a convergent adaptive method for elliptic eigenvalue problems · figotin & goren (2001),...
TRANSCRIPT
![Page 1: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/1.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
A convergent adaptive method for ellipticeigenvalue problems
Stefano Giani and Ivan G. Graham
School of Mathematical SciencesUniversity of Nottingham
14th Leslie Fox PrizeWarwick, 29th June 2009
![Page 2: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/2.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
IntroductionPhotonic Crystal FibersModel Problem
Finite Element Methods (FEMs)Finite Element Methods (FEMs)Adaptivity
Convergence ProofThe Convergent MethodConvergence Proof
NumericsDefect modes
Summary
![Page 3: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/3.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Photonic Crystal Fibers (PCFs)
Applications: communications, filters, lasers, switchersFigotin & Klein (1998), Cox & Dobson (1999), Dobson (1999), Sakoda (2001), Kuchment (2001),Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004),Joannopoulos, Johnson, Winn & Meade (2008),...
S. G.
Convergence of Adaptive Finite Element Methods for EllipticEigenvalue Problems with Applications to Photonic Crystals.
Ph.D. Thesis , University of Bath (2008)
![Page 4: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/4.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Model Problem
Let Ω be a bounded polygonal domain in R2 (or a bounded
polyhedral domain in R3)
Problem: seek eigenpairs (λ,u) of the problem
−∇ · (A ∇u) = λ B u in Ω,
u = 0 on ∂Ω.
We assume that A and B are both piecewise constant on Ω andthat:
∀ξ ∈ Rd with |ξ| = 1, ∀x ∈ Ω, 0 < a ≤ ξTA(x)ξ ≤ a ,
and∀x ∈ Ω, 0 < b ≤ B(x) ≤ b .
![Page 5: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/5.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Variational Formulation
∫
Ω
A u · v dx = λ
∫
Ω
B u v dx .
a(u, v) :=
∫
Ω
A u · v dx ,
‖|v‖|Ω := a(v , v)1/2 ,
b(u, v) :=
∫
Ω
B u v dx .
Variational Problem: seek eigenpairs (λ, u) ∈ R × H10(Ω) such
thata(u, v) = λb(u, v) for all v ∈ H1
0 (Ω) .
![Page 6: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/6.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
The Ritz-Galerkin MethodLet Vn be a finite dimensional space such that: Vn ⊂ H1
0 (Ω):Seek eigenpairs (λn, un) ∈ R × Vn such that
a(un, vn) = λnb(un, vn) for all vn ∈ Vn .
• Tn conforming and shape regular triangulation of Ω,• Vn space of piecewise linear functions over Tn,• Sn is the set of the edges of the triangles of Tn.
![Page 7: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/7.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Standard Convergence Results
For Hmaxn small enough:
|λ − λn| ≤ C2spec(H
maxn )2,
and‖|u − un‖|Ω ≤ CspecHmax
n ,
Strang & Fix (1973), Babuška & Osborn (1991)
![Page 8: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/8.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Adaptivity
ηn(λn, un) :=∑
S∈Sn
ηS,n(λn, un) ,
ηn and ηS,n are explicit andcomputable.
Properties for Hmaxn small enough:
1. ‖|u − un‖|Ω ≤ Crelηn (Reliability)
2. |λ − λn| ≤ C2relηn
2 (Reliability)
3. ηn ≤ Ceff‖|u − un‖|Ω (Efficiency)
Constants Crel and Ceff are independent of Hmaxn .
![Page 9: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/9.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Adaptivity
ηn(λn, un) :=∑
S∈Sn
ηS,n(λn, un) ,
ηn and ηS,n are explicit andcomputable.
Properties for Hmaxn small enough:
C−1eff ηn ≤ ‖|u − un‖|Ω ≤ Crel ηn .
Constants Crel and Ceff are independent of Hmaxn .
![Page 10: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/10.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Marking Strategy 1
Set the parameter 0 < θ < 1:mark the edges (faces) in a minimal subset Sn of Sn such that
(
∑
S∈Sn
η2S,n
)1/2
≥ θηn.
Then construct the set Tn marking all the elements sharing atleast an edge (face) in Sn.
![Page 11: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/11.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Bisection5
@@
@@
@@
@@
@@
@@
@@
@@
1. Split all edges
2. Split one of the new edges
PROS:
1. A new node on each edge
2. A new node in the interior of the element
![Page 12: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/12.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Adaptivity Algorithm
1. Require 0 < θ < 1, tol > 0 and T0
2. Loop
3. Compute (λn, un) on Tn
4. Marking strategy
5. Refine the mesh
6. Continue until ηn > tol.
![Page 13: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/13.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Convergence for Adaptive FEMs
Convergence for Adaptive Finite Element Methods for LinearBoundary Value Problems:Dörfler (1996), Morin, Nochetto & Siebert (2000,2002), Karakashian & Pascal (2003), Mekchay & Nochetto (2005),Mommer & Stevenson (2006), Morin, Siebert & Veeser (2007), Cascon, Kreuzer Nochetto & Siebert (2008), ...
Convergence for Adaptive Finite Element Methods forEigenvalue Problems:G.& Graham (2009), Dai, Xu & Zhou (2008), Carstensen & Gedicke (2009?),Garau, Morin & Zuppa (2009)
![Page 14: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/14.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Oscillations
Oscillations:
osc(vn,Tn) :=
(
∑
τ∈Tn
‖hτ (vn − Pnvn)‖2τ
)1/2
,
where (Pnvn)|τ := 1|τ |
∫
τ vn.
P. Morin, R. H. Nochetto, and K. G. Siebert (2000)Data oscillation and convergence of adaptive FEM.SIAM J. Numer. Anal. 38, 466-488.
![Page 15: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/15.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Marking Strategy 2
Set the parameter 0 < θ < 1:mark the sides in a minimal subset Tn of Tn such that
osc(un, Tn) ≥ θ osc(un,Tn).
Then we take the union of Tn ∪ Tn and we refine all theelements in the union.
![Page 16: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/16.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Adaptivity Algorithm
1. Require 0 < θ < 1, 0 < θ < 1 and T0
2. Loop
3. Compute (λn, un) on Tn
4. Marking strategy 1
5. Marking strategy 2
6. Refine the mesh
7. End Loop
![Page 17: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/17.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Convergence
Theorem (Convergence Result)Provided that λ is a simple eigenvalue and that on the initialmesh Hmax
0 is small enough, there exists a constant p ∈ (0, 1)and constants C0, C1 such that the recursive application of thealgorithm yields a convergent sequence of approximateeigenvalues and eigenvectors, with the property:
‖|u − un ‖|Ω ≤ C0pn ,
|λ − λn| ≤ C20p2n ,
andosc(λnun,Tn) ≤ C1pn.
![Page 18: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/18.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Error Reduction
Theorem (Error Reduction)For each θ ∈ (0, 1), exists a sufficiently fine mesh thresholdHmax
n and constants µ > 0 and α ∈ (0, 1) such that:For any ǫ > 0 then inequality
osc(un,Tn) ≤ µǫ ,
implies either‖|u − un ‖|Ω ≤ ǫ ,
or‖|u − un+1 ‖|Ω ≤ α ‖|u − un ‖|Ω .
![Page 19: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/19.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Oscillations Reduction
Theorem (Oscillations Reduction)There exists a constant α ∈ (0, 1) such that:
osc(un+1,Tn+1) ≤ αosc(un,Tn) + C(Hmaxn )2 ‖|u − un ‖|Ω .
![Page 20: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/20.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Defect modes (I)
0 1 2 3 4 50
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Figure: The structure of the supercell and the trapped mode.
![Page 21: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/21.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Defect modes (II)
0 1 2 3 4 50
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Figure: An adapted mesh for the periodic structure with θ = θ = 0.8.
Refinement in the interior and at the corners of the inclusions.
![Page 22: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/22.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Defect modes (III)
Uniform Adaptive|λ − λn| N β |λ − λn| N β
0.5858 10000 - 0.5858 10000 -0.1966 40000 0.7876 0.1225 20506 2.17910.0653 160000 0.7951 0.0579 44548 0.96590.0188 640000 0.8982 0.0078 220308 1.2541
Table: Comparison between uniform and adaptive refinement (withθ = θ = 0.8) for a trapped mode in the supercell for TE mode problem.
|λ − λn| = O(N−β), N = #DOF .
![Page 23: A convergent adaptive method for elliptic eigenvalue problems · Figotin & Goren (2001), Johnson & Joannopoulos (2002), Ammari & Santosa (2004), Joannopoulos, Johnson, Winn & Meade](https://reader036.vdocuments.site/reader036/viewer/2022071216/6046fd4da6a38826f34be407/html5/thumbnails/23.jpg)
Introduction Finite Element Methods (FEMs) Convergence Proof Numerics Summary
Summary
• We prove the convergence of an adaptive finite elementmethod for elliptic eigenvalue problems,
• The proof exploits reduction results for error andoscillations,
• Consequences:• The computed approximated eigenpairs are approximation
of true eigenpairs,• For any tolerance tol > 0 the adaptive algorithm will end
after a finite number of iterations.