implicit a posteriori error estimation for the time ...web.cs.elte.hu/~izsakf/talk060614.pdf ·...

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

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


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.


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).


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

]

).


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.


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

]

).


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?


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.


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.


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

,


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.


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.


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.


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.


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.


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.


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)

.


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.


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).


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).


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).


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

+.



−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⋅



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).


Test case 2: Global Error for Non-Smooth Solution

Variation of the global error estimate and exact error in the H(curl)-norm.


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.



−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⋅



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)


Test case 3: Fichera Cube

Variation of the global error estimate and exact error in the H(curl)-norm.


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.


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.


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.


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.


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.

implicit a posteriori error estimation for the time ...web.cs.elte.hu/~izsakf/talk060614.pdf ·...

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