A Finite Element approximation of the one-dimensional fractional Poisson equation

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A Finite Element approximation of the one-dimensional fractional Poisson equation

with applications to numerical control

Umberto Biccari

DeustoTech, Universidad de Deusto, Bilbao, Spain

Joint work withVíctor Hernández-Santamaría

DeustoTech, Universidad de Deusto, Bilbao, Spain

Benasque, August 29^th, 2017

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1 Introduction

2 Preliminary theoretical results

3 Development of the numerical scheme 4 Numerical results

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1 Introduction

2 Preliminary theoretical results Elliptic problem

Parabolic problem

3 Development of the numerical scheme Elliptic problem

Parabolic problem

4 Numerical results Control experiments

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Numerical results

Introduction

We present a Finite Element (FE) scheme for the numerical approximation of the solution to the following non-local equation:

Fractional Poisson equation

(−dx²)^su=f, x∈(−L,L), u≡0, x∈R\(−L,L).

As a natural application, we analyze the numerical control problem for the following parabolic equation:

Fractional heat equation







zt+ (−dx²)^sz=g1ω, (x,t)∈(−1,1)×(0,T) z=0, (x,t)∈[R\(−1,1) ]×(0,T) z(x,0) =z0(x), x∈(−1,1)

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Introduction Preliminary theoretical results Development of the numerical scheme Numerical results

Introduction

We present a Finite Element (FE) scheme for the numerical approximation of the solution to the following non-local equation:

Fractional Poisson equation

(−dx²)^su=f, x∈(−L,L), u≡0, x∈R\(−L,L).

As a natural application, we analyze the numerical control problem for the following parabolic equation:







zt+ (−dx²)^sz=g1ω, (x,t)∈(−1,1)×(0,T) z=0, (x,t)∈[R\(−1,1) ]×(0,T) z(x,0) =z0(x), x∈(−1,1)

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Fractional Laplacian

For any functionusufficiently regular and for anys∈(0,1), the s-th power of the Laplace operator is given by

(−d_x²)^su(x) =CsP.V. Z

R

u(x)−u(y)

|x−y|^1+2s dy.

Functional setting: fractional Sobolev spaces

• H^s(−L,L) :=

u ∈L²(−L,L) : ^|u(x^)−u(y)|

|x−y|¹2+s ∈L²((−L,L)²)

.

• kuk_Hs(−L,L):=

Z L

−L

|u|²dx+ Z L

−L

Z L

−L

|u(x)−u(y)|²

|x−y|^1+2s dxdy

!¹₂ .

• H₀^s(−L,L) :=n

u∈H^s(R) : u =0inR\(−L,L)o .

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Existing literature

• R.H. Nochetto, E. Otárola and A.J. Salgado, A. Bonito et al. : FE schemes for the discretization of elliptic and parabolic problems involving thespectralFractional Laplacian (DIFFERENT OPERATOR).

• G. Acosta et al. :

FE schemes for the discretization 2-D elliptic problems involving the Fractional Laplacian.

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Parabolic problem

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1 Introduction

Parabolic problem

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Variational formulation for the elliptic problem

Findu∈H₀^s(−L,L)such that c_1,s

2 Z

R

Z

R

(u(x)−u(y))(v(x)−v(y))

|x−y|^1+2s dxdy

| {z }

a(u,v)

= Z L

−L

fv dx,

for allv ∈H₀^s(−L,L).

Well posedness

a(·,·) :H₀^s(−L,L)×H₀^s(−L,L)→R

continuous and coercive ⇒ Iff ∈H^−s(−L,L), then there exists a unique weak solu- tionu∈H₀^s(−L,L).

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1 Introduction

Parabolic problem

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Control results







zt+ (−d_x²)^sz =g1ω, (x,t)∈(−1,1)×(0,T) z =0, (x,t)∈[R\(−1,1) ]×(0,T) z(x,0) =z0(x), x ∈(−1,1)

Proposition

For all z0∈L²(−1,1)and T >0, the equation is null-controllable with a control function g∈L²(ω×(0,T))if and only if s>1/2.

Proposition

Let s∈(0,1)and T >0. For all z0∈L²(−1,1), there exists a control function g∈L²(ω×(0,T))such that the unique solution z to the fractional Heat equation is approximately controllable.

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Elliptic problem Parabolic problem

Proof of the null controllability (sketch)

The result is equivalent to the existence of a constantC>0 such that the following observability inequality holds

kϕ(x,0)k²_L2(−1,1)≤C Z T

0

Z

ω

ϕ(x,t)g(x,t)dx

2

dt,

whereϕ(x,t)is the unique solution to the adjoint system







−ϕ_t+ (−d_x²)^sϕ=0, (x,t)∈(−1,1)×(0,T)

ϕ=0, (x,t)∈[R\(−1,1) ]×(0,T)

ϕ(x,T) =ϕ^T(x), x ∈(−1,1).

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Spectral expansion ϕ(x,t) =X

k≥1

ϕ_ke^−λ^k^(T^−t)%_k(x), ϕ_k =hϕ^T, %_ki.

The observability inequality becomes

X

k≥1

|ϕ_k|²e^−2λ^k^T ≤C Z T

0

X

k≥1

ϕ_kg_k(t)e^−λ^k^t

2

dt, g_k =hg1ω, %_ki.

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Müntz Theorem: the last inequality is true if and only if

X

k≥1

1

λ_k <+∞.

Eigenvalues of(−d_x²)^s on(−1,1)with DBC¹

λ_k = kπ

2 −(1−s)π 4

2s

+O 1

k

.

The series is convergent if and only ifs>1/2. Therefore, the observability inequality holds whens>1/2, but it is false when s≤1/2.

1M. Kwa´snichi, J. Funct. Anal., 2012

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1 2 3 4 5 6 7 8 9 10

index k 0

2 4 6 8 10 12 14 16

eigenvalue

b=0.5

b=0.4

b=0.3

b=0.2 b=0.1

1 2 3 4 5 6 7 8 9 10

index k 0

50 100 150 200 250

eigenvalue

b=0.6 b=0.7 b=0.8 b=0.9 b=1

First ten eigenvalues of(−dx²)^s on(−1,1)with DBC fors≤1/2 (left) and s>1/2 (right).

• s≤1/2: the equation is not null-controllable.

• s>1/2: the equation is null-controllable.

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Proof of the approximate controllability (sketch)

The result follows from the following property.

Parabolic unique continuation

Givens∈(0,1)andϕ^T₀ ∈L²(−1,1), letϕbe the unique solution to the adjoint equation. Letω⊂(−1,1)be an arbitrary open set. Ifϕ=0 on ω×(0,T), thenϕ=0 on(−1,1)×(0,T).

This, in turn, is a consequence of the Unique Continuation property for the Fractional Laplacian, obtained by Fall and Felli².

2M.M. Fall and V. Felli, Comm. Partial Differential Equations, 2014..

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Parabolic problem

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1 Introduction

Parabolic problem

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Finite element approximation of the elliptic problem

Partition of(−L,L)

−L=x0<x1< . . . <xi <xi+1< . . . <xN+1=L, x_i+1=x_i+h, i =0, . . .N.

• M:={x_i : i=1, . . . ,N}.

• ∂M:={x₀,x_N+1}.

• K_i := [x_i,x_i+1].

[ • • • • • ]

−L=x0 L=x_N+1

x₁ x₂ x_i xi+1 x_N

↑ ↑

∂M ∂M

M

Ki

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Consider the discrete space Vh:=n

v ∈H₀^s(−L,L) v|_K

i ∈ P¹o

,

whereP¹is the space of the continuous and piece-wise linear functions.

Discrete variational formulation Findu_h∈V_hsuch that

c1,s

2 Z

R

Z

R

(uh(x)−uh(y))(vh(x)−vh(y))

|x−y|^1+2s dxdy= Z L

−L

fvhdx, for allv_h∈V_h.

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Given φ_i ^N

i=1any basis ofV_h, it is sufficient that the discrete variational formulation is satisfied for

uh(x) =

N

X

j=1

ujφ_j(x), vh(x) =φ_j(x)

In this way, we are reduced to solve the linear systemA_hu=F

• A_h∈R^N×N: stiffness matrix with components a_i,j =c1,s

2 Z

R

Z

R

(φ_i(x)−φ_i(y))(φ_j(x)−φ_j(y))

|x−y|^1+2s dxdy,

• F ∈R^N given byF = (F₁, . . . ,F_N)with Fi =hf, φ_ii=

Z L

−L

fφ_idx, i =1, . . . ,N.

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Basis functions

We employ the classical basis φ_i ^N

i=1in which eachφ_i is the tent function withsupp(φi) = (xi−1,xi+1)and verifyingφ_i(xj) =δ_i,j.

φ_i(x) =1−|x−xi| h .

(xi−1,0) (x_i,1)

(xi+1,0) (xi,0)

y

x φ_i(x)

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Construction of the stiffness matrix

Remark

A_his symmetric. Therefore, in our algorithm we will only need to compute the values a_i_,j with j ≥i.

Due to the non-local nature of the problem, the matrixA_his full.

While computing the values ai,j, we will only work on the meshM, not considering the points of the set∂M. In this way, we will ensure that the basis functionsφ_i satisfy the zero Dirichlet boundary conditions.

x₀ φ₁

x₂ x₁

φ₂

x₃ φ₃

x₄ xN−1

φ_N

xN+1

x_N y

x 1

. . . .

• •

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The building of the stiffness matrixA_his done it in three steps

1 We fill the upper triangle, corresponding toj ≥i+2.

2 We fill the upper diagonal corresponding toj=i+1.

3 We fill the diagonal.

a1,1 a1,2 a1,3 . . . a1,N

a2,2 a2,3 a2,4 . . . a2,N

. .. . .. . .. ...

. .. . .. aN−2,N

aN−1,N−1 aN−1,N

a_N,N













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Values of a

_i_,j

for j ≥ i + 2

O xi−1

xi−1

xi

x_i+1 xi+1

xj−1

xj

x_j

xj+1

y

x

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Values of a

_i_,i₊₁

O

xi−1

x_i

x_i xi+1

xi+1

x_i+2

xi+2

y

x

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Values of a

_i_,i

O

xi−1

x_i

x_i xi+1

xi+1

y

x

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Entries of the stiffness matrix A

_h

s6=1/2

ai,j=−h^1−2s











4(k+1)^3−2s+4(k−1)^3−2s 2s(1−2s)(1−s)(3−2s)

−6k^3−2s+ (k+2)^3−2s+ (k−2)^3−2s

2s(1−2s)(1−s)(3−2s) , k=j−i,k≥2

3^3−2s−2^5−2s+7

2s(1−2s)(1−s)(3−2s), j=i+1

2^3−2s−4

s(1−2s)(1−s)(3−2s), j=i.

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Entries of the stiffness matrix A

_h

s=1/2

ai,j =











−4(k+1)²log(k +1)−4(k−1)²log(k−1) +6k²log(k) + (k+2)²log(k+2)

+(k−2)²log(k−2), k =j−i,k >2

56ln(2)−36ln(3), j =i+2.

9ln3−16ln2, j =i+1

8ln2, j =i.

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1 Introduction

Parabolic problem

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Penalized Hilbert Uniqueness Method

We have to solve the following minimization problem: find ϕ^T_ε = min

ϕ∈L²(−1,1)Jε(ϕ^T)

where

Jε(ϕ^T) := 1 2

Z T

0

Z

ω

|ϕ|²dxdt+ ε 2

ϕ^T

2

L²(−1,1)+ Z

Ω

z0ϕ(0)dx

and whereϕis the solution to the adjoint problem







−ϕ_t+ (−d_x²)^sϕ=0, (x,t)∈(−1,1)×(0,T)

ϕ=0, (x,t)∈

R\(−1,1)

×(0,T) ϕ(x,T) =ϕ^T(x), x ∈(−1,1).

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The approximate and null controllability properties of the system, for a given initial datumz0, can be expressed in terms of the behavior of the penalized HUM approach. In particular, we have:

Theorem (F. Boyer, ESAIM: PROCEEDINGS, 2013)

The equation is approximately controllable at time T from the initial datum z₀if and only if

ϕ^T_ε →0, as ε→0.

The equation is null-controllable at time T from the initial datum z0

if and only if

M_z²₀ :=2sup

ε>0

inf

L²(0,T;L²(ω))Jε

<+∞.

In this case, we have

kgk_L2(0,T;L²(ω))≤M_z₀, ϕ^T_ε

_L₂_(−L,L) ≤M_z₀√ ε.

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Parabolic problem

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Control experiments

In order to test numerically the accuracy of our method, we use the following problem

(−d_x²)^su=1, x ∈(−L,L) u ≡0, x ∈R\(−L,L).

In this particular case, the solution can be computed exactly and it reads as follows,

Solution

u(x) = 2^−2s√ π Γ ^1+2s₂

Γ(1+s)

L²−x²s

.

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−1 −0.5 0 0.5 1

0 0.2 0.4 0.6 0.8 1

Numerical solution Real solution

s=0.1

−1 −0.5 0 0.5 1

0 0.2 0.4 0.6 0.8 1

s=0.4

−1 −0.5 0 0.5 1

0 0.2 0.4 0.6 0.8 1

s=0.5

−1 −0.5 0 0.5 1

0 0.2 0.4 0.6

s=0.8

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Error analysis

The computation of the error in the spaceH₀^s(−L,L)can be readily done by using the definition of the bilinear form, namely

ku−uhk²_Hs

0(−L,L) =a(u−uh,u−uh) = Z L

−L

f(x) (u(x)−uh(x))dx.

f ≡1 ⇒ ku−uhk_Hs

0(−L,L)= Z L

−L

(u(x)−uh(x))dx

!1/2

. The right-hand side can be easily computed, since we have the closed formula

Z L

−L

u dx = πL^2s+1 2^2sΓ(s+¹₂)Γ(s+³₂) and the term corresponding toRL

−Lu_hcan be carried out numerically.

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Theorem (G. Acosta and J.P. Borthagaray, SIAM J. Numer. Anal., 2017)

For the solution u of the elliptic problem and its FE approximation u_h, if h is sufficiently small, the following estimates hold

ku−u_hk_Hs

0(−L,L) ≤Ch^1/2|lnh| kfk

C¹²^−s(−L,L), ifs<1/2, ku−uhk_Hs

0(−L,L) ≤Ch^1/2|lnh| kfk_L∞(−L,L), ifs=1/2, ku−uhk_Hs

0(−L,L) ≤_2s−1^C h^1/2p

|lnh| kfk_Cβ(−L,L), ifs>1/2, where C is a positive constant not depending on h.

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10⁻³ 10⁻²

10⁻²

10⁻¹ ^{slope 0.5}

h

s=0.1 s=0.3 s=0.5 s=0.7 s=0.9

The convergence rate is maintained also for small values ofs. This confirms that the behavior obtained fors=0.1 is not in contrast with the known theoretical results. Indeed, since it is well-known that the notion of trace is not defined for the spacesH^s(−L,L)withs≤1/2, it is somehow natural that we cannot expect a point-wise convergence in this case.

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As a further validation of this fact, we plot the behavior of theL^∞-norm of the difference between the real and the numerical solution to the fractional Poisson equation.

10⁻³ 10⁻²

10^−0.2 10^−0.1

10⁰ _{slope 0.1}

Increasing the number of point of discretization, theL^∞-norm is decreasing with a rate (inh) of 0.1.

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Controllability of the fractional heat equation

We use the finite-element approximation of(−d_x²)^s for the space discretization and the implicit Euler scheme in the time variable.







M_hzⁿ⁺¹−zⁿ

δt +A_hzⁿ⁺¹=1ωv_hⁿ⁺¹, ∀n∈ {1, . . . ,M−1}

z⁰=z0

We choose the penalization termεas a function ofh.

PRACTICAL RULE: chooseε∼h^2pwherepis the order of accuracy in space of the numerical method used for the discretization of the spatial operator involved. (in this case, we takep=1/2).

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Control experiments

Uncontrolled solution -s=0.8,T =0.2s

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Controlled solution -s=0.8,T =0.2s,ω= (−0.3,0.8)

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10⁻³ 10⁻²

10⁻³ 10⁻² 10⁻¹ 10⁰

slope 0.5

h s=0.8

Cost of the control Size ofy^M Optimal energy

Fors=0.8 we observe that:

The control cost and the optimal energy remain bounded as h→0.

|y^M|_L2(R^M) ∼ Cp

φ(h) =Ch^1/2.

This confirms that the system is null controllable.

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10⁻³ 10⁻²

10⁻¹ 10⁰ 10¹ 10²

slope 0.4 sl.−0.18/−0.3

h s=0.5

Cost of the control Size ofy^M Optimal energy

Fors=0.5 we observe that:

The control cost and the optimal energy do not remain bounded ash→0.

|y^M|_L2(R^M) ∼ Ch^0.4.

This confirms that the system is only approximately controllable.

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A Finite Element approximation of the one-dimensional fractional Poisson equation