Numerical lower bounds on eigenvalues of elliptic operators

Transcription

1 Numerical lower bounds on eigenvalues of elliptic operators Tomáš Vejchodský Institute of Mathematics Czech Academy of Sciences INSTITUTEofMATHEMATICS Czech Academy of Sciences EMS School in Applied Mathematics, Kácov, May 29 June 3, 2016

2 What do we do in numerics?

3 What do we do in numerics? We solve problems approximately.

4 What do we do in numerics? We solve problems approximately. What should we do?

5 What do we do in numerics? We solve problems approximately. What should we do? Solve problems within a given accuracy.

6 Example dumbbell u i = λ i u i in Ω u i = 0 on Ω π π π 4 [Trefethen, Betcke 2006]

7 Example dumbbell u i = λ i u i u i = 0 in Ω on Ω λ λ

8 Example dumbbell u i = λ i u i u i = 0 in Ω on Ω λ λ λ λ

9 Example dumbbell u i = λ i u i u i = 0 in Ω on Ω λ λ λ λ λ λ

10 Example dumbbell u i = λ i u i u i = 0 in Ω on Ω N tri = λ λ λ λ λ λ λ λ

11 Example dumbbell u i = λ i u i u i = 0 in Ω on Ω N tri = λ λ λ λ λ λ λ λ λ λ

12 Example dumbbell u i = λ i u i u i = 0 in Ω on Ω λ λ

13 Example dumbbell u i = λ i u i u i = 0 in Ω on Ω λ λ λ λ

14 Example dumbbell u i = λ i u i u i = 0 in Ω on Ω λ λ λ λ λ λ

15 Example dumbbell u i = λ i u i u i = 0 in Ω on Ω N tri = λ λ λ λ λ λ λ λ

16 Example dumbbell u i = λ i u i u i = 0 in Ω on Ω N tri = λ λ λ λ λ λ λ λ λ λ

17 Upper bounds on eigenvalues Laplace eigenvalue problem u i = λ i u i u i = 0 in Ω on Ω Weak formulation λ i R, u i H0 1 (Ω) : ( u i, v) = λ i (u i, v) v H0 1 (Ω) Finite element method V h = {v h H 1 0 (Ω) : v h K P 1 (K), K T h } Λ h,i R, u h,i V h : ( u h,i, v h ) = Λ h,i (u h,i, v h ) v h V h Lower bound?? λ i Λ h,i, i = 1, 2,..., m

18 Lower bounds on eigenvalues Old problem: Temple 1928, Kato 1949, Lehmann 1949, 1950, Harrell 1978,... Methods based on FEM: 1. Eigenvalue inclusions [Behnke, Mertins, Plum, Wieners 2000] based on [Behnke, Goerish 1994] and [Plum 1997] 2. Crouzeix Raviart elements [Carstensen, Gedicke 2013] 3. Complementarity based [Šebestová, Vejchodský 2016]

19 Method 1. Eigenvalue inclusions Input: Rough lower bounds: λ 2 λ 2,..., λ m+1 λ m+1, Algorithm: FEM eigenpairs: Λ h,i R, u h,i V h, i = 1, 2,..., m Mixed FEM problem: σ h,i W h, q h,i Q h, i = 1, 2,..., m W h = {σ h H(div, Ω) : σ h K RT k (K) K T h } Q h = {q h L 2 (Ω) : q h K P k (K) K T h } (σ h,i, w h ) + (q h,i, div w h ) = 0 w h W h, (div σ h,i, ϕ h ) = ( u h,i, ϕ h ) ϕ h Q h,

20 Method 1. Eigenvalue inclusions Input: Rough lower bounds: λ 2 λ 2,..., λ m+1 λ m+1, Algorithm: FEM eigenpairs: Λ h,i R, u h,i V h, i = 1, 2,..., m Mixed FEM problem: σ h,i W h, q h,i Q h, i = 1, 2,..., m For n = 1, 2,..., m do γ = u h,n + div σ h,n L 2 (Ω), ρ = λ n+1 + γ M ij = ( u h,i, u h,j ) + (γ ρ)(u h,i, u h,j ) N ij = ( u h,i, u h,j ) + (γ 2ρ)(u h,i, u h,j ) + ρ 2 (σ h,i, σ h,j ) +(ρ 2 /γ)(u h,i + div σ h,i, u h,j + div σ h,j ) µ 1 µ n : My i = µ i Ny i, i = 1, 2,..., n If N is s.p.d. and if µ i < 0 then l incl j,n = ρ γ ρ/ (1 µ n+1 j) λ j, j = 1, 2,..., n. end for l incl j = max{l incl j,n, n = j, j + 1,..., m} λ j, j = 1, 2,..., m

21 Method 2. Crouzeix Raviart elements Crouzeix Raviart finite elements V CR h = {v h P 1 (T h ) : v h continuous in midpoints of all γ E h } Find 0 uh,i CR Vh CR, λ CR h,i R : ( u CR h,i, v h ) = λ CR h,i (ucr h,i, v h ) Lower bound (no round-off errors) v h V CR h. l CR i = λ CR h,i 1 + κ 2 λ CR h,i h2 max λ i i = 1, 2,... where κ 2 = 1/8 + j 2 1, h max = max K Th diam K

22 Method 2. Crouzeix Raviart elements Crouzeix Raviart finite elements V CR h = {v h P 1 (T h ) : v h continuous in midpoints of all γ E h } Find 0 uh,i CR Vh CR, λ CR h,i R : ( u CR h,i, v h ) = λ CR h,i (ucr h,i, v h ) v h V CR h. Lower bound (inexact solver: Aũ CR i where l CR i = λ CR h,i 1 + κ 2 ( λ CR h,i r B 1 r B 1 κ 2 = 1/8 + j 2 1, h max = max K Th diam K r = Aũ CR i λ CR h,i BũCR i λ CR h,i BũCR i ) ) h 2 max Provided r B 1 < λ CR h,i λ CR h,i λ i i = 1, 2,... is closer to λ CR h,i than to any other discrete eigenvalue λ CR h,j, j i

23 Method 2. Crouzeix Raviart elements Upper bound T h is the red refinement of T h for i = 1, 2,..., m u h,i = I CM ũ CR h,i S, Q R m m with entries S j,k = ( u h,j, u h,k ) and Q j,k = (u h,j, u h,k ) Sy i = Λ i Qy i, i = 1, 2,..., m Λ 1 Λ 2 Λ m λ i Λ i for i = 1, 2,..., m

24 Method 3. Complementarity based FEM eigenpairs: Λ h,i R, u h,i V h, i = 1, 2,..., m Flux reconstruction: q h,i = z N h q z,i Local mixed FEM: q z,i W z, d z,i P 1 (T z) (q z,i, w h ) ωz (d z,i, div w h ) ωz = (ψ z u h,i, w h ) ωz w h W z (div q z,i, ϕ h ) ωz = (r z,i, ϕ h ) ωz ϕ h P1 (T z ) where ωz is the patch of elements around vertex z N h Tz is the set of elements in ω z Wz = {w h H(div, ω z ) : w h K RT 1 (K) K T z and w h n ωz = 0 on Γ ext } { ω z P {vh P 1 (T z ) : 1 (T z ) = ω z v h dx = 0} for z N h \ Ω P 1 (T z ) for z N h Ω rz,i = Λ h,i ψ z u h,i ψ z u h,i

25 Method 3. Complementarity based FEM eigenpairs: Λ h,i R, u h,i V h, i = 1, 2,..., m Flux reconstruction: q h,i = z N h q z,i Local mixed FEM: q z,i W z, d z,i P 1 (T z) (q z,i, w h ) ωz (d z,i, div w h ) ωz = (ψ z u h,i, w h ) ωz w h W z (div q z,i, ϕ h ) ωz = (r z,i, ϕ h ) ωz ϕ h P1 (T z ) Error estimator: η i = u h,i q h,i L 2 (Ω) ( Lower bound: l cmpl 1 = η 1 + Provided Λ h,i 2 ( λ 1 i l cmpl i = Λ h,i ( 1 + λ 1/2 + λ 1 ) 1 i+1 η Λ h,1) 2 /4 1 η i ) 1, i = 2, 3,...

26 Comparison 1. Inclusions 2. CR elements 3. Complementarity convergence ** *** * generality *** * ** a priori info * *** ** DOFs needed * ** *** algebraic err. ** *** *** adaptivity * ** ***

27 Thank you for your attention Tomáš Vejchodský Institute of Mathematics Czech Academy of Sciences INSTITUTEofMATHEMATICS Czech Academy of Sciences EMS School in Applied Mathematics, Kácov, May 29 June 3, 2016

28 Square λ 1 = 2, u 1 (x, y) = sin(x) sin(y) λ 2 = 5, u 2 (x, y) = sin(2x) sin(y) λ 3 = 5, u 3 (x, y) = sin(x) sin(2y)

29 Two squares λ 1 = 2 λ 2 = 5 λ 3 = 5

30 Two squares λ 1 = 2 λ 2 = 2 λ 3 = 5 λ 4 = 5 λ 5 = 5 λ 6 = 5

31 Dumbbell λ λ λ λ λ λ