Optimized Schwarz Methods for Problems with Discontinuous - PowerPoint PPT Presentation

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 oil 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 oil recovery, earthquake prediction, underground disposal of nuclear waste. ν 1 ν 2 ν 3 ν 4 Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.2/28

Motivation Flow in heterogeneous media has many applications, for example oil recovery, earthquake prediction, underground disposal of nuclear waste. ν 1 ν 2 ν 3 ν 4 ⇒ 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 R 2 , (P) u is bounded at infinity . Ω 1 Ω 2 ν ( x ) = ν 1 ν ( x ) = ν 2 x = 0 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 ∂u u (0 − , y ) = u (0 + , y ) , ∂n (0 − , y ) = ν 2 ∂n (0 + , y ) . ν 1 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 ∂u u (0 − , y ) = u (0 + , y ) , ∂n (0 − , y ) = ν 2 ∂n (0 + , y ) . ν 1 Consider a general Schwarz iteration of the form � −∇ · ( ν 1 ∇ u n +1 ) = f on Ω 1 = ( −∞ , 0) × R , 1 u n +1 ν 1 ∂ ν 2 ∂ � � � � u n ∂x + S 1 = ∂x + S 1 at x = 0 , 2 1 � −∇ · ( ν 2 ∇ u n +1 ) = f on Ω 2 = (0 , ∞ ) × R , 2 u n +1 − ν 2 ∂ − ν 1 ∂ u n � � � � ∂x + S 2 = ∂x + S 2 at x = 0 . 1 2 u n i = approximate solution in subdomain Ω i , at iteration n . S i 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 : � ∞ 1 u ( x, y ) e − iyk dy √ F y [ u ( x, y )] = ˆ u ( x, k ) := 2 π −∞ Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.7/28

Fourier analysis Fourier transform in y : � ∞ 1 u ( x, y ) e − iyk dy √ F y [ u ( x, y )] = ˆ u ( x, k ) := 2 π −∞ Fourier symbols for the transmission operators S i : F y [ S i u ( x, y )] = σ i ( 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 : � ∞ 1 u ( x, y ) e − iyk dy √ F y [ u ( x, y )] = ˆ u ( x, k ) := 2 π −∞ Fourier symbols for the transmission operators S i : F y [ S i u ( x, y )] = σ i ( k )ˆ u ( x, k ) , for i = 1 , 2 . Convergence factor of the Schwarz iteration in Fourier space: u n +1 � � � � ˆ (0 , k ) ( σ 1 − ν 2 | k | )( σ 2 − ν 1 | k | ) i � � � � ρ ( k, σ 1 , σ 2 ) := � = � . � � � � u n − 1 ( σ 1 + ν 1 | k | )( σ 2 + ν 2 | k | ) ˆ (0 , k ) � � i Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.7/28

Fourier analysis Fourier transform in y : � ∞ 1 u ( x, y ) e − iyk dy √ F y [ u ( x, y )] = ˆ u ( x, k ) := 2 π −∞ Fourier symbols for the transmission operators S i : F y [ S i u ( x, y )] = σ i ( k )ˆ u ( x, k ) , for i = 1 , 2 . Convergence factor of the Schwarz iteration in Fourier space: u n +1 � � � � ˆ (0 , k ) ( σ 1 − ν 2 | k | )( σ 2 − ν 1 | k | ) i � � � � ρ ( k, σ 1 , σ 2 ) := � = � . � � � � u n − 1 ( σ 1 + ν 1 | k | )( σ 2 + ν 2 | k | ) ˆ (0 , k ) � � i Optimal choice of operators: σ opt σ opt 1 ( k ) = ν 2 | k | , 2 ( k ) = ν 1 | 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 k 1 ≤ k ≤ k 2 ρ ( k, σ 1 , σ 2 ) max . σ 1 ,σ 2 ∈C 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 k 1 ≤ k ≤ k 2 ρ ( k, σ 1 , σ 2 ) max . σ 1 ,σ 2 ∈C 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. 0.8 0.4 0.6 0.3 ρ 0.4 ρ 0.2 0.2 0.1 0 0 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 k k 1 parameter 2 parameters Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.8/28

Optimized Robin conditions (v. 1) σ 1 ( k ) = σ 2 ( k ) = p ∈ R Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.9/28

Optimized Robin conditions (v. 1) σ 1 ( k ) = σ 2 ( k ) = p ∈ R Convergence factor: � � ( p − ν 1 | k | )( p − ν 2 | k | ) � � ρ ( k, p ) = � . � � ( p + ν 1 | k | )( p + ν 2 | k | ) � Uniform minimization of the convergence factor: � � min k 1 ≤ k ≤ k 2 ρ ( k, p ) max . ( M 1 ) p ∈ R Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.9/28

Optimized Robin conditions (v. 1) σ 1 ( k ) = σ 2 ( k ) = p ∈ R Convergence factor: � � ( p − ν 1 | k | )( p − ν 2 | k | ) � � ρ ( k, p ) = � . � � ( p + ν 1 | k | )( p + ν 2 | k | ) � Uniform minimization of the convergence factor: � � min k 1 ≤ k ≤ k 2 ρ ( k, p ) max . ( M 1 ) p ∈ R We will state our result in terms of µ := max( ν 1 , ν 2 ) k r = k 2 min( ν 1 , ν 2 ) , . k 1 Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.9/28

Optimized Robin conditions (v. 1) Theorem 1. Solution of the min-max problem ( M 1 ) . Let f ( µ ) := ( µ + 1) 2 + ( µ − 1) µ 2 + 6 µ + 1 � . 4 µ If k r ≥ f ( µ ) , then one minimizer of ( M 1 ) is p ∗ = √ ν 1 ν 2 k 1 k 2 . This minimizer p ∗ is unique when p ∗ � � ρ ( k 1 , p ∗ ) ≥ ρ , p ∗ . √ ν 1 ν 2 Otherwise, the minimum is also attained for any p chosen in some closed interval containing p ∗ . If k r < f ( µ ) , then there are two minimizers given by the two positive roots of p 4 + p 2 + ( ν 1 ν 2 k 1 k 2 ) 2 . ν 1 ν 2 ( k 2 1 + k 2 2 ) − k 1 k 2 ( ν 1 + ν 2 ) 2 � � Both of these two values yield equioscillation, i.e. ρ ( k 1 , p ∗ ) = ρ ( k 2 , p ∗ ) . Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.10/28

Optimized Robin conditions (v. 1) k r = 100, µ = 10 k r = 100, µ = 25 0.5 0.5 0.4 0.4 0.3 0.3 | ρ | | ρ | 0.2 0.2 0.1 0.1 0 0 0 20 40 60 80 100 0 20 40 60 80 100 k k k r = 100, µ = 150 k r = 100, µ = 200 1 1 0.8 0.8 0.6 0.6 | ρ | | ρ | 0.4 0.4 0.2 0.2 0 0 0 20 40 60 80 100 0 20 40 60 80 100 k k Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.11/28

Optimized Robin conditions (v. 2) Recall the optimal symbols are σ opt σ opt 1 ( k ) = ν 2 | k | , 2 ( k ) = ν 1 | k | . This suggests a different scaling in the Robin conditions, σ 1 ( k ) = ν 2 p, σ 2 ( k ) = ν 1 p, for p ∈ R . Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.12/28

Optimized Robin conditions (v. 2) Recall the optimal symbols are σ opt σ opt 1 ( k ) = ν 2 | k | , 2 ( k ) = ν 1 | k | . This suggests a different scaling in the Robin conditions, σ 1 ( k ) = ν 2 p, σ 2 ( k ) = ν 1 p, for p ∈ R . Convergence factor: ( p − k ) 2 ρ ( k, p ) = ( p + µk )( p + k/µ ) . Uniform minimization of the convergence factor: � � min k 1 ≤ k ≤ k 2 ρ ( k, p ) max . ( M 2 ) p ∈ R Optimized Schwarz Methods for Problems with Discontinuous Coefficients – p.12/28