Coercivity of high-frequency scattering problems

Coercivity of high-frequency scattering problems Valery Smyshlyaev Department of Mathematics, University College London Joint work with: Euan Spence (Bath), Ilia Kamotski (UCL); Comm Pure Appl Math 2015. Also with: Ivan Graham (Bath), Simon Chandler-Wilde (Reading) Spence, Chandler-Wilde, Graham, S, Comm Pure Appl Math 2011. Durham, 12 July 2016

Summary 1960 s - 1980 s: tremendous interest in rigorous aspects of scattering/ diffraction: decay at t of wave equation 2 w t 2 c2 w = 0 asymptotics as k of Helmholtz equation u + k 2 u = 0 (related in subtle way) 2000 s - present: interest in Numerical Analysis of Helmholtz for k 1 e.g. hybrid asymptotic-numerical methods This talk: boundary integral equations One analysis question: prove relevant operator is coercive Surprise (?) appears that for coercivity need stronger results than those obtained classically (at least in the context of one classic tool Morawetz multipliers)

Obstacle scattering problem: acoustically soft/ TE Maxwell (2d) perfectly conducting boundary u i = e ikx â u s u s + k 2 u s = 0 in Ω ext := R d \ Ω int Ω int Γ u s + u i = 0 on Γ k > 0 Radiation conditions: u s r iku s = o = Uniqueness and existence. ) (r d 1 2 as r.

Boundary integral equations Green s Integral Representation: ( ) u s Φk (x) = n(y) (x, y)us (y) Φ k (x, y) us n (y) ds(y), where Γ Φ k (x, y) := i 4 H(1) 0 (k x y ) (d = 2) e ik x y 4π x y (d = 3) Hence boundary integral equations for v := u n : Single layer: S k v(x) := Φ k (x, y)v(y) ds(y) = u i (x) Γ x Ω ext (uniqueness fails for k 2 = interior Dirichlet eigenvalues) (Adjoint) double layer: ( ) 1 2 I + D k v(x) := 12 v(x) + Φ k ui (x, y)v(y) ds(y) = n(x) n (x) (uniqueness fails for k 2 = interior Neumann eigenvalues) Γ

Combined boundary integral equations Try a combination of a double layer and of a single layer: (Double Layer) iη (Single Layer) with a coupling constant η k (k 1). I.e. let A k = 1 2 I + D k i ηs k. Combined boundary integral equation: ( ) u A k = f n ( ) f = ui n (x) i η ui (x) At high frequencies (k 1) kernel of A k highly oscillatory (and non-linear) in k.

The operator A k ( ) u A k = f n For a fixed k, η > 0: A k : L 2 (Γ) L 2 (Γ) bounded and invertible and u/ n L 2 (Γ) if Γ is Lipschitz (Nečas) Q. What do we want to know about A k? 1. bound on A k (explicit in k) 2. bound on A 1 k (explicit in k) 3. coercivity: γ> 0 such that (A k φ, φ) L 2 (Γ) γ φ 2 L 2 (Γ), φ L 2 (Γ) (γ explicit in k)

The operator A k ( ) u A k = f n For a fixed k, η > 0: A k : L 2 (Γ) L 2 (Γ) bounded and invertible and u/ n L 2 (Γ) if Γ is Lipschitz (Nečas) Q. What do we want to know about A k? 1. bound on A k (explicit in k) relatively easy 2. bound on A 1 k (explicit in k) 3. coercivity: γ> 0 such that (A k φ, φ) L 2 (Γ) γ φ 2 L 2 (Γ), φ L 2 (Γ) (γ explicit in k)

The operator A k ( ) u A k = f n For a fixed k, η > 0: A k : L 2 (Γ) L 2 (Γ) bounded and invertible and u/ n L 2 (Γ) if Γ is Lipschitz (Nečas) Q. What do we want to know about A k? 1. bound on A k (explicit in k) relatively easy 2. bound on A 1 k (explicit in k) much harder (but still not enough!) 3. coercivity: γ> 0 such that (A k φ, φ) L 2 (Γ) γ φ 2 L 2 (Γ), φ L 2 (Γ) (γ explicit in k)

The operator A k ( ) u A k = f n For a fixed k, η > 0: A k : L 2 (Γ) L 2 (Γ) bounded and invertible and u/ n L 2 (Γ) if Γ is Lipschitz (Nečas) Q. What do we want to know about A k? 1. bound on A k (explicit in k) relatively easy 2. bound on A 1 k (explicit in k) much harder (but still not enough!) 3. coercivity: γ> 0 such that (A k φ, φ) L 2 (Γ) γ φ 2 L 2 (Γ), φ L 2 (Γ) (γ explicit in k) even harder!

Why is bounding A 1 not enough? k A k v = f ( v = u ) n Solving numerically using Galerkin method: choose S N L 2 (Γ) (N-dimensional subspace), find v N S N such that (A k v N, φ N ) L 2 (Γ) = (f, φ N ) L 2 (Γ), φ N S N Want quasi-optimality : (Lax-Milgramm + Cea s Lemma) : v v N L2 (Γ) C(k) inf v φ N L2 (Γ) φ N S N ( ) in some sense numerical well-posedness C(k) = A k /γ k S N Bound on A 1 can t give ( ) for important k

Plan Multiplier Methods

Multiplier methods Helmholtz equation u + k 2 u = 0, where u(x), x D R 3, k > 0

Multiplier methods Helmholtz equation D M where u(x), x D R 3, k > 0 integrate by parts ( ) u + k 2 u = 0, M u = (M u) M u get D M u n M u + k 2 M u = 0 D D

Some famous (and not so famous) multipliers radiation condition for u s : us r D ) M ( u s + k 2 u s = 0, iku s = o ) (r d 1 2, d = 2, 3 u s (x) eikr f (ˆx) as r = x r d 1 2

Some famous (and not so famous) multipliers radiation condition for u s : us r D ) M ( u s + k 2 u s = 0, iku s = o ) (r d 1 2, d = 2, 3 u s (x) eikr f (ˆx) as r = x r d 1 2 Green (1828), M = u s Rellich (1940), Morawetz (1968), e.g. M = r us r e.g. M = r us r = x u s ikru s + d 1 u s 2

Classic (high frequency) scattering/ diffraction theory Enormous interest from 1960 s onwards, e.g., USA Keller, Lax, Philips, Morawetz (@ Courant), Melrose... USSR/ Russia - Fock, Buslaev, Babich, Vainberg... 3 main problems 1. Wave equation: behaviour as t 2. Wave equation: propagation of singularities 3. Helmholtz: behaviour as k related in subtle way: 1+2=3 [Vainberg, 1975]

Key concept: (non-)trapping as k Helmholtz in trapping domains has almost eigenvalues/eigenfunctions (resonances)

Key concept: (non-)trapping as k Helmholtz in trapping domains has almost eigenvalues/eigenfunctions (resonances) Classic theory can be translated into results about A 1 k. Expect that 1. For Ω ext certain trapping domains A 1 k n e αkn, 0 < k 1 < k 2 <... some α > 0 2. If Ω ext is non-trapping then A 1 k 1, k k 0 Find 1. Proved 2. Proved ( for star-shaped domains [Chandler-Wilde, Monk 2008] using Rellich M = u ) (N.B. needed extra work to deal with ) r

The operator A k ( ) u A k = f n Spaces: A k : L 2 (Γ) L 2 (Γ) bounded and invertible and u/ n L 2 (Γ) if Γ is Lipschitz Q. What do we want to know about A k? 1. bound on A k (explicit in k) relatively easy 2. bound on A 1 k (explicit in k) use classic high-frequency scattering theory 3. coercivity: γ> 0 such that (A k φ, φ) L 2 (Γ) γ φ 2 L 2 (Γ), φ L 2 (Γ) (γ explicit in k) why?

Quasi-optimality v v N L 2 (Γ) C(k) inf v φ N L φ N S 2 (Γ) N ( ) Want to establish (with explicit k dependence of C) for 1. S N piecewise polynomials 2. S N,k hybrid subspace incorporating asymptotics of v = us n For 1. need N = O(k d 1 ) as k, possibility of 2. giving N = O(1). k-explicit ( ) for 1. classic problem, solved by [Melenk, 2011] (needs bound on A 1 k ) Coercivity (+bound on A k easy) gives ( ) for 1. and 2. k-explicit.

Coercivity γ > 0 such that (A k φ, φ) L2 (Γ) γ φ 2 L 2 (Γ), φ L2 (Γ) Not obvious will hold standard approach to formulations of Helmholtz: prove operator k = coercive + compact k Coercivity for circle (2d) and sphere (3d) k k 0, γ = 1 [Domínguez, Graham, Smyshlyaev, 2007] (Fourier analysis)

Two Coercivity Results using Morawetz Multipliers Result 1. Ω int Lipschitz star-shaped, a specially constructed star-combined A k is coercive k, γ = O(1). [Spence, Chandler-Wilde, Graham, S., Comm Pure Appl Math 2011] Result 2. Ω int smooth convex, the classical combined A k is coercive k k 0, η > η 0 k, γ = 1 2 ε [Spence, I.Kamotski, S. Comm Pure Appl Math 2015 ] (N.B. 1. is first quasi-optimality result for 2nd kind integral equations on L 2 (Γ) for Γ Lipschitz, even for Laplace (k = 0 )

Morawetz - 1 Morawetz & Ludwig (1968): take as a multiplier Mu := x u ikru + d 1 u, 2 Then (the Morawetz-Ludwig identity): 2Re ( Mu( u + k 2 u) ) = [ 2Re ( Mu u ) + ( k 2 u 2 u 2) ] ( x u 2 u r 2) u r iku 2.A corollary: [ ( 2Re Mu u ) + ( k 2 u 2 u 2) ] x ν ds = ν D D [ 2Re ( Mu( u + k 2 u) ) ( + u 2 u r 2) + u r iku 2] dx Let Ω in star-shaped x n β > 0 (n = ν). Take D = Ω ext B(0, R), let R, use radition conditions. = bounds for D-t-N; error bounds for GO/ GTD, etc. k-uniform

Morawetz - 1 Star-combined BIE (Spence, Chandler-Wilde, Graham, S., Comm Pure Appl Math 2011): In the Morawetz identity, choose u = S k φ. Take D = (Ω ext B(0, R)) Ω int, let R. Then, for ( 1 A k := (x n) 2 I + D k ( ) u A k = f n Coercivity: k 0, ) + x Γ S k iηs k, η := kr + i d 1. 2 ( f = x u i (x) i η(x) u i (x) ) Re (A k φ, φ) γ φ 2 L 2, γ = 1 2 ess inf (x n(x)) > 0. x Γ

Classical combined (Spence, I. Kamotski, S., Comm Pure Appl Math 2015) Try as a multiplier, with appropriate vector field Z(x), and scalar functions α(x) and β(x): Mu := Z(x) u ikβ(x)u + α(x)u. Then the following Morawetz-type identity holds: 2Re ( Mu ( + k 2 )u ) = [ 2Re ( Mu u ) + ( k 2 u 2 u 2) ] Z + ( )( 2α Z k 2 u 2 u 2) - 2 Re ( i Z j i u j u ) 2Re (u (ik β + α) u)

For wave equation Morawetz needed: For coercivity of A k we need: Z(x), x Ω ext Z(x), x Ω ext Ω int R ( j Z i ξ i ξ j ) 0, ξ C d, Z.n > 0 on Γ, Z(x) c x as x R ( j Z i ξ i ξ j ) θ ξ 2, ξ C d, Z= n on Γ, Z(x) c x as x (almost enough for A 1 k bound) have for Ω ext non-trapping in 2-d have for Ω int smooth convex in 2 & 3-d...non-trapping?

Summary 1960 s - 1980 s: tremendous interest in rigorous aspects of scattering: decay at t of wave equation 2 w t 2 c2 w = 0 asymptotics as k of Helmholtz equation u + k 2 u = 0 (related in subtle way) 2000 s - present: interest in Numerical Analysis of Helmholtz for k 1 e.g. hybrid asymptotic-numerical methods This talk: boundary integral equations Analysis question: prove relevant operator is coercive Surprise (?) appears that for coercivity need stronger results than those obtained classically (at least in the context of one classic tool Morawetz multipliers)