[PPT] - Systems of convection-diffusion equations Christian Reibiger, Hans-G PowerPoint Presentation

SLIDE 1

Systems of convection-diffusion equations

Christian Reibiger, Hans-G¨

rg Roos

Technical University of Dresden

hans-goerg.roos@tu-dresden.de http://math.tu-dresden.de/˜roos

November 17, 2011– Dresden

SLIDE 2

First •Prev •Next •Last •Full Screen •Close •Quit

DD-MS 60 1 H.-G. Roos, TU Dresden

Outline

motivation: optimal control
weakly coupled systems
strongly coupled systems
systems with two small parameters

SLIDE 3

First •Prev •Next •Last •Full Screen •Close •Quit

DD-MS 60 2 H.-G. Roos, TU Dresden

1.Motivation

Consider: miny,q J(y, q) := miny,q 1

2y − y02 0 + λ 2q2

subject to

Ly := −ǫy′′ + ay′ + by = f + q in (0, 1), y(0) = y(1) = 0. where a, b, y0, f are smooth; 0 < ε << 1 and |a(x)| ≥ α > 0, λ > 0. It is well-known (cf. Troeltzsch 09) that then there is an adjoint state p such that λq + p = 0, (1a) L∗p = −ǫp′′ − ap′ + (b − a′)p = y − y0, p(0) = p(1) = 0. (1b)

SLIDE 4

First •Prev •Next •Last •Full Screen •Close •Quit

DD-MS 60 3 H.-G. Roos, TU Dresden Motivation

Error estimates (Becker, Vexler 07, Hinze, Yan, Zhou 09, Lube,Tews 10): q − qh0 + y − yh0 ≤ C(h3/2 + ε1/2h)(|y|2 + |p|2) based on y − yh2

0 + λq − qh2 0 ≤ 1

λp − ˜ ph2

0 + y − ˜

yh2 (in the second step also estimates in stronger norms) Heinkenschloss, Leykekhman 10: SUPG only first order even away from the boundary layer In sharp contrast to the case of a single equation?? What about the layer structure?

SLIDE 5

First •Prev •Next •Last •Full Screen •Close •Quit

DD-MS 60 4 H.-G. Roos, TU Dresden

2.Weakly coupled systems

−ǫu′′

1

+ a1u′

1

+ b11u1 + b12u2 −ǫu′′

2

− a2u′

2

− b21u1 + b22u2

=
f1

f2

, u1(0) = u1(1)= 0,

u2(0) = u2(1)= 0. Assuming a1, a2 ≥ α > 0, b11, b22 ≥ 0, b12b21 > 0, |b12|, |b21| ≥ β > 0

Linss 07: upwind FDM, information about first order derivatives Linss/Stynes 09: survey sp systems: CMAM, 9(2009), 165-191

Theorem: The solution of the given system can be decomposed in u1 = S1 + E10 + E11, u2 = S2 + E20 + E21 with (for k ≤ 2)

S(k)

1

0 ,
S(k)

2

0 ≤ C,
E(k)

10 (x)

≤ Cǫ1−ke−α x

ǫ,

E(k)

11 (x)

≤ Cǫ−ke−α 1−x

ǫ .

SLIDE 6

First •Prev •Next •Last •Full Screen •Close •Quit

DD-MS 60 5 H.-G. Roos, TU Dresden Weakly coupled systems σ0 = 2ǫ ln N/β σ1 = 1 − 2ǫ ln N/β u1 u2

Used Shishkin mesh Theorem: The solution of the given system is bounded by u − uNǫ ≤ CN −1 ln N Proof is based on standard results for the interpolation error. Especially the estimates |E10 − EI

10|1 ≤ ˜

Ch|E10|2 ≤ Cǫ−1/2N −1, E10 − EI

100 ≤ C min{ǫ1/2N −1, ǫ−1/2N −2} ≤ CN −3/2

for the weak layer term

SLIDE 7

First •Prev •Next •Last •Full Screen •Close •Quit

DD-MS 60 6 H.-G. Roos, TU Dresden Weakly coupled systems

10

1

10

2

10

3

10

4

10

5

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10 N error ε = 1e−4 ||u − uN||0 ||u − uN||ε ||uN − uI||ε ln(N)/N (ln(N)/N)2 10

1

10

2

10

3

10

4

10

5

10

−8

10

−6

10

−4

10

−2

10 N error ε = 1e−8 ||u − uN||0 ||u − uN||ε ||uN − uI||ε ln(N)/N (ln(N)/N)2

Error of the linear FEM on a one-sided Shishkin mesh

SLIDE 8

First •Prev •Next •Last •Full Screen •Close •Quit

DD-MS 60 7 H.-G. Roos, TU Dresden Weakly coupled systems σ0 = 2ǫ ln N/β σ1 = 1 − 2ǫ ln N/β u1 u2

Used two-sided Shishkin mesh

10

1

10

2

10

3

10

4

10

5

10

−10

10

−8

10

−6

10

−4

10

−2

10 N error ε = 1e−4 ||u − uN||0 ||u − uN||ε ||uN − uI||ε ln(N)/N (ln(N)/N)2 10

1

10

2

10

3

10

4

10

5

10

−10

10

−8

10

−6

10

−4

10

−2

10 N error ε = 1e−8 ||u − uN||0 ||u − uN||ε ||uN − uI||ε ln(N)/N (ln(N)/N)2

Error of the linear FEM on a two-sided Shishkin mesh

SLIDE 9

First •Prev •Next •Last •Full Screen •Close •Quit

DD-MS 60 8 H.-G. Roos, TU Dresden

3.Strongly coupled systems

−εu′′

1 − b11u′ 1 − b12u′ 2 + a11u1 + a12u2 = f1,

(2a) −εu′′

2 − b21u′ 1 − b22u′ 2 + a21u1 + a22u2 = f2,

(2b) subject to the boundary conditions u1(0) = u2(0) = u1(1) = u2(1) = 0. Introducing the corresponding matrices B and A and vectors u and f the system takes the form Lu := −εu′′ − Bu′ + Au = f. O’Rordan, Stynes 09: both eigenvalues of B positive (two overlapping layers at x = 0 Linss 09: no information on pointwise derivatives, layer structure ?

SLIDE 10

First •Prev •Next •Last •Full Screen •Close •Quit

DD-MS 60 9 H.-G. Roos, TU Dresden Strongly coupled systems

assumptions: (V1) B is symmetric. (V2) A + 1/2B′ is positive semidefinite. (V3) The eigenvalues of B satisfy λ1(x) > 0, λ2(x) < 0 for all x. We have ε3/2|u|2 + ε1/2|u|1 ≤ f0 . asymptotic approximation: u0

as := w0 + wh˜

c0 + d1

α1

β1

exp(−˜

λ1x/ε) + d2

α2

β2

exp(˜

λ2(1 − x)/ε). The unknown vector ˜ c0 and d1, d2 are determined by the boundary condi- tions. Consequently: The reduced solution does not satisfies the boundary con- ditions, both solution components do have layers at x = 0 and at x = 1.

SLIDE 11

First •Prev •Next •Last •Full Screen •Close •Quit

DD-MS 60 10 H.-G. Roos, TU Dresden Strongly coupled systems

Example: −εu′′ −

−1 1

0 3

u′ =
1

2

with

u0

as :=

x/3 + 1/6

−2x/3 + 2/3

+ d1
1

4

exp(−3x/ε) + d2
1
exp(−(1 − x)/ε).

Using the next term of the asymptotic approximation and the a priori esti- mate above to estimate the remainder we can prove: u allows a decomposition u = S + E1 + E2 with |S|2 ≤ C, where the layer terms satisfy |Ek

1| ≤ Cε−k exp(−˜

λ1x/ε), |Ek

2| ≤ Cε−k exp(−˜

λ2(1 − x)/ε).

SLIDE 12

First •Prev •Next •Last •Full Screen •Close •Quit

DD-MS 60 11 H.-G. Roos, TU Dresden

4.Strongly coupled systems: two small pa- rameters

−ε1u′′

1 + b11u′ 1 − b12u′ 2 + a11u1 + a12u2 = f1,

(3a) −ε2u′′

2 − b21u′ 1 − b22u′ 2 + a21u1 + a22u2 = f2,

(3b) subject to the boundary conditions u1(0) = u2(0) = u1(1) = u2(1) = 0. Introducing the corresponding matrices B and A and vectors u and f the system takes the form Lu := −diag(ε1, ε2)u′′ − Bu′ + Au = f. assumptions:

0 < ε1 < ε2 << 1
B constant, b11 > 0, b22 > 0

SLIDE 13

First •Prev •Next •Last •Full Screen •Close •Quit

DD-MS 60 12 H.-G. Roos, TU Dresden Strongly coupled systems: two small parameters

Example (Linß SINUM 09): −ε1u′′

1 + u′ 1 − u′ 2 = 1

−ε2u′′

2 − 3u′ 2 = 2,

u1(0) = u1(1) = 0, u2(0) = u2(1) = 0 asymptotic approximation: ˜ uas(x) = x

3 + 2 3 1 1+3ε1

ε2

−2

3x + 2 3

−2

3 1 1 + 3 ε1 ε2

1

1 + 3ε1

ε2

e− x

3ε2−(1

3+2 3 1 1 + 3ε1

ε2

)

1
e−1−x

ε1 .

Thus, the ”reduced” solution u0(x) = x

3 + 2 3 1 1+3ε1

ε2

−2

3x + 2 3

varies with ε1/ε2.

Questions:

′′reduced′′ solution?

layer structure?

SLIDE 14

First •Prev •Next •Last •Full Screen •Close •Quit

DD-MS 60 13 H.-G. Roos, TU Dresden Strongly coupled systems: two small parameters

asymptotic approximation: uas(x) = up(x) + uh(x)c + d1V1e−λ1x + d2V2eλ2(1−x)

up(x) + uh(x)c general solution of the reduced system
λ1, λ2 eigenvalues of

˜ B =

−b11/ε1 b12/ε1

b21/ε2 b22/ε2

.
V1, V2 scaled eigenvectors:

V1 :=

b12

b11 + ε1λ1

,

V2 := ε1λ2 − b2

ε1 ε2

b21

ε1 ε2.

Assuming

ε1 << ε2 < 1 we have ε1λ2 = −b11 + 0(ε1 ε2 ), ε2 λ1 = D b11 + 0(ε1

ε2)

SLIDE 15

First •Prev •Next •Last •Full Screen •Close •Quit

DD-MS 60 14 H.-G. Roos, TU Dresden Strongly coupled systems: two small parameters

main result: solution decomposition u1 = S1 + ˜ d1e−λ1x + ˜ d2eλ2(1−x) with S1∞ ≤ C, S′

1∞ ≤ C.

steps of the proof:

study the bvp for the remainder u − uas
apply the strong stability result from Linß 09:

|||u|||∞ ≤ C||f||L1 ≤ C||f||−1,∞ with |||v|||i = max (εiv′

i∞, vi∞)

insert the solution decomposition obtained for u2 into the first equation

and use Kelloggs’s technique

SLIDE 16

First •Prev •Next •Last •Full Screen •Close •Quit

DD-MS 60 15 H.-G. Roos, TU Dresden Strongly coupled systems: two small parameters

References

[1] Roos, H.-G., Reibiger, C.: Numerical analysis of a system of singularly perturbed convection- diffusion equations related to optimal control. accepted in NMTMA [2] Roos, H.-G., Reibiger, C.: Numerical analysis of a strongly coupled system of two convection- diffusion equations with full layer interaction. ZAMM 2011/DOI 10.10002/zamm.201000153 [3] Roos, H.-G.: Special features of strongly coupled systems of convection-diffusion equations with two small parameters. submitted [4] Linß , T.,: Analysis of an upwind finite difference scheme for a system of coupled singularly perturbed convection-diffusion equations. Computing, 79(2007), 23-32 [5] Linß , T.,: Analysis of a system of singularly perturbed convection-diffusion equations with strong

coupling. SINUM, 47(2009), 1847-1862

[6] Linß , T., Stynes, M.: Numerical solution of systems of singularly perturbed differential equations.

Comput. Meth. Appl. Math., 9(2009), 165-191

[7] Linß , T.: Layer adapted meshes for reaction-convection-diffusion problems. Springer 2010 [8] O’Riordan, E., Stynes, M.: Numerical analysis of a system of strongly coupled system of two singularly perturbed convection-diffusion equations. Adv. Comput. Math., 30(2009), 101-121

SLIDE 17

First •Prev •Next •Last •Full Screen •Close •Quit

DD-MS 60 16 H.-G. Roos, TU Dresden