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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
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
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
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
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
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
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.
T. Linß, Dresden 16-18 Nov 2011 – p. 7/24
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
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
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,
T. Linß, Dresden 16-18 Nov 2011 – p. 10/24
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.
T. Linß, Dresden 16-18 Nov 2011 – p. 11/24
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Collocation, a posteriori analysis
Error representation, Γ := G(x, ·):
(u − u∆) (x) =
∫ 1
0L(u − u∆)Γ =
∫ 1
0
(f −Lu∆
)Γ
(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
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
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
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
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
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
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