Interior Dirichlet Problem − ∆ u + ∇ p = k 2 u in Ω ∇ · u = 0 in Ω u = f on Γ Oscillatory � Γ ν · f dS = 0 . Compatibility condiiton Stokes BVPs Interior Neumann Problem − ∆ u + ∇ p = k 2 u in Ω ∇ · u = 0 in Ω σ ( u , p ) = g on Γ
Oscillatory Stokeslets − (∆ + k 2 ) u + ∇ p = δ y ( x ) f in Ω ∇ · u = 0 in Ω
G L ( x , y ) = − log | x − y | ⇒ ∆ G L = δ y ( x ) 2 π Oscillatory Stokeslets − (∆ + k 2 ) u + ∇ p = δ y ( x ) f in Ω ∇ · u = 0 in Ω
G L ( x , y ) = − log | x − y | ⇒ ∆ G L = δ y ( x ) 2 π ↓ ⇒ p = ∇ G L · f Oscillatory Stokeslets − (∆ + k 2 ) u + ∇ p = δ y ( x ) f in Ω ∇ · u = 0 in Ω
G L ( x , y ) = − log | x − y | ⇒ ∆ G L = δ y ( x ) 2 π ↓ ⇒ p = ∇ G L · f ↓ Oscillatory − (∆ + k 2 ) u = ∆ G L f − ∇ ( ∇ G L · f ) Stokeslets − (∆ + k 2 ) u + ∇ p = δ y ( x ) f in Ω ∇ · u = 0 in Ω
G L ( x , y ) = − log | x − y | ⇒ ∆ G L = δ y ( x ) 2 π ↓ ⇒ p = ∇ G L · f ↓ Oscillatory − (∆ + k 2 ) u = ∆ G L f − ∇ ( ∇ G L · f ) Stokeslets ↓ u = (( ∇ ⊗ ∇ − ∆ I ) G BH ) f − (∆ + k 2 ) u + ∇ p = δ y ( x ) f in Ω = − ( ∇ ⊥ ⊗∇ ⊥ G BH ) f ∇ · u = 0 in Ω
G L ( x , y ) = − log | x − y | ⇒ ∆ G L = δ y ( x ) 2 π ↓ ⇒ p = ∇ G L · f ↓ Oscillatory − (∆ + k 2 ) u = ∆ G L f − ∇ ( ∇ G L · f ) Stokeslets ↓ u = (( ∇ ⊗ ∇ − ∆ I ) G BH ) f − (∆ + k 2 ) u + ∇ p = δ y ( x ) f in Ω = − ( ∇ ⊥ ⊗∇ ⊥ G BH ) f ∇ · u = 0 in Ω where � 1 G BH ( x , y ; k ) = 1 2 π log | x − y | + i � 4 H 1 0 ( k | x − y | ) k 2 ∆(∆ + k 2 ) G BH = δ
Stresslet
Stresslet Let G ( k ) ( x , y ) = ( ∇ ⊗ ∇ − ∆ I ) G BH ( x , y ; k ) .
Stresslet Let G ( k ) ( x , y ) = ( ∇ ⊗ ∇ − ∆ I ) G BH ( x , y ; k ) . u ( x ) = G ( k ) ( x , y ) f , p ( x ) = ∇ G L ( x , y ) · f
Stresslet Let G ( k ) ( x , y ) = ( ∇ ⊗ ∇ − ∆ I ) G BH ( x , y ; k ) . u ( x ) = G ( k ) ( x , y ) f , p ( x ) = ∇ G L ( x , y ) · f The stress tensor is σ ( x ) = − p ( x ) I + ∇ u ( x ) + ( ∇ u ( x )) ⊺ =: T ( k ) f
Stresslet Let G ( k ) ( x , y ) = ( ∇ ⊗ ∇ − ∆ I ) G BH ( x , y ; k ) . u ( x ) = G ( k ) ( x , y ) f , p ( x ) = ∇ G L ( x , y ) · f The stress tensor is σ ( x ) = − p ( x ) I + ∇ u ( x ) + ( ∇ u ( x )) ⊺ =: T ( k ) f Stresslet T ijℓ = − ∂ x j G L δ iℓ + ∂ x ℓ − ∆ G BH δ ij + ∂ x i ∂ x j G BH �� � �
Γ = ∂ Ω Ω Layer Potentials ν is normal to boundary
Γ = ∂ Ω Ω Layer Potentials ν is normal to boundary Single � S ( k ) [ µ ]( x ) = G ( k ) ( x , y ) µ ( y ) dS ( y ) Γ
Γ = ∂ Ω Ω Layer Potentials ν is normal to boundary Single � S ( k ) [ µ ]( x ) = G ( k ) ( x , y ) µ ( y ) dS ( y ) Γ Double � � ⊺ � T ( k ) D ( k ) [ µ ]( x ) = · , · ,ℓ ( x , y ) ν ℓ ( y ) µ ( y ) dS ( y ) Γ
S ( k ) single layer σ ( k ) stress of single layer off boundary S D ( k ) double layer off boundary D ( k ) double layer on boundary N ( k ) = D ( k ) ⊺ stress of single layer on boundary
S ( k ) single layer σ ( k ) stress of single layer off boundary S D ( k ) double layer off boundary D ( k ) double layer on boundary N ( k ) = D ( k ) ⊺ stress of single layer on boundary Lemma (Jump conditions) For a given density µ defined on Γ , S µ is continuous across Γ , the exterior and interior limits of the surface traction of D µ are equal, and for each x 0 ∈ Γ , S [ µ ]( x 0 ± h ν ( x 0 )) · ν ( x 0 ) = ∓ 1 h ↓ 0 σ ( k ) 2 µ ( x 0 ) + N ( k ) [ µ ]( x 0 ) lim h ↓ 0 D ( k ) [ µ ]( x 0 ± h ν ( x 0 )) = ± 1 2 µ ( x 0 ) + D ( k ) [ µ ]( x 0 ) . lim
Setting u ( x ) = D ( k ) [ µ ]( x ) the Dirichlet problem becomes − 1 2 µ + D ( k ) µ = f on Γ
Setting u ( x ) = D ( k ) [ µ ]( x ) the Dirichlet problem becomes − 1 2 µ + D ( k ) µ = f on Γ Note: dim( N ( − 1 2 + D ( k ) )) > 0 for any k ( µ , ( − 1 2 + N ( k ) ) ν ) = ( − 1 2 + D ( k ) ) µ , ν ) = 0 for all µ , i.e. ( − 1 2 + N ( k ) ) ν = 0 .
Nullspace Correction Definition W [ µ ]( x ) = 1 � ν ( x )( ν ( y ) · µ ( y )) dS ( y ) | Γ | Γ
Nullspace Correction Definition W [ µ ]( x ) = 1 � ν ( x )( ν ( y ) · µ ( y )) dS ( y ) | Γ | Γ Properties W [ W [ µ ]] = W [ µ ] W [1 / 2 ± D ( k ) ] = 0 W [ S ( k ) ] = 0
Nullspace Correction Definition W [ µ ]( x ) = 1 � ν ( x )( ν ( y ) · µ ( y )) dS ( y ) | Γ | Γ Properties W [ W [ µ ]] = W [ µ ] W [1 / 2 ± D ( k ) ] = 0 W [ S ( k ) ] = 0 Adding W Doesn’t change equation for compatible f
Nullspace Correction Definition W [ µ ]( x ) = 1 � ν ( x )( ν ( y ) · µ ( y )) dS ( y ) | Γ | Γ Properties W [ W [ µ ]] = W [ µ ] W [1 / 2 ± D ( k ) ] = 0 W [ S ( k ) ] = 0 Adding W Doesn’t change equation for compatible f − 1 2 µ + D ( k ) µ + W µ = f
Nullspace Correction Definition W [ µ ]( x ) = 1 � ν ( x )( ν ( y ) · µ ( y )) dS ( y ) | Γ | Γ Properties W [ W [ µ ]] = W [ µ ] W [1 / 2 ± D ( k ) ] = 0 W [ S ( k ) ] = 0 Adding W Doesn’t change equation for compatible f − 1 2 µ + D ( k ) µ + W µ = f W ( − 1 2 µ + D ( k ) µ + W µ ) = W [ f ]
Nullspace Correction Definition W [ µ ]( x ) = 1 � ν ( x )( ν ( y ) · µ ( y )) dS ( y ) | Γ | Γ Properties W [ W [ µ ]] = W [ µ ] W [1 / 2 ± D ( k ) ] = 0 W [ S ( k ) ] = 0 Adding W Doesn’t change equation for compatible f − 1 2 µ + D ( k ) µ + W µ = f W ( − 1 2 µ + D ( k ) µ + W µ ) = W [ f ] W µ = 0
Theorem 2 + D ( k ) + W is not invertible if and For simply-connected Ω , − 1 only if k 2 is an eigenvalue
Theorem 2 + D ( k ) + W is not invertible if and For simply-connected Ω , − 1 only if k 2 is an eigenvalue Proof outline
Theorem 2 + D ( k ) + W is not invertible if and For simply-connected Ω , − 1 only if k 2 is an eigenvalue Proof outline Derive radiation condition for exterior BVPs
Theorem 2 + D ( k ) + W is not invertible if and For simply-connected Ω , − 1 only if k 2 is an eigenvalue Proof outline Derive radiation condition for exterior BVPs Show that layer potentials satisfy this radiation condition
Theorem 2 + D ( k ) + W is not invertible if and For simply-connected Ω , − 1 only if k 2 is an eigenvalue Proof outline Derive radiation condition for exterior BVPs Show that layer potentials satisfy this radiation condition Show that exterior problems are uniquely solvable subject to the radiation condition
Theorem 2 + D ( k ) + W is not invertible if and For simply-connected Ω , − 1 only if k 2 is an eigenvalue Proof outline Derive radiation condition for exterior BVPs Show that layer potentials satisfy this radiation condition Show that exterior problems are uniquely solvable subject to the radiation condition Apply the Fredholm alternative
Theorem 2 + D ( k ) + W is not invertible if and For simply-connected Ω , − 1 only if k 2 is an eigenvalue Proof outline Derive radiation condition for exterior BVPs Show that layer potentials satisfy this radiation condition Show that exterior problems are uniquely solvable subject to the radiation condition Apply the Fredholm alternative Theorem 2 + D ( k ) + iη S ( k ) + W , with η real For multiply-connected Ω , − 1 and positive, is not invertible if and only if k 2 is an eigenvalue
Fredholm Determinant Definition of Trace Class An operator K defined on a Banach space is trace-class if the sum of its singular values is absolutely convergent. We write K ∈ J 1 ( L 2 (Γ)) to denote this class.
Fredholm Determinant Definition of Trace Class An operator K defined on a Banach space is trace-class if the sum of its singular values is absolutely convergent. We write K ∈ J 1 ( L 2 (Γ)) to denote this class. For K ∈ J 1 ( L 2 (Γ)) , can define the Fredholm determinant ∞ � det( I − K ) = (1 − λ j ( K )) j =1
Fredholm Determinant Definition of Trace Class An operator K defined on a Banach space is trace-class if the sum of its singular values is absolutely convergent. We write K ∈ J 1 ( L 2 (Γ)) to denote this class. For K ∈ J 1 ( L 2 (Γ)) , can define the Fredholm determinant ∞ � det( I − K ) = (1 − λ j ( K )) j =1 If K trace-class, det( I − K ) = 0 if and only if I − K is not invertible
Fredholm Determinant Definition of Trace Class An operator K defined on a Banach space is trace-class if the sum of its singular values is absolutely convergent. We write K ∈ J 1 ( L 2 (Γ)) to denote this class. For K ∈ J 1 ( L 2 (Γ)) , can define the Fredholm determinant ∞ � det( I − K ) = (1 − λ j ( K )) j =1 If K trace-class, det( I − K ) = 0 if and only if I − K is not invertible D ( k ) is trace-class, but S ( k ) is not!
Theory Recap
Theory Recap Invertibility of I − 2 D ( k ) − 2 W or I − 2 D ( k ) − 2 i S ( k ) − 2 W indicates eigenvalues
Theory Recap Invertibility of I − 2 D ( k ) − 2 W or I − 2 D ( k ) − 2 i S ( k ) − 2 W indicates eigenvalues f ( k ) = det( I − 2 D ( k ) − 2 W ) is a good, convergent (and analytic) indicator of eigenvalues
Theory Recap Invertibility of I − 2 D ( k ) − 2 W or I − 2 D ( k ) − 2 i S ( k ) − 2 W indicates eigenvalues f ( k ) = det( I − 2 D ( k ) − 2 W ) is a good, convergent (and analytic) indicator of eigenvalues From Bornemann and Zhao-Barnett
Theory Recap Invertibility of I − 2 D ( k ) − 2 W or I − 2 D ( k ) − 2 i S ( k ) − 2 W indicates eigenvalues f ( k ) = det( I − 2 D ( k ) − 2 W ) is a good, convergent (and analytic) indicator of eigenvalues From Bornemann and Zhao-Barnett f N ( k ) = det( I N − 2 D ( k ) N − 2 W N ) given by computing determinant of Nystr¨ om discretization of operator converges to order of accuracy of quadrature
Theory Recap Invertibility of I − 2 D ( k ) − 2 W or I − 2 D ( k ) − 2 i S ( k ) − 2 W indicates eigenvalues f ( k ) = det( I − 2 D ( k ) − 2 W ) is a good, convergent (and analytic) indicator of eigenvalues From Bornemann and Zhao-Barnett f N ( k ) = det( I N − 2 D ( k ) N − 2 W N ) given by computing determinant of Nystr¨ om discretization of operator converges to order of accuracy of quadrature f N ( k ) = det( I N − 2 D ( k ) N − 2 i S ( k ) N − 2 W N ) works ok
Theory Recap Invertibility of I − 2 D ( k ) − 2 W or I − 2 D ( k ) − 2 i S ( k ) − 2 W indicates eigenvalues f ( k ) = det( I − 2 D ( k ) − 2 W ) is a good, convergent (and analytic) indicator of eigenvalues From Bornemann and Zhao-Barnett f N ( k ) = det( I N − 2 D ( k ) N − 2 W N ) given by computing determinant of Nystr¨ om discretization of operator converges to order of accuracy of quadrature f N ( k ) = det( I N − 2 D ( k ) N − 2 i S ( k ) N − 2 W N ) works ok high-order root finding on f N produces high accuracy eigenvalues efficiently
Computational Tools Γ = ∂ Ω Ω
Computational Tools Discretization of curves in panels (O’Neil) Γ = ∂ Ω Ω
Computational Tools Discretization of curves in panels (O’Neil) Singular integrals with Γ = ∂ Ω generalized Gaussian quadrature (Bremer) Ω
Computational Tools Discretization of curves in panels (O’Neil) Singular integrals with Γ = ∂ Ω generalized Gaussian quadrature (Bremer) Ω Fast determinant computation using recursive skeletonization ( FLAM Ho)
Computational Tools Discretization of curves in panels (O’Neil) Singular integrals with Γ = ∂ Ω generalized Gaussian quadrature (Bremer) Ω Fast determinant computation using recursive skeletonization ( FLAM Ho) High order root finding with Chebyshev polynomials ( chebfun Trefethen et al.)
Simply Connected Example f N ( k ) = det( I N − 2 D ( k ) N − 2 W N ) 96 panels 16th order Legendre nodes approximate f N ( k ) by a global chebfun on [0 . 1 , 10] of order 295 (used 513 function evals). basic post-processing 10th eigenfield with vorticity on roots
Determinant 2 , 000 0 − 2 , 000 0 2 4 6 8 10 k
Diagnostics Smallest singular value per root Chebyshev coefficients 10 − 1 10 − 4 10 − 13 10 − 7 10 − 10 10 − 14 10 − 13 0 10 20 30 0 100 200 300
First 30 eigenfunctions (plotting vorticity)
Multiply Connected Example f N ( k ) = det( I N − 2 D ( k ) N − 2 i S ( k ) N − 2 W N ) 192 panels 16th order Legendre nodes approximate f N ( k ) by a global chebfun on [0 . 1 , 10] of order 1024. 10th eigenfield with vorticity
Determinant 10 2 200 10 0 10 − 2 100 10 − 4 0 10 − 6 10 − 8 0 2 4 6 8 10 0 2 4 6 8 10 k
Diagnostics Chebyshev coefficients Smallest singular value per root 10 1 10 − 9 10 − 10 10 − 3 10 − 11 10 − 7 10 − 12 10 − 11 10 − 13 10 − 14 10 − 15 0 20 40 60 80 0 200 400 600 800 1 , 000
First 64 eigenfunctions (plotting vorticity)
Example with More Holes f N ( k ) = det( I N − 2 D ( k ) N − 2 i S ( k ) N − 2 W N ) 368 panels 16th order Legendre nodes approximate f N ( k ) by a piecewise chebfuns on [ j, j + 1] for j = 1 , . . . , 8 of order 51-256 (used 65 to 257 function evals). 10th eigenfield with vorticity basic post-processing on roots
Recommend
More recommend
