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 Science Chinese Academy of Sciences Homepage: lsec.cc.ac.cn/∼hhxie Oct. 17-18 2013
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2013 BCAM Computing Lower and Upper Bounds of Eigenvalues
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2013 BCAM
Computing Lower and Upper Bounds of Eigenvalues
The Second BCAM Workshop on Computational Mathematics
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
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)
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 .
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
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
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).
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)
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
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
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
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
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
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