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Optimized Schwarz Methods for Problems with Discontinuous Coefficients

Olivier Dubois

dubois@math.mcgill.ca

Department of Mathematics and Statistics McGill University

http://www.math.mcgill.ca/∼dubois

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.1/28

Motivation

Flow in heterogeneous media has many applications, for example

• il recovery,

earthquake prediction, underground disposal of nuclear waste.

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.2/28

Motivation

Flow in heterogeneous media has many applications, for example

• il recovery,

earthquake prediction, underground disposal of nuclear waste. ν1 ν2 ν3 ν4

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.2/28

Motivation

Flow in heterogeneous media has many applications, for example

• il recovery,

earthquake prediction, underground disposal of nuclear waste. ν1 ν2 ν3 ν4 ⇒ This suggests a natural nonoverlapping domain decomposition.

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.2/28

Outline

• 1. Motivation
• 2. Introduction to the model problem
• 3. Schwarz iteration and optimal operators
• 4. Optimized transmission conditions

(a) one-sided Robin conditions (two versions) (b) two-sided Robin conditions (c) second order conditions

• 5. Asymptotic performance for strong heterogeneity
• 6. Numerical experiments
• 7. Conclusions and work in progress

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.3/28

Some recent work

• Y. Maday and F

. Magoulès. Multilevel optimized Schwarz methods without overlap for highly heterogeneous media. Research report at Laboratoire Jacques-Louis Lions, 2005.

• Y. Maday and F

. Magoulès. Improved ad hoc interface conditions for Schwarz solution procedure tuned to highly heterogeneous media.

Applied Mathematical Modelling, 30(8):731-743, 2006.

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.4/28

The model problem

We consider a simple diffusion problem with a discontinuous coefficient

• −∇ · (ν(x)∇u) = f on R2,

u is bounded at infinity.

(P)

Ω1 Ω2 x = 0 ν(x) = ν1 ν(x) = ν2

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.5/28

A general Schwarz iteration

The solution to problem (P) satisfies the matching conditions u(0−, y) = u(0+, y), ν1 ∂u ∂n(0−, y) = ν2 ∂u ∂n(0+, y).

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.6/28

A general Schwarz iteration

The solution to problem (P) satisfies the matching conditions u(0−, y) = u(0+, y), ν1 ∂u ∂n(0−, y) = ν2 ∂u ∂n(0+, y). Consider a general Schwarz iteration of the form

• −∇ · (ν1∇un+1

1

) = f

• n Ω1 = (−∞, 0) × R,
• ν1 ∂

∂x + S1

• un+1

1

=

• ν2 ∂

∂x + S1

• un

2

at x = 0,

• −∇ · (ν2∇un+1

2

) = f

• n Ω2 = (0, ∞) × R,
• −ν2 ∂

∂x + S2

• un+1

2

=

• −ν1 ∂

∂x + S2

• un

1

at x = 0. un

i = approximate solution in subdomain Ωi, at iteration n.

Si are linear boundary operators acting in y only

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.6/28

Fourier analysis

Fourier transform in y: Fy[u(x, y)] = ˆ u(x, k) := 1 √ 2π ∞

−∞

u(x, y)e−iykdy

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.7/28

Fourier analysis

Fourier transform in y: Fy[u(x, y)] = ˆ u(x, k) := 1 √ 2π ∞

−∞

u(x, y)e−iykdy Fourier symbols for the transmission operators Si: Fy[Siu(x, y)] = σi(k)ˆ u(x, k), for i = 1, 2.

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.7/28

Fourier analysis

Fourier transform in y: Fy[u(x, y)] = ˆ u(x, k) := 1 √ 2π ∞

−∞

u(x, y)e−iykdy Fourier symbols for the transmission operators Si: Fy[Siu(x, y)] = σi(k)ˆ u(x, k), for i = 1, 2. Convergence factor of the Schwarz iteration in Fourier space: ρ(k, σ1, σ2) :=

• ˆ

un+1

i

(0, k) ˆ un−1

i

(0, k)

• =
• (σ1 − ν2|k|)(σ2 − ν1|k|)

(σ1 + ν1|k|)(σ2 + ν2|k|)

• .

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.7/28

Fourier analysis

Fourier transform in y: Fy[u(x, y)] = ˆ u(x, k) := 1 √ 2π ∞

−∞

u(x, y)e−iykdy Fourier symbols for the transmission operators Si: Fy[Siu(x, y)] = σi(k)ˆ u(x, k), for i = 1, 2. Convergence factor of the Schwarz iteration in Fourier space: ρ(k, σ1, σ2) :=

• ˆ

un+1

i

(0, k) ˆ un−1

i

(0, k)

• =
• (σ1 − ν2|k|)(σ2 − ν1|k|)

(σ1 + ν1|k|)(σ2 + ν2|k|)

• .

Optimal choice of operators: σopt

1 (k) = ν2|k|,

σopt

2 (k) = ν1|k|.

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.7/28

Optimized Schwarz methods

Find the “best” transmission conditions from a class of local operators C, min

σ1,σ2∈C

• max

k1≤k≤k2 ρ(k, σ1, σ2)

• .

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.8/28

Optimized Schwarz methods

Find the “best” transmission conditions from a class of local operators C, min

σ1,σ2∈C

• max

k1≤k≤k2 ρ(k, σ1, σ2)

• .

Equioscillation principle: often, the solution of this min-max problem is characterized by equioscillation of the convergence factor ρ at the local maxima, e.g.

10 20 30 40 50 60 70 80 90 100 0.2 0.4 0.6 0.8

k ρ

10 20 30 40 50 60 70 80 90 100 0.1 0.2 0.3 0.4

k ρ

1 parameter 2 parameters

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.8/28

Optimized Robin conditions (v. 1)

σ1(k) = σ2(k) = p ∈ R

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.9/28

Optimized Robin conditions (v. 1)

σ1(k) = σ2(k) = p ∈ R Convergence factor: ρ(k, p) =

• (p − ν1|k|)(p − ν2|k|)

(p + ν1|k|)(p + ν2|k|)

• .

Uniform minimization of the convergence factor: min

p∈R

• max

k1≤k≤k2 ρ(k, p)

• .

(M1)

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.9/28

Optimized Robin conditions (v. 1)

σ1(k) = σ2(k) = p ∈ R Convergence factor: ρ(k, p) =

• (p − ν1|k|)(p − ν2|k|)

(p + ν1|k|)(p + ν2|k|)

• .

Uniform minimization of the convergence factor: min

p∈R

• max

k1≤k≤k2 ρ(k, p)

• .

(M1)

We will state our result in terms of µ := max(ν1, ν2) min(ν1, ν2) , kr = k2 k1 .

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.9/28

Optimized Robin conditions (v. 1)

Theorem 1. Solution of the min-max problem (M1). Let

f(µ) := (µ + 1)2 + (µ − 1)

• µ2 + 6µ + 1

4µ .

If kr ≥ f(µ), then one minimizer of (M1) is p∗ = √ν1ν2k1k2. This minimizer p∗ is unique when

ρ(k1, p∗) ≥ ρ

• p∗

√ν1ν2 , p∗

• .

Otherwise, the minimum is also attained for any p chosen in some closed interval containing p∗. If kr < f(µ), then there are two minimizers given by the two positive roots of

p4 +

• ν1ν2(k2

1 + k2 2) − k1k2(ν1 + ν2)2

p2 + (ν1ν2k1k2)2.

Both of these two values yield equioscillation, i.e. ρ(k1, p∗) = ρ(k2, p∗).

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.10/28

Optimized Robin conditions (v. 1)

20 40 60 80 100 0.1 0.2 0.3 0.4 0.5

kr = 100, µ = 10 k |ρ|

20 40 60 80 100 0.1 0.2 0.3 0.4 0.5

kr = 100, µ = 25 k |ρ|

20 40 60 80 100 0.2 0.4 0.6 0.8 1

kr = 100, µ = 150 k |ρ|

20 40 60 80 100 0.2 0.4 0.6 0.8 1

kr = 100, µ = 200 k |ρ|

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.11/28

Optimized Robin conditions (v. 2)

Recall the optimal symbols are σopt

1 (k) = ν2|k|,

σopt

2 (k) = ν1|k|.

This suggests a different scaling in the Robin conditions, σ1(k) = ν2p, σ2(k) = ν1p, for p ∈ R.

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.12/28

Optimized Robin conditions (v. 2)

Recall the optimal symbols are σopt

1 (k) = ν2|k|,

σopt

2 (k) = ν1|k|.

This suggests a different scaling in the Robin conditions, σ1(k) = ν2p, σ2(k) = ν1p, for p ∈ R. Convergence factor: ρ(k, p) = (p − k)2 (p + µk)(p + k/µ). Uniform minimization of the convergence factor: min

p∈R

• max

k1≤k≤k2 ρ(k, p)

• .

(M2)

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.12/28

Optimized Robin conditions (v. 2)

Theorem 2. Solution of the min-max problem (M2). The optimization problem (M2) has a unique minimizer, given by

p∗ =

• k1k2.

This value always gives the equioscillation ρ(k1, p∗) = ρ(k2, p∗).

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.13/28

Optimized Robin conditions

Comparison of optimized convergence factors for version 1 and 2: µ = 10 µ = 100

10 20 30 40 50 60 70 80 90 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7

k ρ

• v. 1
• v. 2

10 20 30 40 50 60 70 80 90 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7

k ρ

• v. 1
• v. 2

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.14/28

Asymptotic performance

Theorem 3. When k2 = π

h , and as h → 0 (keeping ν1 and ν2 constant), we find the

asymptotic expansions: Optimized Robin conditions, v. 1

max

k1≤k≤π/h |ρ(k, p∗)| = 1 − 2

√µ + 1 √µ k1h π 1

2

+ O(h)

Optimized Robin conditions, v. 2

max

k1≤k≤π/h |ρ(k, p∗)| = 1 − (µ + 1)2

µ k1h π 1

2

+ O(h)

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.15/28

Optimized two-sided Robin conditions

We now consider two-sided Robin conditions, with two free parameters σ1(k) = ν2p, σ2(k) = ν1q, for p, q ∈ R.

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.16/28

Optimized two-sided Robin conditions

We now consider two-sided Robin conditions, with two free parameters σ1(k) = ν2p, σ2(k) = ν1q, for p, q ∈ R. Convergence factor: ρ(k, p, q) =

• (p − k)(q − k)

(p + ν1

ν2 k)(q + ν2 ν1 k)

• .

Uniform minimization of the convergence factor: min

p,q∈R

• max

k1≤k≤k2 ρ(k, p, q)

• .

(M3)

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.16/28

Optimized two-sided Robin conditions

Let λ := ν1 ν2 .

Theorem 4. Solution of the min-max problem (M3). When λ ≥ 1, the unique minimizing pair (p∗, q∗) of problem (M3) is the unique solution to the system of equations

p∗q∗ = k1k2,

(E1)

|ρ(k1, p∗, q∗)| = |ρ(√p∗q∗, p∗, q∗)|,

(E2) satisfying k1 < p∗ < q∗. When λ ≤ 1, the unique minimizing pair (p∗, q∗) of (M3) is the solution of the above equations (E1)-(E2) satisfying k1 < q∗ < p∗ instead. The optimized convergence factor always has the equioscillation property at the frequencies k1, k2 and kc = √pq.

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.17/28

Optimized two-sided Robin conditions

Computing p∗ reduces to finding the unique real root of the quartic (p + λk1)(p + λk2)(

• k1k2 − p)2 − (p − k1)(k2 − p)(p + λ
• k1k2)2

in the interval (k1, √k1k2) if λ ≥ 1, or in the interval (√k1, k2, k2) if λ ≤ 1.

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.18/28

Optimized two-sided Robin conditions

Computing p∗ reduces to finding the unique real root of the quartic (p + λk1)(p + λk2)(

• k1k2 − p)2 − (p − k1)(k2 − p)(p + λ
• k1k2)2

in the interval (k1, √k1k2) if λ ≥ 1, or in the interval (√k1, k2, k2) if λ ≤ 1.

Theorem 5. When k2 = π

h , and as h approaches 0, the asymptotic performance of the

• ptimized two-sided Robin conditions is

max

k1≤k≤π/h ρ(k, p∗, q∗) = 1

µ − 4(µ + 1) µ(µ − 1)

• k1

π h

1 2 + O(h).

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.18/28

Optimized 2nd order conditions

We can also consider transmission conditions that are second order in the tangential direction

• ν1

∂ ∂x + ν2

• p − q ∂2

∂y2

• un+1

1

=

• ν2

∂ ∂x + ν2

• p − q ∂2

∂y2

• un

2,

• −ν2

∂ ∂x + ν1

• p − q ∂2

∂y2

• un+1

2

=

• −ν1

∂ ∂x + ν1

• p − q ∂2

∂y2

• un

1.

The corresponding symbols in Fourier space are σ1(k) = ν2(p + qk2), σ2(k) = ν1(p + qk2).

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.19/28

Optimized 2nd order conditions

We can also consider transmission conditions that are second order in the tangential direction

• ν1

∂ ∂x + ν2

• p − q ∂2

∂y2

• un+1

1

=

• ν2

∂ ∂x + ν2

• p − q ∂2

∂y2

• un

2,

• −ν2

∂ ∂x + ν1

• p − q ∂2

∂y2

• un+1

2

=

• −ν1

∂ ∂x + ν1

• p − q ∂2

∂y2

• un

1.

The corresponding symbols in Fourier space are σ1(k) = ν2(p + qk2), σ2(k) = ν1(p + qk2). This leads to the min-max problem min

p,q∈R

• max

k1≤k≤k2

• (p + qk2 − k)2

(p + qk2 + µk)(p + qk2 + k/µ)

• .

(M4)

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.19/28

Optimized 2nd order conditions

Conjecture 1. Solution of the min-max problem (M4). The min-max problem (M4) has a unique solution, given by the equioscillation of the convergence factor at the frequencies k1, k2 and kc =

• p

q . This gives the formulas

p∗ = (k1k2)

3 4

• 2(k1 + k2)

, q∗ = 1

• 2(k1 + k2)(k1k2)

1 4 .

When k2 = π

h and h tends to 0, we find the asymptotic performance

max

k1≤k≤π/h ρ(k, p∗, q∗) = 1 −

√ 2

• 2 + µ + 1

µ k1 π 1

4

h

1 4 + O(h 1 2 ).

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.20/28

Comparison of convergence factors

µ = 100 µ = 1000

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

k ρ

• Opt. Robin v.2
• Opt. 2−sided Robin
• Opt. 2nd order

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

k ρ

• Opt. Robin v.2
• Opt. 2−sided Robin
• Opt. 2nd order

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.21/28

Asymptotics for strong heterogeneity

Let us now consider the case when h is small but held fixed, and µ is large µ = max(ν1, ν2) min(ν1, ν2) >> 1. The asymptotics become Optimized Robin conditions, v. 1 max

k1≤k≤π/h |ρ(k, p∗)| = 1 − 2

• k1h

π + . . . − O

• h− 1

2

µ

• .

Optimized Robin conditions, v. 2 max

k1≤k≤π/h |ρ(k, p∗)| =

π k1h 1 µ

• − O

1 µ

• .

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.22/28

Asymptotics for strong heterogeneity

Optimized two-sided Robin conditions max

k1≤k≤π/h |ρ(k, p∗, q∗)| = 1

µ − O

• h

1 2

µ

• .

Optimized second order conditions max

k1≤k≤π/h |ρ(k, p∗, q∗)| =

• π

4k1h 1

4 1

µ

• − O

1 µ

• .

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.23/28

Numerical experiments

• −∇ · (ν(x)∇u(x))

= 1

• n Ω = (0, π)2,

u =

• n ∂Ω.

The domain is divided into two symmetric subdomains and a finite volume discretization is used. For h =

π 300:

10 20 30 40 50 60 70 10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10 10

1

iteration error in ∞ norm

µ = 10

• Opt. Robin v.1
• Opt. Robin v.2
• Opt. 2−sided Robin

10 20 30 40 50 60 70 80 10

−10

10

−8

10

−6

10

−4

10

−2

10 10

2

iteration error in ∞ norm

µ = 100

• Opt. Robin v.1
• Opt. Robin v.2
• Opt. 2−sided Robin

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.24/28

Numerical experiments

Here, we take µ = 10 and vary the grid size h. The table shows the number of iterations to reach a tolerance of 10−6. h

• Opt. Robin v.1
• Opt. Robin v.2
• Opt. 2-sided Robin

π 50

24 24 12

π 100

30 26 12

π 200

54 42 14

π 300

62 48 14

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.25/28

Numerical experiments

Now, let us fix h =

π 300 and vary the heterogeneity ratio µ.

µ

• Opt. Robin v.1
• Opt. Robin v.2
• Opt. 2-sided Robin

101 62 48 14 102 72 16 10 103 180 10 8 104 204 8 6 105 202 6 6

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.26/28

Conclusions

We completely solved the min-max problem for

• ptimized one-sided Robin conditions (2 versions),
• ptimized two-sided Robin conditions.

For one-sided Robin conditions, we showed that the second version leads to much better performance, particularly when the jump in the diffusion coefficient is large. For two-sided Robin conditions, we obtain an optimal asymptotic performance of ρ = O(1/µ), independent of h. For almost all optimized transmission conditions we considered, the convergence is improved as we increase the jump in the diffusion coefficient.

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.27/28

Work in progress

Design of an efficient coarse space correction when using many subdomains. Optimized conditions for the advection-diffusion equation in 2D, −∇ · (ν(x)∇u(x)) + a(x) · ∇u(x) + c(x)u(x) = f(x), with discontinuous coefficients.

Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.28/28