A posteriori error bounds in the
L (L2 )-norm for the wave problem
∞
Emmanuil Georgoulis
Department of Mathematics,
University of Leicester, UK
Joint work with O. Lakkis (Sussex), C. Makridakis (Crete)
European Finite Element Fair, Warwick, 20-21 May 2010
1
Model Problem
Let Ω ⊂ Rd , d = 2, 3, open polygonal domain and T > 0.
Consider the problem
∂tt u − ∇ · (a∇u) = f
u = u0
ut = u1
u=0
in Ω × (0, T ]
in Ω × {0}
in Ω × {0}
on ∂Ω × (0, T ]
for a = a(x) ∈ C (Ω̄) with 0 < αmin ≤ a ≤ αmax and f ∈ L∞ (0, T ; L2 (Ω))
2
Model Problem
Let Ω ⊂ Rd , d = 2, 3, open polygonal domain and T > 0.
Consider the problem
∂tt u − ∇ · (a∇u) = f
u = u0
ut = u1
u=0
in Ω × (0, T ]
in Ω × {0}
in Ω × {0}
on ∂Ω × (0, T ]
for a = a(x) ∈ C (Ω̄) with 0 < αmin ≤ a ≤ αmax and f ∈ L∞ (0, T ; L2 (Ω))
Problem in variational form: Find u as above such that
h∂tt u, v i + ha∇u, ∇v i = hf , v i
∀v ∈ H01 (Ω),
2
Literature review
A posteriori error bounds for FEM for wave problems
Johnson (’93), Süli (’96,’97)
duality methods for wave problem as 1st order system.
Akrivis, Makridakis & Nochetto (’06)
Adjerid (’02), Bernardi & Süli (’05)
time-discrete schemes for 1st order systems.
L∞ (H 1 )-error a posteriori estimates.
Surprisingly few a posteriori results for FEM for 2nd order hyperbolic problems...
3
Literature review
A posteriori error bounds for FEM for wave problems
Johnson (’93), Süli (’96,’97)
duality methods for wave problem as 1st order system.
Akrivis, Makridakis & Nochetto (’06)
Adjerid (’02), Bernardi & Süli (’05)
time-discrete schemes for 1st order systems.
L∞ (H 1 )-error a posteriori estimates.
Surprisingly few a posteriori results for FEM for 2nd order hyperbolic problems...
We are concerned with a posteriori bounds for the L∞ (L2 )-error.
3
FEM for wave problem
Semi-discrete FEM: Find U ∈ Vh :
hUtt , V i + a(U, V ) = hf , V i
∀V ∈ Vh ,
where a(U, V ) = ha∇U, ∇V i.
4
FEM for wave problem
Semi-discrete FEM: Find U ∈ Vh :
hUtt , V i + a(U, V ) = hf , V i
∀V ∈ Vh ,
where a(U, V ) = ha∇U, ∇V i.
Fully-discrete FEM: for n = 1, . . . , N, find U n ∈ Vhn :
h∂ 2 U n , V i + a(U n , V ) = hf n , V i ∀V ∈ Vhn ,
where f n := f (t n , ·), kn = t n+1 − t n and
4
FEM for wave problem
Semi-discrete FEM: Find U ∈ Vh :
hUtt , V i + a(U, V ) = hf , V i
∀V ∈ Vh ,
where a(U, V ) = ha∇U, ∇V i.
Fully-discrete FEM: for n = 1, . . . , N, find U n ∈ Vhn :
h∂ 2 U n , V i + a(U n , V ) = hf n , V i ∀V ∈ Vhn ,
where f n := f (t n , ·), kn = t n+1 − t n and
 n
n−1
U − U
n
n−1
∂U
−
∂U
,
n
2 n
, ∂U :=
∂ U :=
k
 0 n 0
kn
V := π u1
for n = 1, 2, . . . , N,
(1)
for n = 0,
with U 0 := π 0 u0 , and π 0 : L2 (Ω) → Vh0 suitable projection.
4
Dealing with “lower-than-energy” norms
Semi-discrete case We shall make use the ideas of reconstruction, introduced by
Makridakis & Nochetto (’03),
whereby
u−U =
u−w
| {z }
“evolution error”
+
w −U
| {z }
“elliptic error”
with w ∈ H01 (Ω) the elliptic reconstruction of U.
“Dual” concept of the Ritz projection (a.k.a elliptic projection, Wheeler’s Trick)
idea in a-priori analysis of evolution problems.
5
Dealing with “lower-than-energy” norms
Semi-discrete case We shall make use the ideas of reconstruction, introduced by
Makridakis & Nochetto (’03),
whereby
u−U =
u−w
| {z }
“evolution error”
+
w −U
| {z }
“elliptic error”
with w ∈ H01 (Ω) the elliptic reconstruction of U.
“Dual” concept of the Ritz projection (a.k.a elliptic projection, Wheeler’s Trick)
idea in a-priori analysis of evolution problems.
A priori error bounds in the L∞ (L2 ) norm can be found in
projection along with a special choice of test function.
Baker (’76),
utilising Ritz
5
Dealing with “lower-than-energy” norms
Semi-discrete case We shall make use the ideas of reconstruction, introduced by
Makridakis & Nochetto (’03),
whereby
u−U =
u−w
| {z }
“evolution error”
+
w −U
| {z }
“elliptic error”
with w ∈ H01 (Ω) the elliptic reconstruction of U.
“Dual” concept of the Ritz projection (a.k.a elliptic projection, Wheeler’s Trick)
idea in a-priori analysis of evolution problems.
A priori error bounds in the L∞ (L2 ) norm can be found in
projection along with a special choice of test function.
Baker (’76),
utilising Ritz
Fully-discrete case
Suitable space-time reconstruction (cf.
(classical) Baker’s techinique Baker (’76).
Lakkis & Makridakis (’06)),
together with
5
Elliptic Reconstruction
Definition
We define the elliptic reconstruction w ∈ H01 (Ω) of U to be the solution of the
elliptic problem
a(w , v ) = hAU − Πf + f , v i
∀v ∈ H01 (Ω),
where Π : L2 (Ω) → Vh denotes the orthogonal L2 -projection on the finite element
space Vh , and
A : Vh → Vh ,
denotes the (space-)discrete operator defined by
for Z ∈ Vh ,
h−AZ , V i = a(Z , V ) ∀V ∈ Vh .
6
Error equation for ρ
We decompose the error:
e := U − u = ρ − , where := w − U, and ρ := w − u,
7
Error equation for ρ
We decompose the error:
e := U − u = ρ − , where := w − U, and ρ := w − u,
Lemma
We have
h∂tt e, v i + a(ρ, v ) = 0
for all v ∈ H01 (Ω)
7
Error equation for ρ
We decompose the error:
e := U − u = ρ − , where := w − U, and ρ := w − u,
Lemma
We have
h∂tt e, v i + a(ρ, v ) = 0
for all v ∈ H01 (Ω)
or
h∂tt ρ, v i + a(ρ, v ) = h∂tt , v i
7
Error equation for ρ
We decompose the error:
e := U − u = ρ − , where := w − U, and ρ := w − u,
Lemma
We have
h∂tt e, v i + a(ρ, v ) = 0
for all v ∈ H01 (Ω)
or
h∂tt ρ, v i + a(ρ, v ) = h∂tt , v i
Z
Baker’s technique: set v = v (t, ·) =
τ
ρ(s, ·)ds
t
Baker (’76)
7
Abstract a-posteriori bound
Theorem
The following error bound holds:
√ kekL∞ (0,T ;L2 (Ω)) ≤kkL∞ (0,T ;L2 (Ω)) + 2 ku0 − U(0)k + k(0)k
Z T
+2
kt k + Ca,T ku1 − Ut (0)k,
0
where Ca,T := min{2T ,
p
2CΩ /αmin }, where CΩ is the PF constant.
8
Abstract a-posteriori bound
Theorem
The following error bound holds:
√ kekL∞ (0,T ;L2 (Ω)) ≤kkL∞ (0,T ;L2 (Ω)) + 2 ku0 − U(0)k + k(0)k
Z T
+2
kt k + Ca,T ku1 − Ut (0)k,
0
where Ca,T := min{2T ,
p
2CΩ /αmin }, where CΩ is the PF constant.
Controlling , t : use L2 -norm a posteriori bounds for elliptic problems
kk≤ EL2 (U, AU − Πf + f , T ).
kt k≤ EL2 (Ut , (AU − Πf + f )t , T ).
e.g., Verfürth (’96), Ainsworth & Oden (’01)...
8
Time reconstruction
Ideally we want time-reconstruction U(t) ∈ C 1 (0, t N ) such that
Utt = ∂ 2 U n
9
Time reconstruction
Ideally we want time-reconstruction U(t) ∈ C 1 (0, t N ) such that
Utt = ∂ 2 U n
We cannot have that...
9
Time reconstruction
Ideally we want time-reconstruction U(t) ∈ C 1 (0, t N ) such that
Utt = ∂ 2 U n
We cannot have that...
Define
U(t) :=
t − t n−1 n t n − t n−1 (t − t n−1 )(t n − t)2 2 n
U +
U
−
∂ U ,
kn
kn
kn
for t ∈ (t n−1 , t n ], n = 1, . . . , N.
9
Time reconstruction
Ideally we want time-reconstruction U(t) ∈ C 1 (0, t N ) such that
Utt = ∂ 2 U n
We cannot have that...
Define
U(t) :=
t − t n−1 n t n − t n−1 (t − t n−1 )(t n − t)2 2 n
U +
U
−
∂ U ,
kn
kn
kn
for t ∈ (t n−1 , t n ], n = 1, . . . , N.
Note: U(t) ∈ C 1 (0, t N ) and U(t n ) = U n and Ut (t n ) = ∂U n and...
9
Time reconstruction (cont’d)
we have
Utt = (1 + µn (t))∂ 2 U n
10
Time reconstruction (cont’d)
we have
Utt = (1 + µn (t))∂ 2 U n
with
1
µn (t) := −6kn−1 (t − t n− 2 ),
1
t n− 2 :=
1 n
(t + t n−1 )
2
10
Time reconstruction (cont’d)
we have
Utt = (1 + µn (t))∂ 2 U n
with
1
µn (t) := −6kn−1 (t − t n− 2 ),
1
t n− 2 :=
1 n
(t + t n−1 )
2
Remark (vanishing moment property)
The remainder µn (t) satisfies
Z
tn
µn (t) dt = 0,
t n−1
10
Elliptic reconstruction
Definition
Let U n , n = 0, . . . , N, be the fully-discrete solution computed by the method,
Πn : L2 (Ω) → Vhn be the orthogonal L2 -projection, and An : Vhn → Vhn to be the
discrete operator defined by
for q ∈ Vhn ,
hAn q, χi = a(q, χ) ∀χ ∈ Vhn .
We define the elliptic reconstruction w n ∈ H01 (Ω) such that
a(w n , v ) = hg n , v i
∀v ∈ H01 (Ω),
with
g n := An U n − Πn f n + f¯n ,
R tn
where f¯0 (·) := f (0, ·) and f¯n (·) := kn−1 t n−1 f (t, ·)dt for n = 1, . . . , N. (Special
care for n = 1, also)
11
Elliptic reconstruction
Definition
Let U n , n = 0, . . . , N, be the fully-discrete solution computed by the method,
Πn : L2 (Ω) → Vhn be the orthogonal L2 -projection, and An : Vhn → Vhn to be the
discrete operator defined by
for q ∈ Vhn ,
hAn q, χi = a(q, χ) ∀χ ∈ Vhn .
We define the elliptic reconstruction w n ∈ H01 (Ω) such that
a(w n , v ) = hg n , v i
∀v ∈ H01 (Ω),
with
g n := An U n − Πn f n + f¯n ,
R tn
where f¯0 (·) := f (0, ·) and f¯n (·) := kn−1 t n−1 f (t, ·)dt for n = 1, . . . , N. (Special
care for n = 1, also)
w (t) :=
n−1
n
t − t n−1 n t n − t n−1 (t − t n−1 )(t n − t)2 2 n
w +
w
−
∂ w ,
kn
kn
kn
11
A posteriori bound for the fully-discrete wave problem
Theorem
√ kekL∞ (0,t N ;L2 (Ω)) ≤kkL∞ (0,t N ;L2 (Ω)) + 2 ku0 − U(0)k + k(0)k
4
Z tN
X
kt k +
+2
ηi (t N ) + Ca,N ku1 − V 0 k,
0
where Ca,N := min{2t N ,
p
(2)
i=1
2CΩ /αmin }.
12
A posteriori bound for the fully-discrete wave problem
We have made use of the following notation:
mesh change indicator
η1 (τ ) : =
m−1
X Z tj
j=1
+
m−1
X
k(I − Πj )Ut k +
t j−1
Z
τ
k(I − Πm )Ut k
t m−1
(τ − t j )k(Πj+1 − Πj )∂U j k + τ k(I − Π0 )V 0 (0)k
j=1
evolution error indicator
Z
τ
kGk,
η2 (τ ) :=
0
where G : (0, T ] → R with G|(t j−1 ,t j ] := G j , j = 1, . . . , N and
G j (t) :=
with γj := γj−1 +
(t j − t)2 j (t j − t)4
(t j − t)3 2 j
∂g −
−
∂ g − γj ,
2
4kj
3
kj2
j
2 ∂g
+
kj3 2 j
12 ∂ g ,
j = 1, . . . , N, γ0 = 0;
13
A posteriori bound for the fully-discrete wave problem
data error indicator
m−1 Z j
1/2 Z τ
1/2
1 X t 3 ¯j
2
3 ¯m
+
km
kf − f k2
;
kj kf − f k
η3 (τ ) :=
2π
t m−1
t j−1
j=1
time reconstruction error indicator
Z
m−1 Z j
1/2
1 X t 3 j 2 j 2 1/2 τ
3
.
η4 (τ ) :=
kj kµ ∂ U k
+
km
kµm ∂ 2 U m k2
2π
t j−1
t m−1
j=1
14
A posteriori bound for the fully-discrete wave problem
Finally, we bound the and t terms using
Lemma
We have
kkL∞ (0,t N ;L2 (Ω)) +
√
2k(0)k ≤ δ1 (t N ) +
√
2Cel E 0 ,
where
n 8k
1
δ1 (t N ) := max
Cel E(V 0 , ∂g 0 , T 0 ),
27
35 31
o
kj −1 ¯j
+
max
max Cel E j + CΩ αmin
kf − f j k ,
27 27 1≤j≤N kj−1 0≤j≤N
with E j := E(U j , Aj U j − Πj f j + f j , T j ), j = 0, 1, . . . , N.
15
A posteriori bound for the fully-discrete wave problem
Lemma
We have
Z
tN
kt k ≤ δ2 (t N ),
0
where
δ2 (t N ) :=
N
2X
−1
(2kj + kj+1 ) Cel E∂j + CΩ αmin
k∂f j − ∂ f¯j k ,
3
j=0
with
E∂j := E(∂U j , ∂(Aj U j ) − ∂(Πj f j ) + ∂f j , T̂ j ),
j = 0, 1, . . . , N.
16
Outlook
Current work:
(G., Lakkis & Makridakis, in preparation)
explicit methods for wave problem
high order time-stepping for wave problem
17