propagation properties for some discontinuous galerkin and ... · propagation properties for some...

UNIVERSIDAD AUTONOMADE MADRID

www.uam.es/matematicas www.bcamath.org

Propagation properties for some discontinuous Galerkin and higher orderclassical finite element methods of the wave equation

Aurora-Mihaela [email protected]

PhD Thesis defenseAdvisor: Enrique ZUAZUA IRIONDO

Departamento de Matematicas, UAMMadrid, September 30, 2010

Aurora Marica (BCAM, UAM) DG & classical FEM PhD defense, Madrid - Sept.-30-2010 1 / 39

Continuous model

The Cauchy problem for the 1− d wave equation - well posed in the energy space H1 × L2(R):

∂2t u(x , t)− ∂2

x u(x , t) = 0, x ∈ R, t > 0, u(x , 0) = u0(x), ut (x , 0) = u1(x), x ∈ R. (1)

Conservation of the total energy: E(u0, u1) = 12

∫R

(|∂x u(x , t)|2 + |∂t u(x , t)|2

)dx .

Observability inequality (OI): ∀T > 2, ∃C(T ) > 0 s.t. ∀(u0, u1) of finite energy:

E(u0, u1) 6 C(T )

T∫0

EΩ(u0, u1, t) dt,

where Ω = R \ (−1, 1) and EΩ(u0, u1, t) = 12

∫Ω

(|∂x u(x , t)|2 + |∂t u(x , t)|2

)dx .

Bardos, Lebeau, Rauch, Sharp sufficient conditions for the Observation, Control andStabilization of waves form the boundary, SIAM J.Cont.Optim, 1992

E.Zuazua, Exponential decay for the semilinear wave equation with localized damping inunbounded domains, J. Math. Pures Appl., 1991


Proof:

Since R = Ω ∪ I , with I = (−1, 1), the following identities hold:

E(u0, u1) = EΩ(u0, u1, t) + EI (u0, u1, t), ∀t ∈ R,

and

T∫0

EΩ(u0, u1, t) dt = TE(u0, u1)−T∫

0

EI (u0, u1, t) dt.

Set I :=∫R

EI (u0, u1, t) dt = 14π

1∫−1

∫R

(τ2|Fu(x , τ)|2 + |(Fu)x (x , τ)|2) dτ dx .

From the inverse SDFT, u(x , t) = 12π

∫R∑±

12

(u0(ξ)± u1(ξ)

iξ

)exp(±itξ) exp(iξx) dξ,

Fu(x , τ) = A+ + A− :=∑±

1

2

(u0(±τ) +

u1(±τ)

iτ

)exp(±iτx),

(Fu)x (x , τ) = iτ(A+ − A−)

and

τ2|Fu(x , τ)|2 + |(Fu)x (x , τ)|2 = 2τ2(|A+|2 + |A−|2) = 2τ2∑±

1

4|u0(±τ) +

u1(±τ)

iτ|2.

I = 14π

∫I

∫R

12τ2∑±|u0(±τ) + u1(±τ)

iτ|2 dτ dx = |I | 1

4π

∫R

12τ2∑±|u0(τ)± u1(τ)

iτ|2 dτ = |I |E(u0, u1).


Motivation: control of the wave equation

By the Hilbert Uniqueness Method, the observability problem (1) is equivalent to an exactcontrollability problem:

∀T > T∗ = 2, ∀(u0, u1) ∈ H1(R)× L2(R), ∃f ∈ L2(Ω× (0,T )) s.t. the solution of

∂2t u(x , t)− ∂2

x u(x , t) =f (x , t)χΩ(x), x ∈ R, t ∈ (0,T ], u(x , 0) = u0(x), ∂t u(x , 0) = u1(x), x ∈ R

satisfies u(x ,T ) = ∂t u(x ,T ) = 0 for all x ∈ R.

J.-L. Lions, Controlabilite exacte, Tome 1, Masson, 1988.

E. Zuazua, Controllability and Observability of PDEs: Some results and open problems,Elsevier Science, 2006.

Geometric Control Condition (GCC): All rays of Geometric Optics (GO) enter the observationset during the observability time.

F. Macia, E. Zuazua, On the lack of observability for wave equations: a Gaussian beamapproach, Asymptotic Analysis, 2002.

J. Ralston, Gaussian beams and the propagation of singularities, Studies in PDEs, 1983.


A Gaussian beam construction for the continuous case, T ≤ 2

ω(ξ) =: ξ. By means of the FT, equation (1) can be transformed into

utt (ξ, t) + ω2(ξ)u(ξ, t) = 0, ξ ∈ R, t ∈ (0,T ], u(ξ, 0) = u0(ξ), ut (ξ, 0) = u1(ξ), ξ ∈ R.

Solution: u(ξ, t) = 12

(u0(ξ)± u1(ξ)

iω(ξ)

)exp(±itω(ξ)). E(u0, u1) = 1

4π

∫R

(|ξ|2|u0(ξ)|2 + |u1(ξ)|2

)dξ.

Choice of the direction: u1(ξ) = iω(ξ)u0(ξ) or u1(x) = u0x (x).

Solution: u(ξ, t) = u0(ξ) exp(itω(ξ)) or u(x , t) = u0(x + t). E(u0, u1) = 12π

∫Rω2(ξ)|u0(ξ)|2 dξ.

Choice of the scales:A wave number: η0/ε, η0 ∈ R, |η0| = 1, ε << 1. ⇒rays of GO: x(t) = x∗ − tη0.A parameter γ >> 1, which is the support in the Fourier space with εγ << 1.A smooth function σ, compactly supported in (−1, 1). (More generally, σ ∈ S(R)).

Choice of u0: u0(ξ) = γ−1/2σ(γ−1

(ξ − η0

ε

))exp(−iξx∗) 1

|ω(ξ)| .

Properties of these numerical solutions:

a. the total energy E(u0, u1) is of order one.

b. ∃CN (σ, δ) > 0 s.t.T∫0

∫|x−x(t)|≥δ

(|∂t u(x , t)|2 + |∂x u(x , t)|2) dx dt ≤ CN (σ, δ)γ1−2N .

c. For all T ≤ 2 and all β > 0, the observability constant C(T ) satisfies C(T ) ≥ Cβ(σ, δ)γβ .

S. Ervedoza, E. Zuazua, Propagation, observation and numerical approximation of waves.

Proof: Stationary phase lemma (cf. L. Evans, PDEs, 2000)

σ ∈ C∞c (R), ψ ∈ C∞(R) s.t. ψx 6= 0 in supp(σ) ⇒ Iε =∫Rσ(ξ) exp(iψ(ξ)/ε) dξ = O(εN ), ∀N.


The finite difference semi-discrete wave equation

∂2h fj := (fj+1 − 2fj + fj−1)/h2. Consider the finite difference (FD) semi-discrete wave equation:

∂2t uh

j (t)− ∂2huj (t) = 0, j ∈ Z, t ∈ (0,T ], uj (0) = u0

j , ∂t uj (0) = u1j , j ∈ Z. (2)

Well-posedness: ~1(hZ)× `2(hZ). Energy: Eh(−→u 0,−→u 1) = 12

(‖−→u (t)‖2~1(hZ)

+ ‖∂t−→u (t)‖2

`2(hZ)).

For fixed T > 0, consider the discrete OI: Eh(−→u 0,−→u 1) ≤Ch(T )T∫0

EΩ,h(−→u 0,−→u 1, t) dt,

where EΩ,h(−→u 0,−→u 1, t) = h2

∑xj∈Ω

(|∂t uj (t)|2 + |∂huj (t)|2) and ∂hfj := (fj+1 − fj )/h.

Πh := [−π/h, π/h]; ωh(ξ) := 2 sin(ξh/2)/h; uh(ξ, t) = SDFT of −→u (t). It satisfies the ODE:

∂2t uh(ξ, t) + ω2

h(ξ)uh(ξ, t) = 0, ξ ∈ Πh, t ∈ (0,T ], uh(ξ, 0) = uh,0(ξ), ∂t uh(ξ, 0) = uh,1(ξ), ξ ∈ Πh.

Discrete rays of GO: x±h (t) = x∗ ± ∂ξωh(ξ0)t, x∗ ∈ R, ξ0 ∈ Πh. Group velocity: ∂ξωh(ξ).

J. Strikwerda, FD schemes and PDEs, SIAM, 2004.

L.N. Trefethen, Group velocity in FD schemes, SIAM Review, 1982.

Key difference between discrete and continuous velocities: |∂ξω| ≡ 1 versus |∂ξωh(±π/h)| = 0.

A. M., E. Zuazua, Localized solutions for the FD semi-discretization of the wave equation,CRAS, 2010.


00

2

π

π/2 π

Figure: Continuous and discrete dispersion relations versus the corresponding high frequency wave packets.


Exponentially concentrated 1 − d numerical waves

S. Micu, Uniform boundary controllability of a semi-discrete 1-d wave equation, Numer.Math., 2002.

Aim: Exact controllability to rest of

∂2t uh

j − ∂2huh

j = 0, j ∈ ΛN = [−N,N] ∩ Z, uh±(N+1)(t) = vh

±(t), t ∈ (−T/2,T/2), +IC.

By estimates on bi-orthogonal sequences to the family of eigenvalues,

∃vh±(T ) ∈ L2(−T/2,T/2) s.t.

∑±||vh±||L2(−T/2,T/2) ≥ exp(

√N).

For the adjoint problem

∂2t φ

hj − ∂

2hφ

hj = 0, j ∈ ΛN = [−N,N] ∩ Z, φh

±(N+1)(t) = 0, t ∈ (−T/2,T/2), +IC,

C bh (T ) = sup

(φh,0,φh,1)

Eh(φh,0, φh,1)T∫0

(|φhN

(t)

h|2 + |

φh−N

(t)

h|2) dt

= supvh±

∑±||vh±||L2 ≥ exp(

√N).

Conclusion: ∃(φh,0, φh,1) s.t. C bh (T ) corresponding to φh(t) blows-up exponentally.

Set φhext (t) = the extension by zero outside (−1, 1) of φh(t). The associated Ch(T ) blows-up

exponentially!!!

Drawbacks: i) only 1− d ; ii) only for discrete spectrum; iii) no knowledge about the shape ofφh(t) giving exponential blow-up of Ch(T ).


Polynomially concentrated (1 − d) wave packets

Theorem

Fix T > 0, η0 = hξ0 ∈ Π1 and x∗ ∈ (−1, 1) s.t. no GCC, i.e.

|xh(t)| = |x∗ − tω′1(η0)| < 1, ∀t ∈ [0,T ].

σ ∈ C∞c (−1, 1) (or S(R)) and γ = γ(h) > 0 s.t. γ >> 1 and hγ << 1. Explicit initial data in(2):

φh,0(ξ) = γ−1/2σ(γ−1(ξ − ξ0)) exp(−iξx∗)1

iωh(ξ), φh,1(ξ) = iωh(ξ)φh,0(ξ).

Conclusion: ∀α > 0, ∃cα(T , σ, η0) s.t. Ch(T ) ≥ cα(T , σ, η0)γα .

Proof: Aim of the factor γ−1/2 and of 1/iωh(ξ): Eh(φh,0, φh,1) = 12π

1∫−1

|σ(η)|2 dη=constant.

Set Ωδ(t) := x ∈ R : |x − xh(t)| > δ. Auxiliary estimate: ∀ fixed δ, ∀t ∈ [0,T ], ∀N ∈ N∗,

EΩδ(t),h(φh,0, φh,1, t) ≤ CN (T , σ, δ, η0)γ−(2N−1). (&)

If no GCC, ∃δ > 0 s.t. |x − xh(t)| < δ ⊂ (−1, 1), ∀t ∈ [0,T ]. Then Ω× [0,T ] ⊂ Ωδ(0,T ) or

EΩ,h(φh,0, φh,1, t) ≤ EΩδ(t),h(φh,0, φh,1, t), ∀t ∈ [0,T ].


Proof of (&):

Set ψ(η, x , t) := tω1(η) + x(η + η0). By the change of variable η := γ−1(ξ − ξ0), we obtain:

φ(x , t) = γ1/2 1

2π

1∫−1

σ(η) exp

(i

hψ(γhη, x − x∗, t)

)1

iω1(η0 + γhη)dη,

∂tφ(x , t) = γ1/2 1

2π

1∫−1

σ(η) exp

(i


)dη (&&)

and similar for ∂+h φ(x , t) := (φ(x + h, t)− φ(x , t))/h.

A stationary phase - like argument. An identity:

exp

(i


)=

1

iγ

1

∂ηψ(γhη, x − x∗, t)∂η

(exp

(i


)).

Lσ(η, x − x∗, t) := ∂η(

σ(η)∂ηψ(γhη,x−x∗,t)

). By integrations by parts, ∀N ∈ N∗,

∂tφ(x , t) = γ1/2 1

2π

1∫−1

(−

1

iγ

)N

LN σ(η, x − x∗, t) exp

(i


)dη

or |∂tφ(x , t)|2 ≤ γ1−2N

1∫−1

|LN σ(η, x − x∗, t)|2 dη.


Integrability in space of LN σ

We need∫

Ωδ(t)

|∂tφ(x , t)|2 dx <∞. Sufficient∫

Ωδ(t)

|LN σ(η, x − x∗, t)|2 dx <∞.

Lemma

∀N ∈ N, ∃fN (η) ∈ L1(−1, 1) s.t. |LN σ(η, x − x∗, t)|2 ≤ fN (σ)

|x−xh(t)|2N .

Proof for N = 1. |LN σ(η, x − x∗, t)| ≤ |σ′(η)| 1|∂ηψ(γhη,x−x∗,t)| + γh|σ(η)|

|∂2ηψ(γhη,x−x∗,t)|

|∂ηψ(γhη,x−x∗,t)|2 .

Identity: ∂ηψ(γhη, x − x∗, t) = x − xh(t) + tγhη∂2ηω1(η0 + γhη′), η′ ∈ (−1, 1).

Then |∂ηψ(γhη, x − x∗, t)| ≥ |x − xh(t)| − Tγh||∂2ηω1||L∞(η0−γh,η0+γh).

Since γh << 1 and T , δ finite, Tγh||∂2ηω1||L∞(η0−γh,η0+γh) ≤ δ/2 ≤ |x−xh(t)|

2, i.e.

|∂ηψ(γhη, x − x∗, t)| ≥ |x − xh(t)|/2.

Also ∂2ηψ(γhη, x − x∗, t) = t∂2

ηω1(η0 + γhη) does not depend on x .

Open problem

Construct explicit wave packets proving exponential blow-up of the observability constant.


WKB expansions

γh << 1 - the scale indicating the behavior of high frequency wave packets as x →∞.

Close to the ray xh(t) - the right scale seems to be γh1/2 << 1.

−1 0V

−1

0

1

−1 0V

−1

0

1

Figure: Propagation of a Gaussian wave packet with γ = h−3/8 versus γ = h−5/8.

Two ways to analyze this: 1. Gaussian wave packets. A wave packet of the form

φ(x , t) = 12π

∫Πh

√2πγ2 exp

(− (ξ−ξ0)2

2γ2

)exp

(ithω1(ξh) + iξx

)dξ.

Approximate the phase ω1(ξh) ∼ ω1(η0) + (ξ − ξ0)hω′1(η0) + (ξ − ξ0)2h2/2ω′′1 (η0).

The corresponding solution:

ψ(x , t) =γ(t)

γexp

(−γ2(t)

2(tω′1(η0) + x)2

), γ2(t) =

γ2

1− ithγ2ω′′1 (η0).

When γh1/2 >> 1, the Gaussian changes the aperture from γ2 at t = 0 to 1/(hγ)2 at t = 1.


The error:||φ(·, t)− ψ(·, t)||L2(R)

||φ(·, t)||L2(R)

≤ h2γ3t , small if γh2/3 << 1.

2. WKB expansions. φ(x , t) = 12π

∫Rσ(ξ) exp

(ithω(ξh) + iξx

)dξ.

ω(ξh) ∼ ωN (ξh) = ξh + · · ·+(ξh)N

N!

φ(x , t) ∼ φN (x , t).

If N = 2, ∂tφ(x , t)− ∂xφ(x , t) +ih

2∂2

xφ(x , t) = 0 (∗).

Theorem

∃ solutions of (*) s.t. φ(x , t) =∞∑

j=0hj/2aj

(x

h1/2 ,t

h1/2

)exp

(ix

h1/2 + ithω2(h1/2)

).


Another “simple“ schemes providing pathological behavior

Fully discrete FD

00

2

wave number ξ

dis

persio

n r

ela

tio

n

λ=0λ=0.9λ=0.95λ=0.99λ=1

π

ππ/2

FD in higher dimensions

ππ/2

0−π/2

−πππ/2

0−π/2

0

2

23/2

−π


C k -refinements

Spline functions: Sn = f ∈ C n(R) : f χ(j−(n+2)/2,j+(n+2)/2) ∈ Pn+1(j−(n+2)/2, j+(n+2)/2), ∀j

Properties:

supp(Mn) = [−(n + 2)/2, (n + 2)/2]

Mm+n+2 = Mm ∗Mn

Mn(ξ) = sincn+2(ξ/2)

Mn(x − j) basis for Sn

Mn - even

M′n(x) = Mn−1(x + 1/2)−Mn−1(x − 1/2).

Mn(x) =

1(n+1)!

n+2∑j=0

(−1)j(n+2

j

)[(x + (n + 2)/2− j)+]n+1.

Basis functions Mn, n = −1, 0, 1, 2, 3, 4

−3 −2 −1 0 1 2 3−2.5 −1.5 −0.5 0.5 1.5 2.5

0

0.25

0.5

0.75

1

n=−1n=0n=1n=2n=3n=4

Chui, An introduction to wavelets, 1982

Hughes, Reali, Sangalli, Duality and unified analysis of discrete approx. in structuraldynamics and wave propagation..., 2008.

Bowles, Vichenevetsky, Fourier analysis of numerical approx. of hyperbolic equations, 1982.Aurora Marica (BCAM, UAM) DG & classical FEM PhD defense, Madrid - Sept.-30-2010 15 / 39

Despite of their increasing regularity, pathologic behavior is still mantained...

00

0.5

1

1.5

2

2.5

3

3.5

continuousn=0n=1n=2n=3n=4

π/2 π 00

0.5

1

1.5

continuousn=0n=1n=2n=3n=4

π/2 π

Figure: Dispersion versus group velocity for C k -refinements, k = 1, 2, 3, 4.


Filtering mechanisms: Fourier truncation

Fourier truncation, bi-grid algorithm, numerical viscosity, Tychonoff regularization...

E. Zuazua, Propagation, observation and control of waves approximated by finite differencemethods, SIAM Review, 2005.

Πδh := [−πδ/h, πδ/h], δ ∈ (0, 1)

I δh := −→f ∈ `2(hZ) : supp(f h) ⊂ Πδh - discrete functions truncated in Fourier with parameter δ.

Theorem

(−→u h,0,−→u h,1) ∈ I δh × I δh → uniform observability for the FD scheme.

Proof: Aim: Eh(−→u h,0,−→u h,1) ≤ C(T )T∫0

EΩ,h(−→u h,0,−→u h,1, t) dt,

Replace EΩ,h(−→u h,0,−→u h,1, t) =h

2

∑xj∈Ω

(|∂t uj (t)|2 + |∂+h uj (t)|2)

by EΩ,h(−→u h,0,−→u h,1, t) =h

2

∑xj∈Ω

(|∂t uj (t)|2 +

1

2|∂+

h uj (t)|2 +1

2|∂−h uj (t)|2

)

or EΩ,h(−→u h,0,−→u h,1, t) =h

2

∑xj∈Ω

(|∂t uj (t)|2 + |∂huj (t)|2 +h2

4|∂2

huj (t)|2).


Identity:

T∫0

EΩ,h(−→u h,0,−→u h,1, t) dt = TEh(−→u h,0,−→u h,1)−T∫

0

EI ,h(−→u h,0,−→u h,1, t) dt

∫R

EI ,h(−→u h,0,−→u h,1, t) dt = I + II + III .

uj (t) =1

2π

∫Πδ

h

∑±

1

2

(uh,0(ξ)±

uh,1(ξ)

iωh(ξ)

)exp(±itωh(ξ))

exp(iξxj ) dξ

uj (t) =1

2π

∫Πδ

h

(Uh,+j (ξ) + Uh,−

j (ξ)) exp(itωh(ξ)) dξ,

with Uh,±j (ξ) =

1

2

(uh,0(±ξ) +

uh,1(±ξ)

iωh(ξ)

)exp(±iξxj ).

τ = ωh(ξ) → uj (t) =1

2π

∫ωh(Πδ

h)

(Uh,+j (ω−1

h (τ)) + Uh,−j (ω−1

h (τ)))1

ω′h(ω−1h (τ))

exp(itτ) dτ.

(Fuj )(τ) = χωh(Πδh

)(τ)(Uh,+j (ω−1

h (τ)) + Uh,−j (ω−1

h (τ)))1

ω′h(ω−1h (τ))

.


Bi-grid algorithm

R.Glowinski, Ensuring the well-posedness by analogy; Stokes problem and boundary controlfor the wave equation, J. Comput. Phys., 1992.

M. Negreanu, E. Zuazua, Convergence of a multigrid method for the controllability of a1− d wave equation, CRAS, 2004.

P. Loreti, M. Mehrenberger, An Ingham type proof for a two-grid observability theorem,ESAIM COCV, 2008.

L. Ignat, E. Zuazua, Convergence of a two-grid algorithm for the control of the waveequation, JEMS, 2009.

L. Ignat, E. Zuazua, Numerical dispersive schemes for the nonlinear Schrodinger equation,SIAM. J. Numer. Anal., 2009

Description: (fj )j∈Z → (Λf )2j → (ΓΛf )j

Physical space: (Λf )2j = f2j ; Fourier space: Λf2h

(ξ) = f h(ξ) + f h(ξ − sgn(ξ)π/h), ξ ∈ Π2h

Physical space: (Γg)2j = (g2j−1 + g2j+1)/2; Fourier space: Γgh(ξ) = cos2(ξh/2)g2h(ξ)


−12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12−3

−2

−1

0

1

2

3

4

5

−12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12−3

−2

−1

0

1

2

3

4

5

−12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12−3

−2

−1

0

1

2

3

4

5

Figure:−→f versus Λ

−→f versus ΓΛ

−→f

a)

−π −3π/4 −π/2 −π/4 0 π/4 π/2 3π/4 π

b)

−π −3π/4 −π/2 −π/4 0 π/4 π/2 3π/4 π

Figure: f h(ξ) =√

2π/γ exp(−|ξ − ξ0|2/2γ)χΠh(ξ) for ξ0 = π/h, π/2h, 2π/3h (blue, red, green) versus its

projection Λf2h

(ξ), γ = h−1/4.Aurora Marica (BCAM, UAM) DG & classical FEM PhD defense, Madrid - Sept.-30-2010 21 / 39

Theorem

∀(−→u h,0,−→u h,1) ∈ ~1(hZ)× `2(hZ) s.t. ui2j =

ui2j+1+ui

2j−1

2(∗∗), ∀j ∈ Z, i = 0, 1, → uniform

observability property.

Proof: Projection on the low frequency of−→f = (fj )j∈Z: Γδ

−→f = 1

2π

∫Πδ

h

f h(ξ) exp(iξxj ) dξ.

When (−→u h,0,−→u h,1) given by (**), Eh(−→u h,0,−→u h,1) ≤ CEh(Γ1/2−→u h,0, Γ1/2−→u h,1) (∗ ∗ ∗).

Proof of (***): Since ∀−→f s.t. (**), we have f h(ξ) = cos2(ξh/2)f 2h(ξ) and then

Eh(−→u h,0,−→u h,1) =1

4π

∫Πh

(ω2h(ξ)|uh,0(ξ)|2 + |uh,1(ξ)|2) dξ

=1

4π

∫Πh

cos4(ξh/2)(ω2h(ξ)|u2h,0(ξ)|2 + |u2h,1(ξ)|2) dξ

=1

4π

∫Π2h

(ω2h(ξ)cos2(ξh/2)|u2h,0(ξ)|2 + (cos4(ξh/2) + sin4(ξh/2))|u2h,1(ξ)|2) dξ

=1

4π

∫Π2h

(ω2h(ξ)cos−2(ξh/2)|uh,0(ξ)|2 + (1 + tan4(ξh/2))|uh,1(ξ)|2) dξ

≤ 21

4π

∫Π2h

(ω2h(ξ)|uh,0(ξ)|2 + |uh,1(ξ)|2) dξ = 2Eh(Γ1/2−→u h,0, Γ1/2−→u h,1).


Steps in the dyadic decomposition argument(I)

Step I. → (***)

F ∈ C∞c (a, b), 0 ≤ F ≤ 1 and F ≡ 1 in (a + µ, b − µ). Set P(τ) = F (τ) + F (−τ).

Projectors: Pk f (t) :=1

2π

∫R

P(τ/ck )Ff (τ) exp(itτ) dτ .

Hypothesis on a, b, c, µ: 1 < c < b/a < (b − µ)/(a + µ) < ωh((1/2 + ε)π/h)/ωh(π/2h) .

For the solution of the FD system, Pk uj (t) =1

2π

∫Πh

P(ωh(ξ)/ck )uh(ξ, t) exp(iξxj ) dξ .

Eh(Pk uj (t),Pk u′j (t)) conserved in time,

Eh(Pk−→u h,0,Pk

−→u h,1) :=1

4π

∫Πh

P2(ωh(ξ)/ck )(ω2h(ξ)|uh,0(ξ)|2 + |uh,1(ξ)|2) dξ.

∪∞k=k0 (ack , bck ) = (ack0 ,∞) → ∀ωh(ξ) ≥ ack0 belongs to the support of at least one Pk .

Choose kh s.t. (a + µ)ckh ≤ ωh(π/2h) < (a + µ)ckh+1 .

Step II. ∀ωh(ξ) ∈ ((a + µ)ck0 , ωh(π/2h)) belongs to at least one ((a + µ)ck , (b − µ)ck ), i.e.

1 ≤kh∑

k=k0

P2(ωh(ξ)/ck ).


Steps in the dyadic decomposition argument(II)

Step III. Eh(Γ1/2−→u h,0, Γ1/2−→u h,1) ≤ Eh(Γk0−→u h,0, Γk0

−→u h,1) +

kh∑k=k0

Eh(Pk−→u h,0,Pk

−→u h,1) .

Step IV. Pk−→u h,i ∈ I

(1/2+ε)h , so that, for T − 2γ > T 1/2+ε,

Eh(Pk−→u h,0,Pk

−→u h,1) ≤ C 1/2+ε(T − 2γ)

T−γ∫γ

EΩ,h(Pk−→u h,0,Pk

−→u h,1, t) dt.

Step V.∑k≥k0

T−γ∫γ||Pk w(t)||2X dt ≤ C(P, c)

T∫0

||w(t)||2X dt + C(P,T ,γ)

c2k0sup

j||w ||2

L2(jT ,(j+1)T ,X ).

Burq, Controlabilite exacte des ondes dans des ouverts peu reguliers, 1997.

Ignat, Propiedades cualitativas de esquemas numericos de aproximacion de dispersion yde difusion, 2006.

Lebeau, Controle de l’equation de Schrodinger, 1992.

Step VI. Compactness of Eh(Γk0−→u h,0, Γk0

−→u h,1).

Zuazua, Boundary observability for the FD space semi-discretizations of the 2-d waveeqn. in the square, 1999.


−1 0 1 −1

0

1

FD, without bigrid, η0=7π/8

−1 0 1 −1

0

1

With bi−grid of ratio 1/2, η0=7π/8

−1 0 1−1

0

1

FD, without bigrid, η0=255π/256

−1 0 1−1

0

1

With bi−grid of ratio 1/2, η0=255π/256

Figure: Solutions in the physical space for ξ0 = 7π/8h (up) and ξ0 = 255π/256h (down). In blue,continuous/discrete at t = 0, in black, continuous at t = 1, in green, discrete at t = 1.


Some more recent results

A.M., E. Zuazua, High frequency wave packets for the Schrodinger equation and itsnumerical approximations, preprint.

Comparison between the solutions of the continuous linear 1− d Schrodinger equation (CSE)

i∂t u(x , t) + ∂2x u(x , t) = 0, x ∈ R, t ∈ R \ 0, u(x , 0) = ϕ(x), x ∈ R.

and theose of its FD semi-discretization (DSE)

i∂t uj (t) + ∂2huj (t) = 0, j ∈ Z, t ∈ R \ 0, uj (0) = ϕj , j ∈ Z.

Dispersive properties for the solutions of the CSE:

‖u‖Lqt (R,Lp

x (R)) ≤ c(p)‖ϕ‖L2(R), supR

( 1

R

∫R

∫ R

−R|∂1/2

x u(x , t)|2 dx dt)1/2

≤ c‖ϕ‖L2(R),

for (p, q) such that the admissibility conditions 2 ≤ p ≤ ∞ and 2/q = 1/2− 1/p hold.

Cazenave, Semilinear Schrodinger equations, Courant Lecture Notes in Mathematics, 2003.

Linares, Ponce, Introduction to Nonlinear Dispersive Equations, Springer, 2009.

Discrete versions of dispersive properties:do not hold uniformly as h→ 0 for the DSE,hold uniformly in the subclass of bi-grid initial data obtained by linear interpolation from agrid four times coarser.

Ignat, Zuazua, Numerical dispersive schemes for the nonlinear Schrodinger equation, 2009.


00

2

4

6

8

10

−π −π/2 π/2 π

Figure: Discrete versus continuous Fourier symbol


a)

−π −3π/4 −π/2 −π/4 0 π/4 π/2 3π/4 π

b)

−π −3π/4 −π/2 −π/4 0 π/4 π/2 3π/4 π

c)

−π −3π/4 −π/2 −π/4 0 π/4 π/2 3π/4 π

Figure: a) Initial data ϕη0with η0 = π, π/2, 2π/3 and their projections Λr

k

2k hwith b) k = 1 and c) k = 2.

In black, the corresponding weights bk .

a)

−π −3π/4 −π/2 −π/4 0 π/4 π/2 3π/4 π

b)

−π −3π/4 −π/2 −π/4 0 π/4 π/2 3π/4 π

c)

−π −3π/4 −π/2 −π/4 0 π/4 π/2 3π/4 π

Figure: a) Initial data ϕη0with η0 = π, π/2, 2π/3 and their projections Λa

k

2k hwith b) k = 1 and c) k = 2.

In black, the corresponding weights bk . Observation: Λakϕη0

∼ bk (η0)Λrkϕη0

and bk (η0) < 1.


−1

0

1

a)

−π/h −2/h 0 2/h π/h

−1

0

1

b)

−π/h −2/h 0 2/h π/h

−1

0

1

c)

−2π/h 0 2π/h

−1

0

1 d)

0−2sin(2π/3)/h 2sin(2π/3)/h 4π/(3h)−4π/(3h)

Figure: Solutions of both CSE and DSE in the physical space corresponding to the initial data ϕη0using the

projections Λαk , α = r , a, with a) (η0, α) = (π/2, r), b) (η0, α) = (π/2, a), c) (η0, α) = (π, r) and d)(η0, α) = (2π/3, r). Legend: CSE at t = 0, CSE at t = 1, DSE without filtering at t = 1, DSE with bi-grid ofratio 1/2 (k = 1) at t = 1 and DSE with bi-grid of ratio 1/4 (k = 2) at t = 1.

Open problem

Adding non-linearities: transparent BC, splitting methods, fully discrete, non-uniform meshes.


A discontinuous Galerkin (DG) semi-discretization of the wave equation

Average: f (x) = f (x+)+f (x−)2

Jump: [f ](x) = f (x−)− f (x+).

Finite element space: Vh := UAh ⊕ UJ

h , with UAh = spanφA

i , i ∈ Z and UJh = spanφJ

i , i ∈ Z.

−1

0

1

xi−1

xi x

i+1

−1

0

1

−0.5

0.5

xi−1

xi x

i+1

Figure: Typical basis functions for the P1-discontinuous Galerkin methods: φAi (left) and φJ

i (right).

Arnold, Brezzi, Cockburn, Marini, Unified Analysis of DG methods for elliptic problems,2002.

SIPG: ash(u, v) =

∑j∈Z

( xj+1∫xj

ux (x)vx (x) dx−[u](xj )vx(xj )− [v ](xj )ux(xj ) +s

h[u](xj )[v ](xj )

).


SIPG semi-discrete wave equation and its Fourier analysis

ush(x , t) ∈ Vh, s.t.∀t > 0, ∂2

t

∫R

ush(x , t)v(x) dx = as

h(ush(·, t), v), ∀v ∈ Vh + i.c.. (3)

Mh−→U h

tt (t) + Rsh

−→U h(t) = 0,

−→U h(0) =

−→U h,0,

−→U h

t (0) =−→U h,1, where

−→U h(t) = (

−→A h(t),

−→J h(t))

and Mh, Rsh are infinite mass and stiffness matrices.

Ah(ξ, t), Jh(ξ, t):= SDFTs of−→A h(t),

−→J h(t); Uh(ξ, t) := (Ah(ξ, t), Jh(ξ, t))′ verifies the system:

Uhtt (ξ, t) + As

h(ξ)Uh(ξ, t) = 0, Uh(ξ, 0) = Uh,0(ξ), Uht (ξ, 0) = Uh,1(ξ), ξ ∈ Πh, t > 0, (4)

where Ash(ξ) = (Mh(ξ))−1Rs

h(ξ) and Mh(ξ), Rsh(ξ) are the Fourier symbols of Mh and Rs

h .

Total energy:=E sh (−→U h,0,

−→U h,1). DG OI: E s

h (−→U h,0,

−→U h,1) ≤ C s

h (T )

T∫0

E sΩ,h(−→U h,0,

−→U h,1, t) dt.

Uh(ξ, t) =∑±

1

2

[Ps

h(ξ)

(exp(±itλs

ph,h(ξ)) 0

0 exp(±itλssp,h(ξ))

)(Ps

h(ξ))−1

Uh,0(ξ)

+ Psh(ξ)

±exp(±itλs

ph,h(ξ))

iλsph,h

(ξ)0

0 ±exp(±itλs

sp,h(ξ))

iλssp,h

(ξ)

(Psh(ξ)

)−1Uh,1(ξ)

].


Properties of the eigenvalues and eigenvectors of Ash(ξ)

Psph,h(ξ), Ps

sp,h(ξ)=physical, spurious eigenvectors

Λsph,h(ξ), Λs

sp,h(ξ) =physical, spurious eigenvalue, λsph,h(ξ) =

√Λs

ph,h(ξ), λssp,h(ξ) =

√Λs

sp,h(ξ).

Properties of eigenvalues and group velocities:∀s > 1, ∂ξλ

sph,h(ξ)→ 1 and ∂ξλ

ssp,h(ξ)→ 0 as ξ → 0.

∀s ∈ (1,∞) \ 3, ∂ξλsph,h(ξ), ∂ξλ

ssp,h(ξ)→ 0 as ξ → ±π/h.

∂ξλ3ph,h(ξ)→ 1 and ∂ξλ

3sp,h(ξ)→ −1 and ξ ± π/h.

2

0

π

61/2

121/2

0 π/2 π0

121/2

π

2

23/2

π/2 π0

241/2

121/2

π

2

0

0 π/2 π0

2

121/2

π

201/2

481/2

0 π/2 π

Figure: λsph,1(ξ) (black) and λs

sp,1(ξ) (dotted black) for s = 1.5, 2, 3, 5.


Properties of the physical eigenvector

Psph(ξ)→ (1, 0) as ξ → 0

Psph(ξ)→ (1, 0) as ξ → ±π/h and s > 3

Psph(ξ)→ (

√3/7, ±2/i

√7) as ξ → ±π/h and s = 3

Psph(ξ)→ (0, ±1/i) as ξ → ±π/h and s ∈ (1, 3).

Re(F)Im(F)

x2j+2

x2j+1x

2jx

2j−1x

2j−2

0

1

Re(F)Im(F)

x2j−1

x2j+2

x2j

x2j−1

x2j−2

−1

0

1

Re(F)Im(F)

x2j

x2j+1 x

2j+2x

2j−1x

2j−2

−(1/7)1/2

−(3/7)1/2

0

(1/7)1/2

(3/7)1/2

Re(F)Im(F)

x2j+1

x2j+2

x2j

x2j−1

x2j−2

0

1/2

−1/2


Properties of the spurious eigenvector

Pssp(ξ)→ (0, 1) as ξ → 0

Pssp(ξ)→ (0, 1) as ξ → ±π/h and s > 3

Pssp(ξ)→ (±1/i

√3/7, 2/

√7) as ξ → ±π/h and s = 3

Pssp(ξ)→ (±1/i , 0) as ξ → ±π/h and s ∈ (1, 3).

Im(F)Re(F)

1/2

0

−1/2

x2j−2

x2j−1

x2j

x2j+1

x2j+2

Im(F)Re(F)

−1/2

0

1/2

x2j−2

x2j−1

x2j x

2j+1x

2j+2

Im(F)Re(F)

(3/7)1/2

(1/7)1/2

0

−(3/7)1/2

x2j−1

x2j−2

x2j

x2j+1

x2j+2

−(1/7)1/2

Re(F)Im(F)−1

0

1

x2j−1

x2j+1

x2j

x2j+2

x2j−2


Non-uniform observability inequality

When the vector valued initial data Uh,i in (4), i = 0, 1, are of the form

Uh,i (ξ) = Psph,h(ξ)uh,i (ξ), (5)

the corresponding solutions of (4) involve only the physical dispersion relation:

Uh(ξ, t) = Psph,h(ξ)

1

2

∑±

(uh,0(ξ)±

uh,1(ξ)

iλsph,h(ξ)

)exp(±itλs

ph,h(ξ)). (6)

Proposition

T > 0 fixed s.t. the ray xph(t) = x∗ − t∂ξλsph,1(η0) does not enter the observation region before

T ; γ := γ(h) > 0 s.t. γ >> 1 and hγ << 1, σ ∈ S(R) and (4) with Uh,i (ξ) s.t. (5) holds, with

uh,0(ξ) =

√2π

γσ

(ξ − ξ0

γ

)exp(−ix∗(ξ − ξ0))χΠh

(ξ) and uh,1(ξ) = iλsph,h(ξ)uh,0(ξ). (7)

Then ∀α ∈ R+, C sh (T ) in the DG OI satisfies C s

h (T ) > Cα(σ,T , s)γα .


Filtering mechanisms for the DG semi-discrete wave equation

If f h(ξ, t) = f hph(ξ) exp(itλs

ph,h(ξ)) + f hsp(ξ) exp(itλs

sp,h(ξ))

define Γphfj (t) :=1

2π

∫Πh

f hph(ξ) exp(itλs

ph,h(ξ)) exp(iξxj ) dξ

Main results

DG OI holds uniformly as h→ 0 under one of the following hypotheses on the initial data−→U h,i :

Concentration on the physical mode + Fourier truncation: Uh,i (ξ) = Psph,h(ξ)uh,i (ξ) and

−→u h,i ∈ I δh , i = 0, 1,

Concentration on the physical mode + bi-grid algorithm: Uh,i (ξ) = Psph,h(ξ)uh,i (ξ), and

ui2j =

ui2j+1+ui

2j−1

2, i = 0, 1.

Auxiliary result 1: E sh (−→U h,0,

−→U h,1) ≤ CE s

h (Γ1/2h

−→U h,0, Γ

1/2h

−→U h,1)

Null jump part + Fourier truncation of the average part:−→J h,i = 0 and

−→A h,i ∈ I δh , i = 0, 1.


−→U h,1) ≤ CE s

h (Γph−→U h,0, Γph

−→U h,1)

Null jump part + bi-grid algorithm on the average part:−→J h,i = 0 and Ai

2j =Ai

2j+1+Ai2j−1

2,

i = 0, 1.


−→U h,1) ≤ CE s

h (ΓphΓ1/2h

−→U h,0, ΓphΓ

1/2h

−→U h,1)


A. M., E. Zuazua, Localized solutions and filtering mechanisms for the DGsemi-discretizations of the 1− d wave equation, accepted in CRAS.

00

1

2

3

4

5

6

7

ππδ

Figure: The hachured zone corresponds to frequencies eliminated by the filtering mechanism Concentration onthe physical mode + Fourier filtering


−1 0 1−1

0

1Physical mode, without bigrid, s=5

−1 0 1−1

0

1Null jumps, without bigrid, s=5

−1 0 1−1

0

1Physical mode, bigrid of ratio 1/2, s=5

Figure: Average and jumps parts of the SIPG solution for ξ0 = 255π/256h, s = 5: blue=continuous/discreteat t = 0, black=continuous at t = 1, green=averages at t = 1, red=jumps at t = 1.


Conclusions and open problems:

Conclusions:DG provides a rich class of schemes allowing to modify the physical component of thesystem by means of s in order to fit better the behavior of the continuous wave equation.Despite of this, these schemes generate high frequency wave packets propagating at anarbitrarily low velocity or even traveling in the wrong sense.By taking the jumps of the initial data to be zero and filtering their averages by a bi-gridalgorithm, we recover the uniform observability property of the numerical scheme.This kind of behavior is not specific to non-conforming approximation methods. It can bealso found for the quadratic classical finite element semi-discretization of the wave equation.

Open problems:propagation properties of other more sophisticated DG schemes: discontinuousPetrov-Galerkin, hybridizable DG.

Cockburn, Gopalakrishnan, Lazarov, Unified hybridization of DG, mixed, and continuousGalerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 2009

Calo, Demkowicz, Gopalakrishnan et al., A class of DPG metrods, Part IV: Wavepropagation, IECS 10-17, 2010.

fully discrete DG approximations of the wave equation.DG approximations of the wave equations in higher dimensions on uniform gridsDG semi-discretizations of the 1− d wave equation on a bounded interval and uniform grids.DG approximations of the 1− d wave equation on non-uniform grids.Dispersive properties of the DG semi-discretization of the Schrodinger equation.

Thank you very much for your attention!


Conclusions and open problems:

Conclusions:DG provides a rich class of schemes allowing to modify the physical component of thesystem by means of s in order to fit better the behavior of the continuous wave equation.Despite of this, these schemes generate high frequency wave packets propagating at anarbitrarily low velocity or even traveling in the wrong sense.By taking the jumps of the initial data to be zero and filtering their averages by a bi-gridalgorithm, we recover the uniform observability property of the numerical scheme.This kind of behavior is not specific to non-conforming approximation methods. It can bealso found for the quadratic classical finite element semi-discretization of the wave equation.

Open problems:propagation properties of other more sophisticated DG schemes: discontinuousPetrov-Galerkin, hybridizable DG.

Cockburn, Gopalakrishnan, Lazarov, Unified hybridization of DG, mixed, and continuousGalerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 2009

Calo, Demkowicz, Gopalakrishnan et al., A class of DPG metrods, Part IV: Wavepropagation, IECS 10-17, 2010.

fully discrete DG approximations of the wave equation.DG approximations of the wave equations in higher dimensions on uniform gridsDG semi-discretizations of the 1− d wave equation on a bounded interval and uniform grids.DG approximations of the 1− d wave equation on non-uniform grids.Dispersive properties of the DG semi-discretization of the Schrodinger equation.

Thank you very much for your attention!Aurora Marica (BCAM, UAM) DG & classical FEM PhD defense, Madrid - Sept.-30-2010 39 / 39

propagation properties for some discontinuous galerkin and ... · propagation properties for some...

Documents