/2

)j

(−1)r+1

Ji

(ILr q

)(ξ)

(Γ − IM

r Γ)(ξ) dξ

=

(ILr q

)(r)

i−1/2

(2r + 1)!

Ji

dr−1

dξr−1

(pr,i(ξ)(ξ − xi−1/2)

)Γ′′(ξ) dξ

+

(ILr q

)(r−1)

i−1/2

(2r)!

Ji

dr−1

dξr−1pr,i(ξ)Γ

′′(ξ) dξ,

pr,i(ξ) := (ξ − xi)r (ξ − xi−1)

r

T. Linß, Dresden 16-18 Nov 2011 – p. 13/24

Page 17: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

FEM, a posteriori analysis

Constants

αr := maxξ∈[0,1]

∣∣∣∣dr−1

dξr−1

(ξr(ξ − 1)r

)∣∣∣∣

βr := maxξ∈[0,1]

∣∣∣∣dr−1

dξr−1

(ξr(ξ − 1)r(ξ − 1/2)

)∣∣∣∣

Then,∥∥∥∥

dr−1

dξr−1

(pr,i(ξ)(ξ − xi−1/2)

)∥∥∥∥∞,Ji

≤ βihr+2i

∥∥∥∥dr−1

dξr−1pr,i(ξ)

∥∥∥∥∞,Ji

≤ αihr+1i

T. Linß, Dresden 16-18 Nov 2011 – p. 14/24

Page 18: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

FEM, a posteriori analysis

Constants

αr βr

r = 1 14

√3

36

r = 2√

39

116

r = 3 38

(3√

30+9)√

525−70√

3

2450

r = 4(12

√30+36)

√525−70

√3

122538

r = 5 154 ≈ 1.434081520

T. Linß, Dresden 16-18 Nov 2011 – p. 14/24

Page 19: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

FEM, a posteriori analysis

(ILr q

)(r−1)

i−1/2,(ILr q

)(r)

i−1/2: qi−(r+ℓ)/r := q (xi−1 + tℓhi), tℓ = ℓ/r

Dr−1− qi :=

(r

hi

)r−1 r−1∑

j=0

(r − 1

j

)(−1)jqi−(1+j)/r

Dr−1+ qi :=

(r

hi

)r−1 r−1∑

j=0

(r − 1

j

)(−1)jqi−j/r.

(ILr q

)(r−1)

i−1/2=

Dr−1+ qi + Dr−1

− qi

2

(ILr q

)(r)

i−1/2=

r(Dr−1

+ qi − Dr−1− qi

)

hi

T. Linß, Dresden 16-18 Nov 2011 – p. 15/24

Page 20: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

FEM, a posteriori analysis

Theorem 1.

‖u − u∆‖∞ ≤∥∥∥∥q − IL

r q

c

∥∥∥∥∞

+ maxi=1,...,N

{hr+1

i

ε2

(αr

(2r)!

∣∣∣Dr−1+ qi + Dr−1

− qi

∣∣∣

+2rβr

(2r + 1)!

∣∣∣Dr−1+ qi − Dr−1

− qi

∣∣∣)}

.

T. Linß, Dresden 16-18 Nov 2011 – p. 16/24

Page 21: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

FEM, a posteriori analysis

Theorem 1.

‖u − u∆‖∞ ≤∥∥∥∥q − IL

r q

c

∥∥∥∥∞

+ maxi=1,...,N

{hr+1

i

ε2

(αr

(2r)!

∣∣∣Dr−1+ qi + Dr−1

− qi

∣∣∣

+2rβr

(2r + 1)!

∣∣∣Dr−1+ qi − Dr−1

− qi

∣∣∣)}

.

Remark: If c and f smooth, then

limhi→0

∣∣∣Dr−1+ qi − Dr−1

− qi

∣∣∣ = 0

T. Linß, Dresden 16-18 Nov 2011 – p. 16/24

Page 22: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

FEM, a posteriori analysis

Theorem 1.

‖u − u∆‖∞ ≤∥∥∥∥q − IL

r q

c

∥∥∥∥∞

+ maxi=1,...,N

{hr+1

i

ε2

(αr

(2r)!

∣∣∣Dr−1+ qi + Dr−1

− qi

∣∣∣

+2rβr

(2r + 1)!

∣∣∣Dr−1+ qi − Dr−1

− qi

∣∣∣)}

.

Remark: q = ε2u′′∆ =⇒ Dr−1

+ q,Dr−1− q = ε2u

(r+1)∆

Interpolation:∥∥u − IL

r u∥∥

Ji

≤ Chr+1i

∥∥u(r+1)∥∥

Ji

T. Linß, Dresden 16-18 Nov 2011 – p. 16/24

Page 23: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

FEM, a posteriori analysis

Theorem 1.

‖u − u∆‖∞ ≤∥∥∥∥q − IL

r q

c

∥∥∥∥∞

+ maxi=1,...,N

{hr+1

i

ε2

(αr

(2r)!

∣∣∣Dr−1+ qi + Dr−1

− qi

∣∣∣

+2rβr

(2r + 1)!

∣∣∣Dr−1+ qi − Dr−1

− qi

∣∣∣)}

.

Remark: On a badly adapted mesh: η ∼ ε−2.

T. Linß, Dresden 16-18 Nov 2011 – p. 16/24

Page 24: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

FEM, a posteriori analysis

alternative estimate: trade hi for ε−1

(ILr q

)(r)

i−1/2

Ji

dr−1

dξr−1

(pr,i(ξ)(ξ − xi−1/2)

)Γ′′(ξ) dξ

= −(ILr q

)(r)

i−1/2

Ji

dr

dξr

(pr,i(ξ)(ξ − xi−1/2)

)Γ′(ξ) dξ,

(ILr q

)(r−1)

i−1/2

Ji

dr−1

dξr−1pr,i(ξ)Γ

′′(ξ) dξ

= −(ILr q

)(r−1)

i−1/2

Ji

dr

dξrpr,i(ξ)Γ

′(ξ) dξ

T. Linß, Dresden 16-18 Nov 2011 – p. 17/24

Page 25: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

FEM, a posteriori analysis

Theorem 1′.

‖u − u∆‖∞ ≤∥∥∥∥q − IL

r q

c

∥∥∥∥∞

+2

(2r)!max

i=1,...,Nhr+1

i

[αr

∣∣∣Dr−1+ qi + Dr−1

− qi

∣∣∣

+r

2r + 1βr

∣∣∣Dr−1+ qi − Dr−1

− qi

∣∣∣

]

.

with

αr := min

{2αr

ε2,

r!

hiεγ

}, βr := min

{2βr

ε2,

r!

2hiεγ

}

T. Linß, Dresden 16-18 Nov 2011 – p. 18/24

Page 26: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

Collocation

Find: u∆ ∈ S1k+1(∆), k = r − 1, such that

(Lu∆) (τi,j) = f(τi,j), i = 1, . . . , N, j = 1, . . . , k, (D)

τi,j – Gauß-Legendre points, i.e., zeros of

Mk,i(ξ) := p(k)r,i (ξ),

[pk,i(ξ) := (ξ − xi)

k (ξ − xi−1)k]

T. Linß, Dresden 16-18 Nov 2011 – p. 19/24

Page 27: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

Collocation

Find: u∆ ∈ S1k+1(∆), k = r − 1, such that

(Lu∆) (τi,j) = f(τi,j), i = 1, . . . , N, j = 1, . . . , k, (D)

τi,j – Gauß-Legendre points, i.e., zeros of

Mk,i(ξ) := p(k)r,i (ξ),

[pk,i(ξ) := (ξ − xi)

k (ξ − xi−1)k]

Interpolation: I−1k−1 : ϕ 7→ I−1

k−1ϕ ∈ S−1k−1(∆) with

ϕ(τi,j) =(I−1k−1ϕ

)(τi,j), j = 1, . . . , k, i = 1, . . . , N.

Then (D) ⇔I−1k−1 (Lu∆ − f) ≡ 0 (D’)

T. Linß, Dresden 16-18 Nov 2011 – p. 19/24

Page 28: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

Collocation, a posteriori analysis

Error representation, Γ := G(x, ·):

(u − u∆) (x) =

∫ 1

0L(u − u∆)Γ =

∫ 1

0

(f −Lu∆

(D’):

∫ 1

0I−1k−1 (f − Lu∆) Γ = 0.

u′′∆ ≡ I−1

k−1u′′∆

=⇒ (u − u∆) (x) =

∫ 1

0

(I−1k−1q − q

)Γ, q := f − ru∆

T. Linß, Dresden 16-18 Nov 2011 – p. 20/24

Page 29: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

Collocation, a posteriori analysis

Interpolation: I0k+1 : ϕ 7→ I0

k+1ϕ ∈ S0k+1(∆) with

ϕ(xi−1) =(I0k+1ϕ

)(xi−1), ϕ(xi) =

(I0k+1ϕ

)(xi),

ϕ(τi,j) =(I0k+1ϕ

)(τi,j), j = 1, . . . , k,

Then

(u − u∆) (x) =

∫ 1

0

(q − I0

k+1q)Γ +

∫ 1

0

(I0k+1q − I−1

k−1q)Γ

T. Linß, Dresden 16-18 Nov 2011 – p. 21/24

Page 30: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

Collocation, a posteriori analysis

Interpolation: I0k+1 : ϕ 7→ I0

k+1ϕ ∈ S0k+1(∆) with

ϕ(xi−1) =(I0k+1ϕ

)(xi−1), ϕ(xi) =

(I0k+1ϕ

)(xi),

ϕ(τi,j) =(I0k+1ϕ

)(τi,j), j = 1, . . . , k,

Then

(u − u∆) (x) =

∫ 1

0

(q − I0

k+1q)Γ +

∫ 1

0

(I0k+1q − I−1

k−1q)Γ

∣∣∣∣∫ 1

0

(q − I0

k+1q)Γ

∣∣∣∣ ≤∥∥∥∥∥q − I0

k+1q

c

∥∥∥∥∥∞

T. Linß, Dresden 16-18 Nov 2011 – p. 21/24

Page 31: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

Collocation, a posteriori analysis

(I0k+1q − I−1

k−1q)(τi,j) = 0, j = 1, . . . , k

Thus

(I0k+1q − I−1

k−1q)(ξ)

=Mk,i(ξ)

Mk,i(xi)

(Q+

k,i

ξ − xi−1

hi− (−1)kQ−

k,i

ξ − xi

hi

)

Q−k,i := qi−1 −

(I−1k−1q

)(xi−1)

Q+k,i := qi −

(I−1k−1q

)(xi)

(discrete derivativesof q of order k) × hk

i

T. Linß, Dresden 16-18 Nov 2011 – p. 22/24

Page 32: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

Collocation, a posteriori analysis

Theorem 2.

‖(u − u∆)‖∞ ≤ η(ru∆ − f,∆)

with η(q,∆) = ηI(q,∆) + η2(q,∆) + η3(q,∆),

ηI(q,∆) :=

∥∥∥∥∥q − I0

k+1q

c

∥∥∥∥∥∞

η2(q,∆) :=3

2ρ2max

i=1,...,N

[max

{∣∣Q−k,i

∣∣,∣∣Q+

k,i

∣∣}

min

{2,

hiρ

ε,h2

i ρ2

ε2

}],

η3(q,∆) :=3

2ρ2max

i=1,...,N

[∣∣Q+k,i − (−1)kQ−

k,i

∣∣ min

{1,

hiρ

ε

}].

T. Linß, Dresden 16-18 Nov 2011 – p. 23/24

Page 33: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

Outlook

numerical experiments (in progress)

adaptivity, mesh equidistribution

reaction-convection-diffusion

−εu′′ + νbu′ + cu = f in (0, 1), u(0) = u(1) = 0

(SD)FEM :)(upwind) collocation :(

time-dependent problems (in progress → Natalia)

T. Linß, Dresden 16-18 Nov 2011 – p. 24/24

Page 34: A posteriori error estimators for higher-order methods ... fileVariational formulation Boundary value problem: Lu := −ε2u′′ + cu = f in (0,1), u(0) = u(1) = 0 Variational formulation:

Outlook

numerical experiments (in progress)

adaptivity, mesh equidistribution

reaction-convection-diffusion

−εu′′ + νbu′ + cu = f in (0, 1), u(0) = u(1) = 0

(SD)FEM :)(upwind) collocation :(

time-dependent problems (in progress → Natalia)

Thank you.

T. Linß, Dresden 16-18 Nov 2011 – p. 24/24