implicit a posteriori error estimation for the time ...web.cs.elte.hu/~izsakf/talk060614.pdf ·...
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Implicit a posteriori error estimation for the
time-harmonic Maxwell equations
Davit Harutyunyan, Ferenc Izsak∗ and Jaap van der Vegt
Numerical Analysis and Computational Mechanics Group
Department of Applied Mathematics
University of Twente, The Netherlands
MAFELAP Conference, Brunel University, London
14 June, 2006
∗ Also at Department of Applied Analysis and Computational Mathematics, ELTE, Budapest
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University of Twente - Chair Numerical Analysis and Computational Mechanics 1
Outline of Presentation
• Time harmonic Maxwell equations
• Definition of the local error equations
• Numerical solution of the local error equations
• Theoretical results for the local error equations and implicit error estimates
• Numerical examples
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Time Harmonic Maxwell Equations
• Time harmonic Maxwell equations - function spaces with perfectly conducting
boundary conditions:
curl curl E − k2E = J , in Ω ⊂ R
3
E × ν = 0, on ∂Ω,(1)
with J ∈ [L2(Ω)]3 a given source term and ν the outward normal on ∂Ω.
• The Hilbert space corresponding to the Maxwell equation is:
H(curl, Ω) = u ∈ [L2(Ω)]3: curl u ∈ [L2(Ω)]
3,
H0(curl, Ω) = u ∈ H(curl, Ω) : ν × u|∂Ω = 0
equipped with the curl norm:
‖u‖curl,Ω = (‖u‖2[L2(Ω)]3 + ‖curl u‖2
[L2(Ω)]3)1/2.
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Definitions
• Define the bilinear form BK on H(curl, K) × H(curl, K) and the trace
operators γτ and πτ with
BK(u, v) = (curl u, curl v)K − k2(u, v)K
γτv = ν × v|∂K and πτv = (ν × v|∂K) × ν.
• Weak formulation of (1): Find an E ∈ H0(curl, Ω) such that
BΩ(E, v) = (J , v), ∀ v ∈ H0(curl, Ω).
• Discrete weak form of (1): Find an Eh ∈ Hh0 (curl, Ω) such that
BΩ(Eh, vh) = (J , vh), ∀ vh ∈ Hh0 (curl, Ω),
where Hh0 is the Nedelec edge element space (e.g. of order 1).
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Error Estimation: Basic Setting
• Define the computational error on an element K as:
eh = (E − Eh)|K.
• Overall aim: estimate ‖eh‖curl,K.
• Informations at hand:
⊲ J , Eh with boundary conditions, the subdomain K.
⊲ The element residual on K
rK := J − curl curl Eh + k2Eh in K.
⊲ The tangential jump of the curl at the element face lj between K and Kj
Rlj=
1
2(νj ×
[
curl Eh|K − curl Eh|Kj
]
).
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A Posteriori Error Estimates
• Objective: estimation of the error for an existing approximation.
• Main tool: weak form for the error, localization to the subdomains.
• Two types of a posteriori error estimates are frequently used:
Explicit error estimates, which use data provided by the numerical solution;
practically a “simple formula” depending on known data.
Implicit error estimates, which solve local error equations with the data provided
by the numerical solution.
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Local Error Equation
• Variational formulation for the local error: Find an eh ∈ H(curl, K) such that:
BK(eh, v) = (curl eh, curl v)K − k2(eh, v)K
= (curl E, curl v)K − k2(E, v)K − ((curl Eh, curl v)K − k
2(Eh, v)K)
= (J, v)K − (γτcurl E, πτv)∂K − BK(Eh, v), ∀v ∈ H(curl, K).
• At the faces lj of element K we use the approximation:
γτcurl E|lj ≈ γτcurl E|lj =1
2(νj ×
[
curl Eh|K + curl Eh|Kj
]
).
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Numerical Solution of Local Problems
• Local error formulation:
On each element K, find eh ∈ Vh,K, the implicit error estimate such that
∀ w ∈ Vh,K the following relation is satisfied:
(curl eh, curl w)K − k2(eh, w)K = (J , w)K − (curl Eh, curl w)K
+k2(Eh, w)K − 1
2(νj ×
(
curl Eh|K + curl Eh|Kj
)
, w)∂K.
• Chief question: how to choose Vh,K?
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Basis Functions for the Local Error Equation
On the reference element K = (0, 1)3 define Vh,K = span(φ0
k9k=1), where
φ01 = ((1 − ξ)(1 − η)η(1 − ζ)ζ, 0, 0)
T, φ
02 = (ξ(1 − η)η(1 − ζ)ζ, 0, 0)
T,
φ07 = ((1 − ξ)ξ(1 − η)η(1 − ζ)ζ, 0, 0)
T, φ
03 = (0, (1 − ξ)ξ(1 − η)(1 − ζ)ζ, 0)
T,
φ04 = (0, (1 − ξ)ξη(1 − ζ)ζ, 0)
T, φ
08 = (0, (1 − ξ)ξ(1 − η)η(1 − ζ)ζ, 0)
T,
φ05 = (0, 0, (1 − ξ)ξ(1 − η)η(1 − ζ))
T, φ
06 = (0, 0, (1 − ξ)ξ(1 − η)ηζ)
T,
φ09 = (0, 0, (1 − ξ)ξ(1 − η)η(1 − ζ)ζ)
T.
The space Vh,K is obtained with an affine transformation; K is rectangular.
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Well Posedness of Local Problems
• Theorem 1 Assume that k is not a Maxwell eigenvalue in the sense that the
problem: Find u ∈ H(curl, K) such that
B(u, v) = 0, ∀ v ∈ H(curl, K)
has only the trivial (u = 0) solution. Then the variational problem:
Find eh ∈ H(curl, K) such that
B(eh, v) = (J , v), ∀ v ∈ H(curl, K) (2)
has a unique solution for all J ∈ [L2(K)]3.
• Using lifting, the variational problem for the error can be rewritten into (2).
• This is also the case for any reasonable choice of γτcurl E.
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Eigenvalues for the Local Error Equation on a Rectangular
Domain
• Theorem 2 The eigenfunctions in the Maxwell eigenvalue problem
(corresponding to (2)):
Find v ∈ H(curl, K) and k ∈ R such that
curl curl v − k2v = 0 in K,
ν × curl v = 0 on ∂K
with K = (0, a) × (0, b) × (0, c) are:
v(x, y, z) =
C1 sink1π
a x cosk2π
b y cosk3π
c z
C2 cosk1π
a x sink2π
b y cosk3π
c z
C3 cosk1π
a x cosk2π
b y sink3π
c z
,
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for any k1, k2, k3 ∈ N with
k1π
aC1 +
k2π
bC2 +
k3π
cC3 = 0 and C1, C2, C3 ∈ R.
The appropriate eigenvalues are k2 =(
k1π
a
)2
+(
k2π
b
)2
+(
k3π
c
)2
with
k1, k2, k3 ∈ N arbitrary.
⋄ Example: a = b = c = 1 (unit cube).
Smallest eigenvalue: π2(12 + 12 + 12) ≈ 29.6.
Corollary 1 (2) is well posed whenever k is not a Maxwell eigenvalue in the sense
of Theorem 2.
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Implicit Error Estimate as Lower Bound of the Error
• Introduce ηK as error indicator - an explicit error estimate on K:
ηK = (h2‖r‖2[L2(K)]3 + h‖R‖2
[L2(∂K)]3)12 .
• Theorem 3 The implicit a posteriori error estimate eh can be used as a lower
bound for the exact error with respect to the curl norm as follows:
‖eh‖curl,K ≤ CηK ≤√
DC(1 + k2)‖eh‖curl,K + Res(r, R)
with C, D constants independent of the mesh size h and frequency k and
Res(r, R) = (h2‖r − r‖[L2(K)]3 + h‖R − R‖[L2(∂K)]3)1/2
arising from the interpolation errors of r and R.
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Solution of the local problems
K: cubic subdomain with edge length h.
The bilinear form for the bubble function space span(φk9k=1) on K:
• B1,K - stiffness matrix of size 9 × 9.
• B1,K = B1,curl − k2B1,0, where
B1,curl[i][j] =
∫
K
curl φi · curl φj, B1,0[i][j] =
∫
K
φi · φj.
• Scaling: B1,K = 1hB1,curl − k2hB1,0.
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Properties of the local problems
Ill conditioned matrices arise:
cond(B1,K) ∼ 1
h2.
The solution of the local problem seems to become less precise as h → 0.
Theorem 4 The following inf-sup condition holds:
There is a C ∈ R such that for any u ∈ span(φj9j=1)
sup‖v‖curl,K=1
v∈span(φj9j=1)
|B1,K(u, v)| ≥ C‖u‖curl,K,
where C does not depend on u and the finiteness parameter h.
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Main difficulties and tasks
• Challenges in the analysis and computations
Non-coercive bilinear forms in 3D.
Non-smooth solutions arise.
Non-convex domains with edges, corners arise.
Presence of big or even small wave numbers (k).
• The performance of the implicit error estimator is evaluated by:
checking if the estimator provides the right type of error distribution
comparing the magnitude of the global error estimate and its convergence under
mesh refinement with the exact error.
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Definition of Error Contributions
• The exact error δK and the implicit local error estimate δK on element K are
defined as:
δK = ‖E − Eh‖curl,K, δK = ‖eh‖curl,K.
• The exact global error δ and the implicit global error estimate δh are obtained by
summing the local contributions
δ =
∑
K∈Th
δ2K
1/2
, δh =
∑
K∈Th
δ2K
1/2
.
• In all cases: comparison of
analytic error : ‖E − Eh‖curl,K = ‖eh‖curl,K and implicit error : ‖eh‖curl,K.
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Test case 1: Smooth Solution
• The Maxwell equations on Ω = (0, 1)3 with given source term
J(x, y, z) = (π2(p2 + m2) − 1)
sin(πpy) sin(πmz)
sin(πpz) sin(πmx)
sin(πpx) sin(πmy)
with p, m ∈ N, which has a smooth solution:
E(x, y, z) =
sin(πpy) sin(πmz)
sin(πpz) sin(πmx)
sin(πpx) sin(πmy)
.
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Test case 1: Locations for Local Error Computations
0 0.2 0.4 0.6 0.8 1
0
0.5
10
0.2
0.4
0.6
0.8
1
2⋅
7⋅16⋅
4⋅
9⋅
14⋅5⋅
1⋅
6⋅
12⋅3⋅
8⋅
Location of some elements where the implicit error estimation was conducted.
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Test case 1: Error Distribution for Smooth Solution
1 2 3 4 5 6 7 8 9 12 14 16
3
6
8x 10
−3
loca
l err
or e
stim
ator
implicit error
Error distribution in H(curl)-norm (p = m = 1, h = 116).
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Test case 1: Error distribution for Smooth Solution
1 2 3 4 5 6 7 8 9 12 14 16
0.02
0.04
0.06
0.08
0.1
0.13
loca
l err
or e
stim
ator
implicit error
Error distribution in H(curl) norm (p = 5, m = 1, h = 116).
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Test case 1: Global Error for Smooth Solution
Variation of the global error estimate and exact error in the
H(curl)-norm (p = m = 1).
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Test case 2: Non-Smooth Solution
• Consider the domain Ω = (−1, 1)3 and the function:
f : Ω → R with f = max|x|, |y|, |z|.
• Define E : Ω → R as E := −∇f(x, y, z).
Then E solves the following Maxwell problem:
curl curl E − E = ∇f in Ω,
E × ν = 0 on ∂Ω.
• The right hand side is in [L2(Ω)]3, but the exact solution is not smooth,
E 6∈ [H12(Ω)]3, but E ∈ [H
12−ǫ(Ω)]3 for any ǫ ∈ R
+.
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Test case 2: Locations for Local Error Computations
−1 −0.5 0 0.5 1
−1
0
1−1
−0.5
0
0.5
1
2⋅
7⋅
4⋅
9⋅
5⋅
14⋅
1⋅
6⋅
12⋅
17⋅
3⋅
8⋅
Location of some elements where the implicit error estimation was conducted.
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Test case 2: Error Distribution for Non-Smooth Solution
1 2 3 4 5 6 7 8 9 12 14 170
0.006
0.009
0.011
0.014
loca
l err
or e
stim
ator
implicit error
Error distribution in H(curl)-norm (h = 116).
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Test case 2: Global Error for Non-Smooth Solution
Variation of the global error estimate and exact error in the H(curl)-norm.
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Test case 3: Fichera Cube
• Consider a Fichera cube Ω = (−1, 1)3 \ [−1, 0]3.
• The solution on this domain has corner and edge singularities and can serve as an
interesting test case.
• Boundary conditions and source term in the Maxwell equations are chosen such
that the exact solution is E = grad(r2/3 sin(23φ)) in spherical coordinates, with
r =√
x2 + y2 + z2, φ = arccosr
z.
• Note, E does not belong to [H1(Ω)]3.
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Test case 3: Locations for Local Error Computations
−1−0.5
00.5
1
−1
0
1−1
−0.5
0
0.5
1
25⋅
17⋅8⋅
29⋅
10⋅
1234567
⋅⋅⋅⋅⋅⋅⋅⋅15⋅14
31⋅
27⋅
Location of some elements where the implicit error estimation was conducted.
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Test case 3: Error Distribution for Fichera Cube Solution
1 2 3 4 5 6 7 8 10 1415 17 25 27 29 31
0.002
0.006
0.01
0.016
0.02
loca
l err
or e
stim
ator
implicit error
Error distribution in H(curl)-norm (h = 18)
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Test case 3: Fichera Cube
Variation of the global error estimate and exact error in the H(curl)-norm.
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Comparisons with some existing schemes
• Beck, Hiptmair et al. (M2AN 2000) consider the following elliptic boundary value
problem
curl (curl E) + βE = J in Ω = (0, 1)3,
E × ν = 0 on ∂Ω,
where β is a given positive function on Ω.
• The exact solution is given by E = (0, 0, sin(πx)).
• Note, the bilinear form for this problem is coercive, contrary to the bilinear form
discussed for the Maxwell equations.
• We make comparisons in terms of the effectivity index εh :=δh
δ. This quantity
merely reflects the quality of the global estimate.
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University of Twente - Chair Numerical Analysis and Computational Mechanics 31
Effectivity Index
β
Level0 1 2 3 4 5
10−4 4.05 8.05 8.18 8.24 8.27 8.29
10−2 4.05 8.05 8.17 8.23 8.27 8.29
1 3.01 7.64 7.78 7.84 7.87 7.89
102 2.29 4.27 4.70 4.95 5.20 5.26
104 2.33 4.23 4.66 4.86 4.95 5.00
Effectivity index εh for the error estimator given by Beck et. al.
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University of Twente - Chair Numerical Analysis and Computational Mechanics 32
Effectivity Index
β
h 1
4
1
8
1
16
1
32
1
6410−4 0.67 0.67 0.67 0.67 0.67
10−2 0.67 0.67 0.67 0.67 0.67
1 0.67 0.67 0.67 0.67 0.67
102 0.63 0.65 0.67 0.67 0.67
104 0.44 0.37 0.40 0.53 0.63
Effectivity index εh for the new implicit error estimator.
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University of Twente - Chair Numerical Analysis and Computational Mechanics 33
Conclusions
• We have developed and analyzed an implicit a posteriori error estimation technique
for the time harmonic Maxwell equations.
• The algorithm is well suited both for cases where the bilinear form is coercive and
the more complicated indefinite case.
• The algorithm is tested on a number of increasingly complicated test cases and the
results show that it gives an accurate prediction of the error distribution and also
the local and global error.
• Also, in comparison with other a posteriori estimation techniques, it gives for the
cases considered a sharper estimate of the error and the error distribution.
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University of Twente - Chair Numerical Analysis and Computational Mechanics 34
Further works
• The same procedure on a tetrahedral mesh (in progress).
• Chief question: how to use the local basis?
The choice span(Φj9j=1) does the job, but what else could be chosen?
• A meaningful adaptation strategy using the error estimates.