2013 bcam computing lower and upper bounds of eigenvalues
TRANSCRIPT
2013 BCAM
Computing Lower and Upper Bounds of Eigenvalues
The Second BCAM Workshop on Computational Mathematics
Hehu Xie
Email: [email protected]
LSEC, Academy of Mathematics and System ScienceChinese Academy of Sciences
Homepage: lsec.cc.ac.cn/∼hhxie
Oct. 17-18 2013
Hehu Xie
(LSEC, Academy of Mathematics and System Science Chinese Academy of Sciences Homepage: lsec.cc.ac.cn/∼hhxie )Lower and Upper Bounds Oct. 17-18 2013 1 / 40
Contents
1 Nonconforming Finite element method for eigenvalue problem
2 Nonconforming multilevel methodOne correction stepMultilevel schemeSuperclose analysisNumerical results
3 Lower bound of eigenvalue by nonconforming finite element
4 Upper bound by conforming element postprocessing
5 Concluding remarks
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 2 / 40
Contents
1 Nonconforming Finite element method for eigenvalue problem
2 Nonconforming multilevel methodOne correction stepMultilevel schemeSuperclose analysisNumerical results
3 Lower bound of eigenvalue by nonconforming finite element
4 Upper bound by conforming element postprocessing
5 Concluding remarks
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 2 / 40
Contents
1 Nonconforming Finite element method for eigenvalue problem
2 Nonconforming multilevel methodOne correction stepMultilevel schemeSuperclose analysisNumerical results
3 Lower bound of eigenvalue by nonconforming finite element
4 Upper bound by conforming element postprocessing
5 Concluding remarks
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 2 / 40
Contents
1 Nonconforming Finite element method for eigenvalue problem
2 Nonconforming multilevel methodOne correction stepMultilevel schemeSuperclose analysisNumerical results
3 Lower bound of eigenvalue by nonconforming finite element
4 Upper bound by conforming element postprocessing
5 Concluding remarks
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 2 / 40
Contents
1 Nonconforming Finite element method for eigenvalue problem
2 Nonconforming multilevel methodOne correction stepMultilevel schemeSuperclose analysisNumerical results
3 Lower bound of eigenvalue by nonconforming finite element
4 Upper bound by conforming element postprocessing
5 Concluding remarks
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 2 / 40
Eigenvalue problem
Eigenvalue problem
Eigenvalue problem has wide application
Schodinger equation
Material science =⇒ Density Functional Theory (DFT) =⇒Eigenvalue problems
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 3 / 40
Eigenvalue problem
Eigenvalue problem
Find (λ, u) such that−∆u = λu, in Ω,
u = 0, on ∂Ω,∫Ω u
2dΩ = 1.
Weak form
Find (λ, u) such that b(u, u) = 1 and
a(u, v) = λb(u, v), ∀v ∈ V := H10 (Ω),
where
a(u, v) =
∫Ω∇u∇vdΩ, b(u, v) =
∫ΩuvdΩ.
Eigenpair series
(λ1, u1), · · · , (λj , uj), · · · with limj→∞
λj =∞ and b(ui, uj) = δij .
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 4 / 40
Nonconforming Finite element discretization
Mesh
Th: quasi-uniform decomposition of Ω into triangles or rectangles
hK : the diameter of a cell K ∈ Th and the mesh diameter h describesthe maximum diameter of all cells K ∈ ThEh denotes the edge set of Th and Eh = E ih ∪ Ebh, where E ih denotesthe interior edge set and Ebh denotes the edge set lying on theboundary ∂Ω
Define V := H10 (Ω) and W = L2(Ω).
Nonconforming finite element space
CR element is defined on the triangular partition (Crouzeix-Raviart)
Vh :=v ∈ L2(Ω) : v|K ∈ span1, x, y,
∫`v|K1ds =
∫`v|K2ds,
when K1 ∩K2 = ` ∈ E ih and
∫`v|Kds = 0, if ` ∈ Ebh
,
where K, K1, K2 ∈ Th.
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 5 / 40
Nonconforming Finite element discretization
Nonconforming finite element space
The finite element space Vh is the corresponding nonconforming finiteelement space on the partition, i.e. Vh * V .
EQrot1 : rectangle element [Lin, Tobiska and Zhou (2005)]
Vh :=v ∈ L2(Ω) : v|K ∈ span1, x, y, x2, y2,
∫`v|K1ds =
∫`v|K2ds,
if K1 ∩K2 = `, and
∫`v|Kds = 0, if K ∩ ∂Ω = `
,
where K, K1, K2 ∈ Th.
ECR element is defined on the triangular partition (Hu-Huang-Lin,Lin-Xie-Luo-Li-Yang)
Vh :=v ∈ L2(Ω) : v|K ∈ span1, x, y, x2 + y2,
∫`v|K1ds =
∫`v|K2ds,
when K1 ∩K2 = ` ∈ E ih, and
∫`v|Kds = 0, if ` ∈ Ebh
,
where K, K1, K2 ∈ Th.
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 6 / 40
Nonconforming finite element method
Nonconforming finite element method
The nonconforming finite element approximation is defined as follows:Find (λh, uh) ∈ R× Vh such that b(uh, uh) = 1 and
ah(uh, vh) = λhb(uh, vh), ∀vh ∈ Vh,
where the bilinear form ah(·, ·) is defined as
ah(uh, vh) =∑K∈Th
∫K∇uh∇vhdK.
The bilinear form ah(·, ·) is Vh-elliptic on V + Vh. Thus we define thenorms ‖ · ‖a,h and ‖ · ‖b on Vh + V by
‖v‖2a,h = ah(v, v), ‖v‖2b = b(v, v) for v ∈ V + Vh.
Eigenpair series
(λ1,h, u1,h), · · · , (λN,h, uN,h) with N = dimVh, λ1,h ≤ λ2,h ≤ · · · ≤ λN,hand b(ui,h, uj,h) = δij .
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 7 / 40
Notation
Notation
Define the operator T : W 7−→ V by
a(Tf, v) = b(f, v), ∀v ∈ V and ∀f ∈W
As we know, the operator T is compact
Define the corresponding discrete operator Th : W 7−→ Vh by
ah(Thf, vh) = b(f, vh), ∀vh ∈ Vh and ∀f ∈W
M(λj) denote the eigenfunction set corresponding to the eigenvalueλj which is defined by
M(λj) =w ∈ V : w is an eigenfunction corresponding to
λj and ‖w‖b = 1.
Define the following notation
δh(λj) = ‖(T − Th)|M(λj)‖a,h, ρh(λj) = ‖(T − Th)|M(λj)‖b.
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 8 / 40
Error estimates
Lemma (MercierOsbornRappazRaviart,Rannacher,YangChen)
Let (λj,h, uj,h) ∈ R× Vh be the j-th nonconforming finite elementeigenpair approximation. Then λj,h → λj and there exist uj ∈M(λj) suchthat ‖uj − uj,h‖a,h ≤ Cjδh(λj),
‖uj − uj,h‖b ≤ Cjρh(λj),
|λj − λj,h| ≤ Cjρh(λj),
where the constants Cj depend on the j-th eigenvalue λj .
Theorem (Lin-Xie)
‖uj − uj,h‖a,h ≤ Cjδh(λj) ≤ Cjhγ ,‖uj − uj,h‖b ≤ Cjhγδh(λj) ≤ Cjh2γ ,
|λj − λj,h| ≤ Cjhγδh(λj) ≤ Cjh2γ ,
where 0 < γ ≤ 1 denotes the regularity of the eigenfunctionuj ∈ H1+γ(Ω).
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 9 / 40
Outline
1 Nonconforming Finite element method for eigenvalue problem
2 Nonconforming multilevel methodOne correction stepMultilevel schemeSuperclose analysisNumerical results
3 Lower bound of eigenvalue by nonconforming finite element
4 Upper bound by conforming element postprocessing
5 Concluding remarks
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 10 / 40
Multilevel correction method
Idea
Solving eigenvalue problems is more difficult than solving boundaryvalue problems which have many efficient method to solve theboundary value problems (multigrid and DDM)
The idea for correction methods is to transform solving eigenvalueproblem to the corresponding boundary value problem solving
Two-grid method (Xu-Zhou), Multilevel correction method (Lin-Xie)Here we only consider the nonconforming multilevel correction method
For simplicity of describing, we only consider the simple eigenvaluesλj . But the the multiple eigenvalues can be given similarly
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 11 / 40
One correction step
Setting
Assume we have obtained the eigenpair approximation(λj,hk , uj,hk) ∈ R× VhkLet Vhk+1
6⊂ V be the nonconforming finite element space based onthe finer mesh Thk+1
which is produced by refining Thk in the regularway with the index β (β is the index of regular refinement, i.e.,hk+1 = hk/β, always β = 2)
We define the conforming linear (or bilinear) finite element space WH
on the coarsest mesh TH (Thk is from TH by refinement)
One correction step is to improve the accuracy of the currenteigenpair approximation (λj,hk , uj,hk)
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 12 / 40
One correction step
Algorithm (One Correction Step)
1 Define the following boundary value problem:Find uj,hk+1
∈ Vhk+1such that
ahk+1(uj,hk+1
, vhk+1) = λj,hkb(uj,hk , vhk+1
), ∀vhk+1∈ Vhk+1
.
Solve this equation with some efficient numerical method to obtain anew eigenfunction approximation uj,hk+1
∈ Vhk+1.
2 Define a new finite element space VH,hk+1= WH + spanuj,hk+1
and solve the following small scale eigenvalue problem:Find (λj,hk+1
, uj,hk+1) ∈ R× VH,hk+1
such thatb(uj,hk+1
, uj,hk+1) = 1 and
ah(uj,hk+1, vH,hk+1
) = λj,hk+1b(uj,hk+1
, vH,hk+1), ∀vH,hk+1
∈ VH,hk+1
Define (λj,hk+1, uj,hk+1
) = Correction(WH , λj,hk , uj,hk , Vhk+1)
Step 1 is the same as two-grid method. But step 2 is new and is thekey point for the multilevel correction method
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 13 / 40
Error estimates
Theorem
Assume the given eigenpairs (λj,hk , uj,hk) in One Correction Step have thefollowing error estimates
‖uj − uj,hk‖a,h . εhk(λj),
‖uj − uj,hk‖b . Hγεhk(λj),
|λj − λj,hk | . Hγεhk(λj).
Then after One Correction Step, the resultant eigenpair approximation(λj,hk+1
, uj,hk+1) has the following error estimates
‖uj − uj,hk+1‖a,h . εhk+1
(λj),
‖uj − uj,hk+1‖b . Hγεhk+1
(λj),
|λj − λj,hk | . Hγεhk+1(λj).
where εhk+1(λj) := Hγεhk(λi) + hγk+1 (a better accuracy than εhk(λj))
Two-grid method has no the estimate ‖uj − uj,hk+1‖b
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 14 / 40
Outline
1 Nonconforming Finite element method for eigenvalue problem
2 Nonconforming multilevel methodOne correction stepMultilevel schemeSuperclose analysisNumerical results
3 Lower bound of eigenvalue by nonconforming finite element
4 Upper bound by conforming element postprocessing
5 Concluding remarks
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 15 / 40
Nonconforming multilevel method
Algorithm (Nonconforming Multilevel Scheme)
1 Construct a coarse nonconforming finite element space Vh1 on Th1and solve the following eigenvalue problem:Find (λh1 , uh1) ∈ R× Vh1 such that b(uh1 , uh1) = 1 and
ah1(uh1 , vh1) = λh1b(uh1 , vh1), ∀vh1 ∈ Vh1 .
Choose the eigenpair (λj,h1 , uj,h1) which approximates the desiredeigenvalue λj and its eigenfunction
2 Construct a series of finer finite element spaces Vh2 , · · · , Vhn on thesequence of nested meshes Th2 , · · · , Thn
3 Do k = 1, · · · , n− 1Obtain new eigenpair approximation (λj,hk+1
, uj,hk+1) by One
Correction Step(λhj,k+1
, uj,hk+1) = Correction(WH , λj,hk , uj,hk , Vhk+1
).end Do
Finally, we obtain the eigenpair approximation (λj,hn , uj,hn) ∈ R× Vhn .
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 16 / 40
Error estimate and computational work
Theorem (Error estimate)
The resultant eigenpair approximations (λj,hn , uj,hn) obtained byNonconforming Multilevel Scheme have the following error estimates
‖uj − uj,hn‖a,h .n∑k=1
hγkHγ(n−k),
‖uj − uj,hn‖b .n∑k=1
hγkHγ(n−k+1),
|λj − λj,hn | .n∑k=1
h2γk H
2γ(n−k) + h2γn .
Computational work
If the computation work in the multilevel method for the boundary valueproblem by the nonconforming elements is O(N), the computational workof the algorithm here for the eigenvalue problem is O(N).
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 17 / 40
Error estimate for convex domain
Corollary (Error estimate)
Assume Ω is convex and the eigenfunction u ∈ H2(Ω). Under thecondition CHβ < 1, the resultant eigenpair approximation (λj,hn , uj,hn)has the following optimal error estimates
‖uj − uj,hn‖a,h . hn,
‖uj − uj,hn‖b . Hhn,
|λj − λj,hn | . h2n.
Remark
The condition CHβ < 1 (for example β = 2) is easy to satisfy
The errors of the eigenpair approximation (λj,hn , uj,hn) are optimal(i.e. it has the same accuracy as we solve the eigenvalue problemdirectly by the nonconforming elements)
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 18 / 40
Outline
1 Nonconforming Finite element method for eigenvalue problem
2 Nonconforming multilevel methodOne correction stepMultilevel schemeSuperclose analysisNumerical results
3 Lower bound of eigenvalue by nonconforming finite element
4 Upper bound by conforming element postprocessing
5 Concluding remarks
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 19 / 40
Superclose analysis
Direct nonconforming finite element method
Let (λj,h, uj,h) denote the eigenpair approximation by the direct eigenvaluesolving which is defined as follows:Find (λj,h, uj,h) ∈ R× Vh such that b(uj,h, uj,h) = 1 and
ah(uj,h, vh) = λj,hb(uj,h, vh), ∀vh ∈ Vh.
Lemma (Eigenvalue expansion)
For the eigenvalue approximations λj,h and λj,h, the following expansionholds
λj,h − λj,h =ah(uj,h − uj,h, uj,h − uj,h)− λj,hb(uj,h − uj,h, uj,h − uj,h)
b(uj,h, uj,h)
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 20 / 40
Superclose analysis
Theorem (Superclosesness)
Let (λj,hn , uj,hn) denote the eigenpair approximation by directnonconforming finite element method. Then we have the followingsuperclose properties
‖uj,hn − uj,hn‖a,h .n∑k=2
Hγ(n−k)(‖uj,hk−1− uj,hk‖b + |λj,hk−1
− λj,hk |),
‖uj,hn − uj,hn‖b . Hγ‖uj,hn − uj,hn‖a,h,|λj,hn − λj,hn | . ‖uj,hn − uj,hn‖2a,h.
Assume the series of the meshes satisfies the following estimatesn∑k=1
Hγ(n−k)hγk . hn,
and the eigenvalue approximation λj,hn has the lower-bound property:λj,hn < λj . Then the eigenvalue λj,hn also has the lower-bound property:
λj,hn < λj .
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 21 / 40
Superclose analysis
Corollary (Superclosesness)
Assume Ω is convex and the eigenfunction uj ∈ H2(Ω). Let (λj,hn , uj,hn)denote the eigenpair approximation by the direct nonconforming elements.Under the condition CHβ < 1, we have the following superclose properties
‖uj,hn − uj,hn‖a,h . h2n,
‖uj,hn − uj,hn‖b . H‖uj,hn − uj,hn‖a,h,|λj,hn − λj,hn | . h4
n.
If the eigenvalue approximation λj,hn has the lower-bound property:
λj,hn < λj ,
the eigenvalue λj,hn also has the lower-bound property:
λj,hn < λj .
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 22 / 40
Outline
1 Nonconforming Finite element method for eigenvalue problem
2 Nonconforming multilevel methodOne correction stepMultilevel schemeSuperclose analysisNumerical results
3 Lower bound of eigenvalue by nonconforming finite element
4 Upper bound by conforming element postprocessing
5 Concluding remarks
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 23 / 40
Numerical example 1
Model example on unit square
Ω = (0, 1)× (0, 1) with the regularity index γ = 1
Here, we adopt the meshes which are produced by regular refinementfrom the initial mesh generated by Delaunay method to investigatethe convergence behaviors.
We checked the numerical results for two regular refinement wayswith hk+1 = hk/2 and hk+1 = hk/4 (k = 1, · · · , n− 1), respectively.Furthermore, we choose TH = Th1 with H = 1/4.
From Theorem Error Estimate Theorem, we have the following errorestimates for these two refinement ways
‖uhn − u‖a,h . hn, ‖uhn − u‖b . Hhn, |λhn − λ| . h2n,
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 24 / 40
Errors for model problem
Figure: The errors for the first eigenvalue 2π2 by the multilevel method withhk+1 = hk/2, where (λh, uh) is produced by the multilevel method and(λdirh , udirh ) by the direct nonconforming finite element method
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 25 / 40
Errors for model problem
Figure: The errors for the first eigenvalue 2π2 by the multilevel method withhk+1 = hk/4, where (λh, uh) is produced by the multilevel method and(λdirh , udirh ) by the direct nonconforming finite element method
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 26 / 40
Lower bound of eigenvalue by nonconforming finite element
Lower bound
The conforming finite element method can only obtain the upperbounds of the eigenvalues
The nonconforming finite element method gives a way to get thelower bounds of the eigenvalues
Armentano and Duran [2004] propose an eigenvalue expansion whichcan be used to prove the lower bounds of CR element for the singulareigenvalue problems
Zhang, Yang and Chen [2007] gives a full version of the eigenvalueexpansion for the nonconforming eigenvalue approximation
Tomas Vejchodsky get the lower bound of the first eigenvalue by aposteriori error estimate method [2012]
In the following parts of this lecture, we set V NCH and V C
h to denotethe nonconforming element space and conforming element space,respectively
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 27 / 40
Bounds of convergence rate
Error estimates
The following basic error estimates for the ECR element hold
|λ− λh| ≤ Ch2γ‖u‖21+γ ,
‖u− uh‖a,h ≤ Chγ‖u‖1+γ ,
‖u− uh‖b ≤ Chγ‖u− uh‖a,h ≤ Ch2γ‖u‖1+γ ,
where 0 < γ ≤ 1 denotes the regularity of the eigenfunction u ∈ H1+γ(Ω).
Lemma (Lower bound: Krızek, Roos, and Chen 2011, Lin, Xu and Xie2011)
If we solve the eigenvalue problem by ECR, bilinear or linear elements,the following lower bound for the convergence rate holds
‖u− uh‖a,h ≥ Ch.
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 28 / 40
Interpolation for ECR
Interpolation
The corresponding interpolation operator corresponding to ECR can bedefined in the same way:∫
`(u−Πhu)ds = 0, ∀` ∈ Eh,∫
K(u−Πhu)dK = 0, ∀K ∈ Th,
Interpolation properties
For any u ∈ V , we have the following properties
ah(u−Πhu,Πhu) = 0,
‖Πhu‖a,h ≤ ‖u‖a ,‖u−Πhu‖b ≤ Ch‖u −Πhu‖a,h ,‖u−Πhu‖b ≤ Ch‖u − uh‖a,h .
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 29 / 40
Lower bound results
Lemma (Armentano and Duran, 2004, Zhang, Yang and Chen 2007)
Suppose (λ, u) is the exact eigenpai, (λh, uh) ∈ R× V NCh is the eigenpair
of the discrete problem, the following expansion holds
λ− λh = ‖u− uh‖2a,h − λh‖vh − uh‖2b+λh(‖vh‖2b − ‖u‖2b) + 2ah(u− vh, uh), ∀vh ∈ V NC
h .
Lower bound results [Lin-Luo-Xie 2012]
Let λj and λj,h be the j-th exact eigenvalue and its correspondingnumerical approximation by ECR or EQrot
1 element. When h is smallenough, we have
0 ≤ λj − λj,h ≤ Ch2γ‖u‖21+γ .
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 30 / 40
Lower bound approximation
For ECR element.
Choose vh = Πhuj . For the second and fourth terms
‖uj,h −Πhuj ‖2b ≤ Ch2γ‖uj − uj ,h‖2a,h ,
ah(uj −Πhuj , vh) = 0, ∀vh ∈ V NCh .
For the third term
‖vj,h‖2b − ‖uj,h‖2b = (Πhuj − uj,h,Πhuj + uj,h)
=(Πhuj − uj , (Πhuj,h + uj)−Π0 (Πhuj + uj)
)≤ Ch‖Πhuj − uj‖b ≤ Ch1+γ‖uj − uj,h‖a,h,
where Π0 denote piece constant interpolation.
Since ‖uj − uj,h‖ ≥ Ch, the first term in the eigenvalue expansion is thedominant term and then λj,h ≤ λj .
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 31 / 40
Upper bound by postprocessing
Upper bound
We use the correction method to get the upper bound
The correction step just need to solve the corresponding boundaryvalue problems in the conforming finite element space V C
h
Postprocessing method
Based on the current obtained eigenpair approximation(λh, uh) ∈ R× V NC
h .
1 Utilize the conforming finite element (linear or bilinear) to solve thefollowing source problem: Find uh ∈ V C
h such that
a(uh, vh) = λhb(uh, vh), ∀vh ∈ V Ch
2 Calculate the following Rayleigh quotient as an approximation of λ
λh :=a(uh, uh)
b(uh, uh).
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 32 / 40
Upper bound
Theorem
Assume u ∈ H1+γ(Ω) with 1/2 < γ ≤ 1. Then when h is small enough,we have
0 ≤ λh − λ ≤ Ch2γ
and
max|λ− λh|, |λh − λ| ≤ λh − λh ≤ Ch2γ .Then
λ ∈ [λh, λh].
Proof.
λh − λ =‖uh − u‖2a‖uh‖2b
− λ‖uh − u‖2b‖uh‖2b
and
‖uh − u‖2a,h ≥ Ch2, and ‖uh − u‖2b ≤ Ch4γ .
So the first term is dominant and positive.
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 33 / 40
Better accuracy correction
Two-Grid methods
Aim here: Not only obtain higher order convergence but also upperbound of eigenvalue approximations with correction method
Two-grid method: J. Xu ([1992]), J. Xu and A. Zhou([1999,2001,2002])
V HCh means the finer conforming finite element space has the
accuracy
infvh∈V HC
h
‖u− vh‖a,h = infv∈V NC
h
‖u− vh‖2a,h
Produce the space V HCh by refining the mesh or improving the order
of the finite element space
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 34 / 40
Better accuracy correction
Algorithm (Higher order postprocessing method)
Assume we have obtained eigenpair approximations(λ1,h, u1,h), · · · , (λm,h, um,h) by the nonconforming elements
1 For j = 1, 2, · · · ,mFind uj,h ∈ V HC
h such that
a(uj,h, vh) = λj,hb(uj,h, vh), ∀vh ∈ V HCh .
2 Construct the finite dimensional space Vh = spanu1,h, · · · , um,hand solve the following small eigenvalue problem (m×m) in thespace Vh:
Find (λh, uh) ∈ R× Vh such that
a(uh, vh) = λhb(uh, vh), ∀v ∈ Vh.
Finally, we obtain the new eigenpair approximations(λ1,h, u1,h), · · · , (λm,h, um,h).
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 35 / 40
Better accuracy and upper bound
Theorem
For the eigenpair approximations (λ1,h, u1,h), · · · , (λm,h, um,h) obtained bythis algorithm, we have the following estimates
0 ≤ λj,h − λj ≤ Ch4γ , [better accuracy and upper bound]
‖uj,h − u‖a ≤ Ch2γ , [better accuracy]
Also we haveλ ∈ [λh, λh].
Proof.
With minimum-maximum principleλj = min
Vj⊂V,dimVj=jmaxv∈Vj
R(v),
and the discrete version in the space Vhλj,h = min
Vj,h⊂Vh,dimVj,h=jmaxv∈Vj,h
R(v).
Since Vh ⊂ V = H10 (Ω), λj ≤ λj,h.
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 36 / 40
Numerical results: ECR
Numerical example
Model eigenvalue problem on the unit square
λh denote the eigenvalue by linear element postprocess
λh denote the eigenvalue by better accuracy correction method withthe quadratic element
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 37 / 40
Errors of ECR and EQrot1
Figure: The errors for the eigenvalue approximations on unit square by ECR (left)
and EQrot1 (right), where Err1 =
∑6j=1(λj − λj,h), Err2 =
∑6j=1(λj,h − λj) and
Err3 =∑6
j=1(λj,h − λj)Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 38 / 40
Concluding remarks
Concluding remarks
1 Propose a type of multilevel method for the nonconforming finiteelement eigenvalue problem
2 We can obtain the lower bounds of the eigenvalues by some type ofnonconforming elements
3 The nonconforming multilevel method can also obtain the lowerbound of the eigenvalue problem
4 First we can use the nonconforming multilevel method to obtain thelower bounds of the eigenvalue (with optimal computation work) andthen use some simple calculation to produce the upper bounds of theeigenvalue (with conforming elements)
5 The total computation work for two sides of the eigenvalue needs onlythe optimal computation work
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 39 / 40
Thanks
Thank You Very Much!
Hehu Xie (LSEC, AMSS) Lower and Upper Bounds Oct. 17-18 2013 40 / 40