remarks on asymptotic expansions for singularly perturbed second - - PowerPoint PPT Presentation

workshop in honor of Martin Stynes

November 16–18, 2011

remarks on asymptotic expansions for singularly perturbed second order ODEs with multiple scales J.M. Melenk

joint work with

• L. Oberbroeckling
• B. Pichler
• C. Xenophontos

TU Wien Institut f¨ ur Analysis und Scientific Computing

Intro weakly coupled strongly coupled DG conclusions

1

Introduction

2

weakly coupled elliptic-elliptic case

3

strongly coupled systems

4

DG

5

Conclusions

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

1

Introduction

2

weakly coupled elliptic-elliptic case

3

strongly coupled systems

4

DG

5

Conclusions

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

setting of the talk

−Eu′′ + Bu′ + Au = f in I = (0, 1), u(0) = u(1) = 0. E = ε1 ε2

• ,

0 < ε1 ≤ ε2 ≤ 1 questions: identification of the correct limit problem for “small” E layer structure of the solution asymptotic expansions and their regularity continuous dependence of solution on data for small E

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

“E small” =?

E = ε1 ε2

• key ingredient of asymptotic analysis:

identify the length scales assume scale separation

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

“E small” =?

E = ε1 ε2

• key ingredient of asymptotic analysis:

identify the length scales assume scale separation

expect here: solution features on scales O(ε1), O(ε2), O(1)

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

“E small” =?

E = ε1 ε2

• key ingredient of asymptotic analysis:

identify the length scales assume scale separation

expect here: solution features on scales O(ε1), O(ε2), O(1) = ⇒ scale separation governed by ε1 ε2 and ε2 1

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

“E small” =?

E = ε1 ε2

• key ingredient of asymptotic analysis:

identify the length scales assume scale separation

expect here: solution features on scales O(ε1), O(ε2), O(1) = ⇒ scale separation governed by ε1 ε2 and ε2 1 = ⇒ for the phrase “E small” to be meaningful, have to specify which quantities ε1/ε2 and/or ε2/1 are small (→ different asymptotic regimes!) example

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

−Eu′′ +

• 3

−2 −2

• u′ =

1 1

• ,

u(0) = u(1) = 0.

ε1 < < ε2 < < 1

limit boundary conditions

ε1/ε2 → 0 and ε2/1 → 0: u1(0) − 2 3u2(0) = 0, u2(1) = 0

ε1 = 10−9, ε2 = 100ε1

more details

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

−Eu′′ +

• 3

−2 −2

• u′ =

1 1

• ,

u(0) = u(1) = 0.

ε1 < < ε2 < < 1 ε1 = ε2 < < 1

limit boundary conditions

ε1/ε2 → 0 and ε2/1 → 0: u1(0) − 2 3u2(0) = 0, u2(1) = 0 ε1 = ε2 → 0: 2u1(0) − u2(0) = 0, u1(1) + 2u2(1) = 0

ε1 = 10−9, ε2 = 100ε1

more details

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

1

Introduction

2

weakly coupled elliptic-elliptic case

3

strongly coupled systems

4

DG

5

Conclusions

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

weakly coupled system

− Eu′′ + A(x)u = f in I = (0, 1), u(0) = u(1) = 0

Assumptions

A, f analytic on I, A uniformly positive definite on I. E = ε1 ε2

• ,

0 < ε1 ≤ ε2 ≤ 1. features: well-posedness and continuous dependence of solution on data (e.g., in energy norm) asymptotic expansions are designed to yield small residual → truncated expansion yields small error (in appropriate norms) solution u has features on 3 scales: O(ε1), O(ε2), O(1)

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

expansions

correct asymptotic expansion ansatz hinges on the presence of scale separation, i.e., whether ε2 1 and/or ε1 ε2 is small → 4 different cases! focus here:

ε1 ε2 and ε2 1 small

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

expansions

make the ansatz u(x) ∼

• i,j

ε2 1 iε1 ε2 j + + +

• singularly perturbed coupled systems

J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

expansions

make the ansatz u(x) ∼

• i,j

ε2 1 iε1 ε2 j uij(x) + + +

• uij: functions on the O(1) scale

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

expansions

make the ansatz u(x) ∼

• i,j

ε2 1 iε1 ε2 j uij(x) + uij( x) + +

• uij: functions on the O(1) scale
• x = x/ε2
• uij:

functions on (0, ∞) for the O(ε2) scale exponentially decaying

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

expansions

make the ansatz u(x) ∼

• i,j

ε2 1 iε1 ε2 j uij(x) + uij( x) + uij( x) +

• uij: functions on the O(1) scale
• x = x/ε2
• uij:

functions on (0, ∞) for the O(ε2) scale exponentially decaying

• x = x/ε1
• uij:

functions on (0, ∞) for the O(ε1) scale exponentially decaying

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

expansions

make the ansatz u(x) ∼

• i,j

ε2 1 iε1 ε2 j uij(x) + uij( x) + uij( x) +

uR

ij(

xR) + uR

ij(

xR)

• uij: functions on the O(1) scale
• x = x/ε2
• uij:

functions on (0, ∞) for the O(ε2) scale exponentially decaying

• x = x/ε1
• uij:

functions on (0, ∞) for the O(ε1) scale exponentially decaying

• uR

ij,

uR

ij: analogous boundary layer fct at x = 1

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

computation of asymptotic expansion

insert ansatz into the differential equation separate scales, i.e., view variables x, x, x as independent variables equate like powers of ε1/ε2 and ε2/1 this yields a recurrence relation of DAEs for the functions uij,

• uij,

uij insertion of the b.c. at x = 0 and x = 1 closes the DAEs

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

uij = uij vij

• ,
• uij =

uij

• vij
• ,
• uij =

uij

• vij
• ,

convention: functions with negative subscript vanish

recursions for A = constant

− u′′

i−2,j−2

v′′

i−2,j

• + Auij

= Fij = δ(i,j),(0,0)f (1a) − ( ui,j−2)′′ ( vi,j)′′

• + A

ui,j = 0, (1b) −

• (

ui,j)′′ ( vi,j+2)′′

• + A

ui,j = 0, (1c) boundary conditions: uij(0) + uij(0) + uij(0) = 0, (1d) decay conditions for uij, uij at +∞ (1e)

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

properties of the functions uij, uij, uij

in general, the equations are systems of DAEs all arising DAEs are successively solvable the regularity of the functions uij, uij, uij can be controlled explicitly in terms of i, j. Define approximation to u by truncated expansion: uM1,M2 :=

• i≤M1

j≤M2

ε2 1 iε1 ε2 j uij(x) + uij( x) + uij( x) +

uR

ij(

xR) + uR

ij(

xR)

• =:

w(x) smooth part + u( x) + u( x)

• layer parts

+

• uR(

xR) + u( xR)

• layer parts

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

uM1,M2 :=

• i≤M1

j≤M2

ε2 1 iε1 ε2 j uij(x) + uij( x) + uij( x) +

uR

ij(

xR) + uR

ij(

xR)

• =:

w(x) + u( x) + u( x) +

• uR(

xR) + u( xR)

Theorem (optimal truncation)

Select, for implied constants depending on f and A: M1 ∼ 1 ε2 , M2 ∼ ε2 ε1 Then: u − uM1,M2E,I ≤ C

• e−b/ε2 + e−bε2/ε1

w is analytic (with control uniformly in ε1, ε2)

• u is typical boundary layer function on ε2-scale
• u is typical boundary layer function on ε1-scale

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

remarks

analogous results for the cases that either ε1/ε2 or ε2/1 is not small → talk by C. Xenophontos regularity permits design and analysis of hp-FEM → talk by

• C. Xenophontos

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

1

Introduction

2

weakly coupled elliptic-elliptic case

3

strongly coupled systems

4

DG

5

Conclusions

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

strongly coupled model problem

−Eu′′ + Bu′ + Au = f in I = (0, 1), u(0) = u(1) = 0. E = ε1 ε2

• ,

0 < ε1 ≤ ε2 ≤ 1 B, A ∈ R2×2

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

strongly coupled model problem

−Eu′′ + Bu′ + Au = f in I = (0, 1), u(0) = u(1) = 0. E = ε1 ε2

• ,

0 < ε1 ≤ ε2 ≤ 1 B, A ∈ R2×2

assumptions

non-degeneracy of B: B invertible, B = LDU with L = 1 l21 1

• ,

D = d1 d2

• ,

U = 1 u12 1

• ,

scale separation: ε1 ε2

• suff. small,

ε2 1 suff. small

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

expansions

make the ansatz u(x) ∼

• i,j

ε2 1 iε1 ε2 j uij(x) + uij( x) + uij( x) +

• uR

ij(

xR) + uR

ij(

xR)

• uij: functions on the O(1) scale
• x = x/ε2
• uij:

functions on (0, ∞) for the O(ε2) scale exponentially decaying

• x = x/ε1
• uij:

functions on (0, ∞) for the O(ε1) scale exponentially decaying

• uR

ij,

uR

ij: analogous boundary layer fct at x = 1

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

asymptotic expansion

inserting the ansatz into the differential equation yields recursion of DAEs insertion of boundary conditions yields systems closes the system of DAEs

• bservation: the procedure reveals the correct limit problem

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

asymptotic expansion

inserting the ansatz into the differential equation yields recursion of DAEs insertion of boundary conditions yields systems closes the system of DAEs

• bservation: the procedure reveals the correct limit problem

procedure so far is formal = ⇒

1 well-posedness: it is not clear that we can solve for the BVPs

appearing in the recursion

2 justification of asymptotic expansion: truncating the

asymptotic expansion yields small residual → small error?

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

towards a canonical form and stability

recall B = LDU = 1 l21 1 d1 d2 1 u12 1

• −Eu′′ + Bu′ + Au

= f −EU−1(Uu)′′ + LD(Uu)′ + AU−1(Uu) = f − L−1EU−1

• =:

E

(Uu)′′ + L−1LD(Uu)′ + L−1AU−1

• =:

A

(Uu) = L−1f after relabelling u and f, we consider the problem − Eu′′ + Du′ + Au = f, u(0) = u(1) = 0.

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

− Eu′′ + Du′ + Au = f

facts:

• E is no longer symmetric

for ε1/ε2 and ε2/1 sufficiently small, we have ξ⊤ Eξ ∼ ξ⊤Eξ ∀ξ ∈ R2 in this new variable u, limit problem as (ε1/ε2, ε2/1) → 0 is: Du′ + Au = f i = 1, 2 : ui = 0 at x = 0 or x = 1 depending on sign of di example in the case of d1, d2 having different sign, it is not clear that the limit problem is well-posed (BVP!) upshot: we have to require well-posed of the limit problem

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

Theorem (stability)

Assume: the limit problem is well-posed. D−1 A has two distinct eigenvalues. Then: If ε1/ε2 and ε2/1 are sufficiently small, the singularly perturbed BVP is well-posed and for a C > 0 independent of E 1 (u′)⊤Eu′ dx + u2

L∞(I) ≤ Cf2 L2(I)

Corollary (asymptotic expansion meaningful)

(i) the asymptotic expansion is well-defined (ii) the truncated asymptotic expansion yields a small (in ε1/ε2 and ε2/1) residual and thus a small error

remark:

• assump. that the EVs of D

A be distinct can likely be relaxed

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

remarks on the structure of the layers

if f is sufficiently smooth, then: u has two layers (length scales ε1 and ε2). Their location depends on the sign pattern of D:

if d1 > 0, then the ε1-layer is at x = 1, otherwise it is at x = 0 if d2 > 0, then the ε2-layer is at x = 1, otherwise it is at x = 0

some components of the layers are small:

the first component of the ε2-layer is weak:

• u1 = O(ε2) + O(ε1/ε2)

the second component of the ε1-layer is weak:

• u2 = O(ε1/ε2)

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

remarks on lack of scale separation

−Eu′′ + Bu′ + Au = f consider the case ε1 ≈ ε2.

1 scale by diag(1, ε1/ε2) so that with ε = ε1 we have

−εIu′′ + Bu′ + Au = f

2 diagonalize

B: with B = S−1DS we get −ε SS−1

=I

(Su)′′ + SS−1

=I

D(Su)′ + S AS−1

= A

(Su) = S−1f

3 =

⇒ system where coupling is only through the lowest order terms.

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

remarks on lack of scale separation

−Eu′′ + Bu′ + Au = f, E =   ε1 ε2 ε2   where 0 < ε1 < < ε2 ≤ 1

1 seek block LDU-decomposition of B such that

B = 1 ∗ I d1

• B

1 ∗ I

• 2 diagonalize

B to get form − Eu′′ + Du′ + Au = f

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

1

Introduction

2

weakly coupled elliptic-elliptic case

3

strongly coupled systems

4

DG

5

Conclusions

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

DG

− Eu′′ + Du′ + Au = f, u(0) = u(1) = 0.

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

DG

− Eu′′ + Du′ + Au = f, u(0) = u(1) = 0. setting here: ε1/ε2 < < 1 and ε2/1 < < 1 limit problem is assumed to be well-posed

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

DG

− Eu′′ + Du′ + Au = f, u(0) = u(1) = 0. setting here: ε1/ε2 < < 1 and ε2/1 < < 1 limit problem is assumed to be well-posed recall: ξ⊤ Eξ ∼ ξ⊤Eξ ∀ξ ∈ R2 let S := 1

2

• E +

E⊤ be the symmetric part of E

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

DG

− Eu′′ + Du′ + Au = f, u(0) = u(1) = 0. setting here: ε1/ε2 < < 1 and ε2/1 < < 1 limit problem is assumed to be well-posed recall: ξ⊤ Eξ ∼ ξ⊤Eξ ∀ξ ∈ R2 let S := 1

2

• E +

E⊤ be the symmetric part of E T = {K | K ∈ T } is a mesh on I = (0, 1) with nodes 0 = x0 < x1 < · · · < xN = 1 average and jump are defined by [ [u] ](xi) = u(xi−)−u(xi+), { {u} }(xi) = 1 2 (u(xi+) + u(xi−))

[ [u] ](x0) = −u(x0+), [ [u] ](xN ) = u(xN −), { {u} }(x0) = u(x0+), { {u} }(xN ) = u(xN −) singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

DG

Sp,0(T ) = piecewise discontinuous polynomials of degree p on T a(u, v) = atrans(u, v) + aell(u, v),

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

DG

Sp,0(T ) = piecewise discontinuous polynomials of degree p on T a(u, v) = atrans(u, v) + aell(u, v), atrans(u, v) =

• K∈T
• K

v⊤(Du′ + Au) dx + v⊤D( u − u)|∂K

• ui =
• u(xi+)

if di < 0 u(xi−) if di > 0

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

DG

Sp,0(T ) = piecewise discontinuous polynomials of degree p on T a(u, v) = atrans(u, v) + aell(u, v), atrans(u, v) =

• K∈T
• K

v⊤(Du′ + Au) dx + v⊤D( u − u)|∂K aell(u, v) =

• K∈T
• K

(v′)⊤ Eu′ dx +

• i

[ [v⊤] ]{ { Eu′} } + θ[ [u⊤] ]{ { E⊤v′} } + σ hi [ [v⊤] ]S[ [u] ] where hi = min{hK, hK′} (here, xi = node shared by K, K′), σ > 0 sufficiently large, θ = 1 (“SIP”)

• ui =
• u(xi+)

if di < 0 u(xi−) if di > 0

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

Theorem (stability of DG)

Assume:

• A is positive definite

σ is sufficiently large ε1/ε2 and ε2/1 are sufficiently small. Then: a(u, u) ≥ Cu2

DG

∀u ∈ Sp,0(T ) u2

DG

=

• K
• K

(u′)⊤ Eu′ + u⊤ Au dx + 1 2

• i

[ [u⊤] ] |D| [ [u] ] +

• i

σ hi [ [u⊤] ]S[ [u] ]

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

Shishkin mesh for layer of scale O(ε)

τεln N 1 N nodes each

Corollary (convergence)

Use: Shishkin mesh Tε1,ε2 with N nodes in each of the ε1- and ε2-layers piecewise linears (p = 1) Then, for sufficiently large transition parameter τ (appearing in the definition of the Shishkin mesh): u − uNE,I N−1 ln3/2 N + O

• ε2M1 + (ε1/ε2)M2

where the parameters M1, M2 depend on the regularity of f.

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

1

Introduction

2

weakly coupled elliptic-elliptic case

3

strongly coupled systems

4

DG

5

Conclusions

singularly perturbed coupled systems J.M. Melenk

Intro weakly coupled strongly coupled DG conclusions

Conclusions

the correct ansatz of asymptotic expansions depends on scale separation the correct asymptotic ansatz reveals the limit problem for strongly coupled systems, stability of the limit problem may have to be stipulated for meaningful asymptotic expansions good canonical forms permit identifying the correct upwinding for stable DG-formulations

singularly perturbed coupled systems J.M. Melenk

notes

−Eu′′ + Du′ + Au = f, u(0) = u(1) = 0

u1 u2

D = 1 −1

• A

=

• 1

1

• E

= 10−2 10−1

• f

= 1 1

• limit problem

Du′ + Au = f, u1(0) = 0, u2(1) = 0 back

singularly perturbed coupled systems J.M. Melenk

notes

details for the example

B =

• 3

−2 −2

• =

LDU =

• 1

−2/3 1 3 −4/3 1 −2/3 1

• =

V −1 4

• V−1

= 1 2 2 −1 −1 4 1 5 1 2 2 −1

• limit b.c. for ε1/ε2 → 0:

(note: 3 > 0, −4/3 < 0) (Uu)1(0) = 0, (Uu)2(1) = 0 limit b.c. for ε1 = ε2 → 0: (note: −1 < 0, 4 > 0) (V−1u)1(1) = 0, (V−1u)2(0) = 0 back

singularly perturbed coupled systems J.M. Melenk