2013 BCAM Computing Lower and Upper Bounds of Eigenvalues

2013 BCAM

Computing Lower and Upper Bounds of Eigenvalues

The Second BCAM Workshop on Computational Mathematics

Hehu Xie

Email: hhxie@lsec.cc.ac.cn

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

Eigenvalue problem

Eigenvalue problem has wide application

Schodinger equation

Material science =⇒ Density Functional Theory (DFT) =⇒Eigenvalue problems

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),

a(u, v) =

∫Ω∇u∇vdΩ, b(u, v) =

∫ΩuvdΩ.

Eigenpair series

(λ1, u1), · · · , (λj , uj), · · · with limj→∞

λj =∞ and b(ui, uj) = δij .

Nonconforming Finite element discretization

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.

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 = `

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

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 .

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.

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).

Outline

Multilevel correction method

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

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)

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

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

Outline

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 .

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).

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)

Outline

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)

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 .

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

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 .

Outline

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,

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

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

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

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.

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 .

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

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 .

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 .

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

Postprocessing method

Based on the current obtained eigenpair approximation(λh, uh) ∈ R× V NC

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).

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

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

‖uh − u‖2a,h ≥ Ch2, and ‖uh − u‖2b ≤ Ch4γ .

So the first term is dominant and positive.

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

‖u− vh‖a,h = infv∈V NC

‖u− vh‖2a,h

Produce the space V HCh by refining the mesh or improving the order

of the finite element space

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).

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

and the discrete version in the space Vhλj,h = min

Vj,h⊂Vh,dimVj,h=jmaxv∈Vj,h

Since Vh ⊂ V = H10 (Ω), λj ≤ λj,h.

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

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

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

Thanks

Thank You Very Much!

2013 BCAM Computing Lower and Upper Bounds of Eigenvalues

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