1 domain-based methods: convergence analysis for hp-fem. 2 discussion of some non-standard FEMs. 3 BEM for Helmholtz problems.

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1 outline Helmholtz problems at large wavenumber k joint work with S. Sauter (Zürich) M. Löhndorf (Vienna) S. Langdon (Reading) 1 domain-based methods: convergence analysis for hp-fem 2 discussion of some non-standard FEMs 3 BEM for Helmholtz problems TU Wien Institut für Analysis und Scientific Computing 2 references used in the talks F. Ihlenburg: FE Analysis of acoustic scattering, Springer Verlag, 1998 M. Ainsworth: Discrete dispersion relation for hp-version finite element approximation at high wave number, SINUM 42 (2004), , S. Sauter: Convergence Analysis for Finite Element Discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions, Math. Comp., 79 (2010), , S. Sauter: Wave-Number Explicit Convergence Analysis for Galerkin Discretizations of the Helmholtz Equation, ASC Rep. 31/2009 I. Babuška, S. Sauter: Is the pollution effect of the FEM avoidable for the Helmholtz equation?, SIREV, 42 (2000), E. Perrey-Debain, O. Laghrouche, P. Bettess, J. Trevelyan: Plane wave basis finite elements and boundary elements for three-dimensional wave scattering,, Phil. Trans. R. Soc. Lond. A, 262 (2004), R. Tezaur and C. Farhat, three-dimensional discontinous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems, IJNME, 66 (2006), C. Gittelson, R. Hiptmair, I. Perugia: Plane wave Discontinuous Galerkin methods, M 2 AN, 43 (2009), A. Buffa, P. Monk, error estimates for the UWVF of the Helmholtz equation, M 2 AN, 42 (2008), R. Hiptmair, A. Moiola, I. Perugia, Approximation by plane waves, SAM Report , ETH Zürich. R. Hiptmair, A. Moiola, I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SAM Report , ETH Zürich A. Moiola, Approximation properties of plane wave spaces and application to the analysis of the plane wave discontinuous Galerkin method, SAM Report , ETH Zürich Melenk: mapping properties of combined field Helmholtz boundary integral operators, ASC Rep. 01/2010 M. Löhndorf, : wavenumber-explicit hp-bem for high frequency scattering, ASC Rep. 02/2010 S. Chandler-Wilde, P. Monk: wave-number-explicit bounds in time-harmonic scattering, SIMA 39 (2008), S. Chandler-Wilde, I. Graham: Boundary integral methods in high frequency scattering in: highly oscillatory problems, Cambridge University Press, B. Engquist, A. Fokas, E. Hairer, A. Iserles eds., 2009 T. Betcke, S. Chandler-Wilde, I. Graham, S. Langdon, M. Lindner: condition number estimates for combined potential integral operators in acoustics and their boundary element discretization, to appear in Numer. Meths. PDEs 3 1 Introduction 2 classical hp-fem convergence of hp-fem regularity 3 some nonstandard FEM 4 boundary integral equations (BIEs) introduction to BIEs hp-bem regularity through decompositions numerical examples (classical hp-bem) example of a non-standard BEM 4

2 Helmholtz model problem u k 2 u = f in, b.c. on, (radiation condition at ) Examples: acoustic scattering problems electromagnetic scattering problems goals: efficient and reliable numerical methods also for large k > 0. Discretization domain-based discretizations FEM integral equation based discretizations BEM the discretizations have the abstract form: find u N V N s.t. a k (u N, v) = l(v) v W N. how to choose V N, W N, a k for large k? fundamental issues approximability: approximate u from V N well: standard (polynomial based) approximation nonstandard approximation (e.g., info from asymptotics) stability: ideally: (asymptotic) quasi-optimality, i.e., u u N C inf v V N u v 5 with some C independent of k in a norm of interest 6 model problem Lu := u k 2 u = f in, R d bounded, Bu := n u iku = 0 on. hp-fem spaces V N abstract FEM discretization given V N H 1 () find u N V N s.t. a(u N, v) = l(v) v V N. weak formulation find u H 1 () s.t. a(u, v) = l(v) where a(u, v) := l(v) := u v k 2 fv. v H 1 () uv ik uv, K F K K T = triangulation of R d with element maps F K diam K h for all K T V N := S p (T ) := {u H 1 () u F K P p } 7 N := dim V N h d p d 8

3 x 10 4 real part, k=50 1/h=40 FEM exact x x x 10 4 real part, k=50 1/h=50 FEM exact rel. H 1 error stability/onset of asymptotic quasioptimality p=1; u k 2 u = 1 k=10 k=100 k=1000 kh = constant is not sufficient to obtain given (relative) accuracy ( pollution ) 2 x real part, k=50 1/h=100 FEM exact DOF per wave length u k 2 u = 1 in (0, 1) u(0) = 0, u (1) iku(1) = 0 mathematical analysis: Babuška & Ihlenburg: 1D, uniform meshes, arbitrary, fixed p Ainsworth: d > 1, infinite tensor product mesh, p-explicit x 9 10 dispersion analysis: p.w. linears, uniform mesh u k 2 u = f, u(0) = 0, u (1) iku(1) = 0 cont. Green s fct: G(x, y) = k 1 { sin kxe iky disc. Green s fct: G h (x, y) = where 0 < x < y sin kye ikx y < x < 1 { sin k x (A sin k y + cos k y) 1 h sin k h A = A(k, k, h) C is a constant k = discrete wave number 0 < x < y sin k y (A sin k x + cos k x) y < x < 1 dispersion relation cos k h = cos 6 2k2 h k 2 h 2 For kh small, we get k h = kh 1 24 (kh)3 + and therefore k = k 1 24 k3 h discrete dispersion analysis, I Theorem (Babuska & Ihlenburg) For the 1D model problem and piecewise polynomial approximation of degree p l on a uniform mesh, there holds for kh < π and solution u H l+1 (0, 1): ( ) kh p ] u u N H 1 (0,1) C p,l [1 ( ) h l + k u H l+1(0,1) } 2p {{ } pollution 2p }{{} best approximation error conclusion the phase error ( pollution ) is not as pronounced for higher order elements as for lower order elements. suggests that ( ) kh p k small σp is an important condition details 12

4 rel. H 1 error p=1; u k 2 u = 1 k=10 k=100 k=1000 rel. H 1 error p=2; u k 2 u = 1 k=10 k=100 k=1000 rel. H1 error p=1 rel. H1 error p= DOF per wave length DOF per wave length k=4 k=40 k=100 k= DOF per wavelength k=4 k=40 k=100 k= DOF per wavelength p=3; u k 2 u = 1 k=10 k=100 k=1000 p=4; u k 2 u = 1 k=10 k=100 k=1000 p=3 rel. H 1 error rel. H 1 error rel. H1 error DOF per wave length DOF per wave length 13 k= k=40 k=100 k= DOF per wavelength 14 relative H1 Error k=4 k=40 k=400 p=1; quarter ring relative H1 Error p=2; quarter ring k=4 k=40 k=400 relative H1 Error 10 2 p=1; half ring k=4 k=40 k=400 relative H1 Error 10 2 p=2; half ring k=4 k=40 k= points per wave length points per wave length 10 2 points per wave length points per wave length p=3; quarter ring 10 2 p=3; half ring relative H1 Error 10 4 k=4 k=40 k=400 relative H1 Error 10 4 k=4 k=40 k= points per wave length points per wave length 16

5 1 Introduction 2 classical hp-fem convergence of hp-fem regularity 3 some nonstandard FEM 4 boundary integral equations (BIEs) introduction to BIEs hp-bem regularity through decompositions numerical examples (classical hp-bem) example of a non-standard BEM 17 stability analysis of hp-fem goals: show that the scale resolution conditions kh p small together with p C log k is sufficient to guarantee quasi-optimality of the hp-fem no uniform meshes ( no discrete Green s function) use only stability of the continuous problem assumptions: geometry is (piecewise) analytic solution operator f u grows only polynomially in k (in a suitable norm) techniques: view Helmholtz problems as H 1 -elliptic plus compact perturbation study regularity of suitable adjoint problems 18 Theorem stability of the continuous problem u k 2 u = f in n u iku = g on Let R d, d {1, 2, 3} be a bounded Lipschitz domain. Then u 1,k Ck 5/2 [ ] f L 2 () + g L 2 ( ) where v 2 1,k := v 2 H 1 () + k2 v 2 L 2 () remarks on stability of the continuous problem one option to get a priori estimates is the use of judicious test functions. For star shaped domains, an interesting test fct is v = x u and then clever integration by parts (Rellich identities) here: use estimates for layer potentials remark If is star-shaped with respect to a ball, then u 1,k C [ ] f L 2 () + g L 2 ( ) 19 20

6 Theorem (quasioptimality of hp-fem) Let be analytic. Then there exist c 1, c 2, C > 0 independent of h, p, k s.t. for kh p c 1 and p c 2 log k there holds: a(u, v) := the adjoint problem u v k 2 uv ik uv. Remark u u N 1,k C inf v V N u v 1,k where v 2 1,k = v 2 H 1 () + k2 v 2 L 2 (). if p = O(log k) then there is no pollution choice p log k and h p/k leads to quasioptimality for a fixed number of points per wavelength generalization to polygonal possible (see below) 21 adjoint solution operator Sk : u = Sk (f) solves a(v, u ) = strong formulation: u k 2 u = f in, n u + iku = 0 on. vf v H 1 () 22 notation adjoint solution operator Sk : u = Sk (f) solves a(v, u ) = vf v H 1 () notation k-dependent norm: v 2 1,k := v 2 L 2 () + k2 v 2 L 2 () continuity: a(u, v) C c u 1,k v 1,k (C c indep. of k) adjoint approximation property: η N := sup f L 2 () Sk inf f v 1,k v V N f L 2 () η N = sup f L 2 () Sk inf f v 1,k, v 2 1,k = v 2 L v V 2 () + k2 v 2 L 2 () N f L2 () Theorem (quasioptimality) If 2C c kη N 1 then the Galerkin-FEM is quasi-optimal and e := u u N satisfies e 1,k 2C c inf u v 1,k, v V N e L 2 () C c η N e 1,k study adjoint approximation property η N need regularity for S k 23 24

7 quasioptimality: proof a(u, v) = ( u, v) L 2 () k 2 (u, v) L 2 () ik(u, v) L 2 ( ) v 2 1,k = v 2 L + 2 k2 v 2 L = Re a(v, v) + 2 2k2 v 2 L2 Gårding ineq. Sk η N = sup inf f v 1,k v V N f L 2 f L 2 assumption: C c kη N 1/2 define ψ by a(, ψ) = (, e) L 2, i.e. ψ = S k e e 2 L 2 = a(e, ψ) = a(e, ψ ψ N ) C c e 1,k ψ ψ N 1,k = e 2 L 2 C c e 1,k η N e L 2 = e L 2 C c η N e 1,k e 2 1,k = Re a(e, e) + 2k 2 e 2 L 2 = e 1,k 2C c inf v V N u v 1,k. Re a(e, u v N ) + 2k 2 (C c η N ) 2 e 2 1,k C c e 1,k u v N 1,k e 2 1,k 25 Theorem (k-explicit regularity by decomposition) Let be analytic. Then u = Sk (f) can be written as u = u H 2 + u A, where for C, γ > 0 independent of k: u H 2 H 2 () C f L 2 (), n u A L 2 () Ck 3/2 γ n max{n, k} n f L 2 () n N 0. implication for adjoint approximation η N : ( kh inf k u H 2 v 1,k v S p (T h ) p + k2 h 2 p 2 [ ( kh inf k u A v 1,k k 7/2 v S p (T h ) σp ) f L2 () ) p + ] f L2 () = kη N small, if kh p + k7/2 ( kh σp ) p small 26 proof the decomposition result properties of the Newton potential N k 1 analyze the Newton potential, i.e., the full space problem 2 analyze the bounded domain case by a fixed point argument u = N k (f) := G k f u solves u k 2 u = f eik z 2ik d = 1, i G k (z) := 4 H(1) 0 (k z ) d = 2, e ik z 4π z d = 3. 1 Ĝ k (ξ) = c d ξ 2 k 2, c d R in R d key ingredient of analysis: study the symbol of N k, i.e., Ĝk(ξ) 27 28

8 Theorem properties of the Newton potential, I Let u = N k (f) and supp f B R. Then: k 1 u H 2 (B R ) + u H 1 (B R ) + k u L 2 (B R ) C f L 2 (B R ) proof: localize the represen. u = G k f and analyze the symbol for χ C0 (Rd ) with χ 1 on B 2R set u R (x) := G(x y)χ(x y)f(y) dy = (G k χ) f. R d Then: u R = u on B R analyze symbol Ĝ k χ. Parseval gives estimates for u R L 2 (R d ), u R H 1 (R d ), u R H properties of the Newton potential N k, II Theorem (decomposition lemma) Let supp f B R and u = N k (f) = G k f. Then, for every η > 1 the function u BR can be written as u = u H 2 + u A where ( s u H 2 L 2 (B R ) C ) η 2 (ηk) s 2 f 1 L 2, s {0, 1, 2}, s u A L 2 (B R ) Cη ( dηk ) s 1 f L 2 s N 0. Corollary For every q (0, 1), one can decompose u BR = u H 2 + u A s.t. k u H 2 L 2 (B R ) + u H 2 H 1 (B R ) qk 1 f L 2 u H 2 H 2 (B R ) C f L 2 u A analytic 30 proof key idea: decomposition u = u H 2 + u A follows from decomposition of f in Fourier space (recall: u = N k (f)) select η > 1. write f = L ηk f + H ηk f, where the low pass filter L ηk and the high pass filter H ηk are F(L ηk f) = χ Bηk f, define u H 2 := N k (H ηk f). define u A := N k (L ηk f). F(Hηk f) = χ R d \B ηk f, u H 2 = N k (H ηk f) = bounds for u H 2 ( ) χ R û H 2 = Ĝk χ R d \B ηk f = c d \B ηk d ξ 2 k f 2 observe that for ξ ηk (recall: η > 1) 1 ξ 2 k 2 ξ 2 ξ 2 k 2 η2 1 η 2 1 (ηk) 2, η2 1 η 2 1 (ηk) 0 hence, for s {0, 1, 2}: ξ ξ 2 k 2 η2 1 η 2 1 (ηk) 1 u H 2 H s (R d ) = ξ s û H 2 L 2 (R d ) C(ηk) s 2 f L 2 (R d ) 31 32

9 estimating u A u A := N k (L ηk f). Hence, u A solves u A k 2 u A = L ηk f on R d Lemma (frequency splitting) key ingredient of the proof: The decomposition f = H ηk f + L ηk f has the following properties: Paley-Wiener: L ηk f is analytic and, since supp χ Bηk f Bηk, s L ηk f L 2 (R d ) C( dηk) s f L 2 (R d ) s N 0 recall a priori estimate: u A H 1 (B 2R ) + k u A L 2 (B 2R ) C L ηk f L 2 C f L 2 (i) L ηk is analytic (ii) H ηk f satisfies H ηk f L 2 (R d ) f L 2 (R d ) H ηk f H 1 (R d ) C(ηk) 1 f L 2 (R d ) use elliptic regularity (induction argument) to get s u A L 2 (B R ) C(γηk) s 1 f L 2 (for suitable γ > 0) 33 H ηk f H s (R d ) C s,s (ηk) (s s ) f H s (R d ) for s s. note: analogous splittings possible on manifolds (e.g., ) 34 decomposition theorem for bounded domains recall: aim is the proof of the following theorem: Theorem (k-explicit regularity) Let be analytic. Then the solution u = S k (f) of u k 2 u = f in n u iku = 0 on can be written as u = u H 2 + u A, where for C, γ > 0 independent of k: u H 2 H 2 () C f L 2 (), n u A L 2 () Ck 3/2 γ n max{n, k} n f L 2 () n N key steps of the decomposition (general case) u k 2 u = f L 2 () n u iku = 0 goal: write u = u I H + u I 2 A + δ, where u I H 2 H 2 () C f L 2 () in on u I A is analytic with s u I A L 2 () Ck 3/2 max{s, k} s f L 2 () s N 0 δ solves δ k 2 δ = f δ L 2 () n δ ikδ = 0 in on where, for a q (0, 1), f δ L 2 () q f L 2 () then: obtain decomposition of u by a geometric series argument 36

10 key steps of the decomposition u k 2 u = f L 2 () n u iku = 0 in on Let S k denote the solution operator for this problem. 1 decompose f = L ηk f + H ηk f. Set u A,1 := S k (L ηk f). Then, u A,1 1,k Ck 5/2 L ηk f L 2. Furthermore, u A,1 is analytic and satisfies the desired bounds (elliptic regularity) u H 2,1 := N k (H ηk f). u H 2,1 H2 () C H ηk f L2 () C f L2 (). u H2,1 H 1 () Ck 1 H ηk f L 2 () Ck 1 f L 2 (). 2 the remainder δ := u (u H 2,1 + u A,1 ) solves δ k 2 δ = 0 n δ ikδ = n u H 2,1 iku H 2,1 =: g together with g H 1/2 ( ) C u H 2,1 H 2 () C f L 2 () 37 δ solves key steps of the decomposition δ k 2 δ = 0 n δ ikδ = g, g H 1/2 ( ) C f L 2 () 3 Decompose g as g = L ηk g + H ηk g with L ηk g analytic H ηk g H 1/2 ( ) C 1 ηk g H 1/2 ( ) C 1 ηk f L 2 () 4 define u A,2 and u H 2,2 as solutions of u H2,2 + k 2 u H 2,2 = 0 u A,2 k 2 u A,2 = 0 n u A,2 iku A,2 = L ηk g n u H 2,2 iku H 2,2 = H ηk g 38 key steps of the decomposition u A,2 k 2 u A,2 = 0 n u A,2 iku A,2 = L ηk g 5 L ηk u H2,2 + k 2 u H2,2 = 0 n u H 2,2 iku H 2,2 = H ηk g is analytic = u A,2 is analytic with appropriate bounds 6 standard a priori bounds for u H 2,2 ( positive definite Helmholtz problem ): u H2,2 1,k C H ηk g H 1/2 ( ) C 1 ηk f L 2 (), u H 2,2 H 2 C H ηk g H 1/2 ( ) C f L 2 (). 7 δ := u (u A,1 + u A,2 + u H 2,1 + u H 2,2) = δ (u A,2 + u H 2,2) solves δ k 2 δ = 2k 2 u H 2,2 =: f δ, n δ ikδ = 0, and f δ L 2 = 2k 2 u H 2,2 L 2 C 1 η f L 2. = selecting η sufficiently large concludes the argument. 39 Theorem (decomposition for convex polygons) Let R 2 be a convex polygon. Then the solution u of u k 2 u = f in, n u iku = 0 on can be written as u = u H 2 + u A, where u H 2 H 2 () C f L 2 () Φ β,n n+2 u A L 2 () Cγ n max{n, k} n+1 f L 2 () n N 0 for some C, γ > 0 and β [0, 1). r(x) := min. distance to vertices r c := min{1, n + 1 k } 1 ( ) if r c 1 Φ β,n = n+β if r c < 1 r min{1, n+1 k } Φ β,n 1 40

11 Theorem (quasi-optimality of hp-fem, polygons) Let R 2 be a convex polygon. Let T L h be a mesh s.t.: the restriction of Th L to \ J j=1 B ch(a j ) is quasi-uniform restricted to B ch(a j ) is a geometric mesh with L layers. T L h Then: the hp-fem is quasi-optimal under the condition kh p sufficiently small and p L c log k. exterior Dirichlet problem u k 2 u = f in R d \, u = 0 on r u iku = o(r (1 d)/2 ). Assume additionally supp f B R. c R B R L = 2 σ = 0.2 O(h) Φ β,n Theorem (exterior Dirichlet problem) Assume: is analytic the solution operator f u satisfies k u L 2 ( c R ) + u H 1 ( c R ) Ck α f L 2 ( c R ) Then: the solution u can be written as u = u H 2 + u A with n u A L 2 ( c R ) Cγ n max{n, k} n k α 1 f L 2 ( c R ) n N 0 u H 2 H 2 ( c R ) C f L 2 ( c R ). c R B R quasi-optimality of hp-fem hp-fem is quasi-optimal under the scale resolution condition, if the DtN-operator on B R is realized exactly. 43 basic mechanisms: summary 1 operator has the form elliptic + compact perturbation : u k 2 u = ( u + k 2 u) 2k 2 u (Gårding inequality) 2 asymptotic quasi-optimality 3 k-explicit regularity for the (adjoint) problem in the form of an additive splitting permits to be explicit about k-dependence of onset of quasi-optimality pollution free methods by changing the discretization and/or ansatz functions? 1D is possible: one can devise nodally exact methods ( pollution-free). This is due to the fact that the space of homogeneous solutions is finite dimensional (dim: 2) pollution unavoidable for d > 1 for methods with fixed stencil (Babuska & Sauter) 44

12 1 Introduction 2 classical hp-fem convergence of hp-fem regularity 3 some nonstandard FEM 4 boundary integral equations (BIEs) introduction to BIEs hp-bem regularity through decompositions numerical examples (classical hp-bem) example of a non-standard BEM Trefftz type ansatz functions idea: approximate solutions of u k 2 u = 0 with functions that solve the equation as well. examples (2D): plane waves: W (p) := span{e ikωn (x,y) n = 1,..., p}, ω n = (cos 2πn 2πn p, sin cylindrical waves: V (p) := span{j n (kr) sin(nϕ), J n (kr) cos(nϕ) n = 0,..., p} fractional Bessel functions (near corners of polygons): span{j nα (kr) sin(nαϕ) n = 1,..., N}, α = π ω fundamental solutions: x i span{g k ( x x i ) i = 1,..., N} more generally: discretized potentials Γ p ) ω 45 O 46 reasons: Trefftz type ansatz functions improved approximation properties (error vs. DOF) greater potential for adaptivity (directionality) hope of reduction of pollution stability analysis not clear that approach coercive + compact perturbation can be made to work for interesting cases often, different, stable numerical formulations used such as least squares DG 47 Approximation properties of systems of plane waves for the approximation of u satisfying u k 2 u = 0 on R 2 W (p) := span{e ikωn (x,y) n = 1,..., p}, ω n = (cos 2πn 2πn, sin p p ) Theorem (h-version: Moiola, Cessenat & Després) Let K be a shape regular element with diameter h. Let p = 2µ + 1. Then there exists v W (2µ + 1) s.t. u v j,k,k C p h µ j+1 u µ+1,k,k, 0 j µ + 1 where v 2 j,k,k = jm=0 k2(j m) v 2 H m (K). Remarks: Extension to 3D possible analogous results for cylindrical waves 48

13 Approximation properties of systems of plane waves II Theorem (p-version, exponential convergence) Let R 2,. Then: inf u v H v W (p) 1 ( ) Ce bp/ log p, Theorem (p-version, algebraic conv.) Let be star shaped with respect to a ball and satisfy an exterior cone condition with angle λπ. Let u H k (), k 1. Then: ( log 2 ) λ(k 1) inf u v (p + 2) H v W (p) 1 () C. p + 2 Remarks: simultaneously h and p-explicit bounds possible (2D, Moiola) extension to 3D possible (Moiola) 49 approximation methods using special ansatz functions 1 partition-of-unity methods (Babuška & Melenk, Bettes & Laghrouche, Astley, etc.) employ standard variational formulation and construct H 1 -conforming ansatz spaces based on the chosen ansatz function 2 Least squares method: approximate with special fcts elementwise and penalize jumps across interelement boundaries (Trefftz, Stojek, Monk & Wang, Betcke, Desmet) 3 Discontinuous enrichement method (Farhat et al.): approximate with plane waves elementwise and enforce interelement continuity by a Lagrange multiplier 4 ultra weak formulation: (Cessenat & Després, Monk & Huttunen, Hiptmair & Moiola & Perugia, Feng et al.) DG-like variational formulation that is only posed on the skeleton ; solution defined as L-harmonic extension into the elements 50 u k 2 u = 0 on = (0, 1) 2, n u + iku = g, on exact solution: u(x, y) = e ik(cos θ,sin θ) (x,y), θ = π 16, k = 32. partition of unity: bilinears ϕ i on uniform n n grid Note: dim V (p) = 2p + 1, rel. error in H approximation with general. harm. polyn problem size N dim W (p) = p n=8 n=4 n=2 rel. error in H approximation with plane waves n=8 n=4 n= problem size N 51 performance of PUM: scattering by a sphere scattering by a sphere B 1 (radius 1) of an incident plane wave sound hard b.c. on Γ = B 1 (i.e., Neumann b.c.) computational domain: ball of diameter 1 + 4λ (λ = 2π/k) b.c. n u s + ( 1 r iκ)us = 0 on outer boundary mesh: 4 layers in rad. dir., 8 5 elem./layers; 160 elem.; 170 nodes k number DOF L 2 (Γ) DOF per waves error wave per node length π % π % π % π % π % 2.48 taken from: Perrey-Debain, Laghrouche, Bettess, Trevelyan, 03 52

14 interelement continuity by penalty (Least Squares) model problem: u k 2 u = 0 on, n u iku = g on Γ, notation: T = mesh, E = internal edges, E Γ = edges on Γ, approximation space: V N {v L 2 () ( v k 2 v ) K = 0 K T } Cost functional J(u) := e E I k 2 [u] 2 L 2 (e) + [ u] 2 L 2 (e) + numerical method: minimize J over V N. e E Γ n u iku g 2 L 2 (e) 1 existence and uniqueness of minimizer u N V N is guaranteed 2 consistency: if exact solution u is sufficiently smooth, then J(u) = interelement continuity by penalty: error estimates can get bounds for J(u N ) from a elementwise approximation properties of V N given above for plane waves: J(u N ) inf J(v)= inf k 2 [u v] 2 v V N v V L 2 (e) + [ h(u v)] 2 L 2 (e) N e E I + e E Γ n (u v) ik(u v) 2 L 2 (e) extract L 2 -error estimates from J(u N ): Theorem (Monk & Wang) Let be convex. Let T be a quasi-uniform mesh with mesh size h. Let u u N satisfy the homogeneous Helmholtz equation elementwise. Then: u u N 2 L 2 () C,kh 1 J(u N ) proof: duality argument (later) numerics 54 Lagrange multiplier technique for interelement continuity original problem: find u H 1 () s.t. a(u, v) = l(v) v H 1 () notation: T = mesh, E = set of internal edges/faces spaces: X = {u L 2 () u K H 1 (K) K T }, M = ( H (E)) 1/2, E E define b(u, µ) = E E [u], µ define a T (u, µ) = K T a K (u, v), a K (u, v) = u v k 2 uv ± ik uv K K Let X N X, M N M: Find (u N, λ N ) X N M N s.t. a T (u N, v) + b(v, λ N ) = l(v) v X N b(u N, µ) = 0 µ M N Discontinuous enrichement method of Farhat et al. (IJNME 06) Ansatz space for solution u: X N := K T W K, W K := span{e ikdn x n = 1,..., N u } Ansatz space for Lagrange multiplier M N := E E W E, W E := span{e ikcnωn t n = 1,..., N λ } where the parameters c n are between 0.4 and 0.8 and are obtained from a numerical study of a test problem 55 56

15 performance of DEM: scattering by a sphere scattering by a sphere B 1 (radius 1) of an incident plane wave sound hard b.c. on Γ = B 1 (i.e., Neumann b.c.) computational domain: ball of diameter 2 b.c. n u s iκu s = 0 on outer boundary DG approach: continuous flux formulation model problem: u k 2 u = 0 on, n u+iku = g on reformulation as a first order system by setting σ := u/(ik): ikσ = u on iku σ = 0 on ikσ n + iku = g on weak elementwise formulation: for every K T there holds ikσ τ + u τ uτ n = 0 τ H(div, K) K K K ikuv + σ v σ nv = 0 v H 1 (K) K K K legend: dashed lines = standard Q 2, Q 3, Q 4 elements; solid lines = new elements; taken from Tezaur & Farhat, 57 Helmholtz IJNME problems 06 at large k 58 discretization: H 1 (K) V N (K), DG: discrete flux formulation H(div, K) Σ N (K) u u N (elementwise) σ σ N (elementwise) (multivalued) traces u and σ on the skeleton are replaced with (single-valued) numerical fluxes û N, σ N discrete formulation (flux formulation) ikσ N τ + iku N v + K K K K u N τ σ N v K K û N τ n = 0 τ Σ N (K) σ N nv = 0 v V N (K) DG: from the flux formulation back to the primal formulation elimination of the variable σ N by 1 requiring V N (K) Σ N (K) for all K T 2 selecting test fct τ = v and integrating by parts gives u N v k 2 u N v (u N û N ) n v ik σ N n = 0 K T K K Since V N consists of discontinuous functions, this is equivalent to: DG formulation Find u N V N s.t. for all v V N u N v k 2 u N v + K T K K (û N u N ) v n ik σ N nv = 0 K 59 60

16 for interior edges DG: special choices of fluxes σ N = 1 ik { h u } α[u N ] û N = {u N } β 1 ik [ h u N ] for boundary edges σ N = 1 ik hu N 1 δ ik ( hu N + iku N n gn) û N = u N δ ik ( hu n + iku N g) 1 α = β = δ = 1/2: UWVF (Cessenat/Després, Monk et al.) 2 α = O(p/(kh log p)), β = O((kh log p)/p), δ = O((kh log p)/p): Hiptmair/Moiola/Perugia if V N (K) = space of elementwise solutions of homogeneous Helmholtz eqn volume contribution can be made to vanish by further integration by parts 61 DG formulation find u N V N s.t. A N (u N, v) = i 1 k Γ δg nv + Γ (1 δ)gv v V N A N (u, v) = {u }[ h v ] + i 1 β [ h u][[ h v ] E I k E I { h u }[v ] + ik α[u][[v ] E I E I + δ)u n v + i Γ(1 1 δ n u n v k Γ δ n uv + ik (1 δ)uv Γ coercivity α, β, δ > 0 = Im A(u, u) > 0 0 u V N 62 Γ convergence theory: coercivity properties I u 2 DG := 1 k β1/2 [ h u] 2 L 2 (E I ) + α1/2 [u] 2 L 2 (E I ) + 1 k δ1/2 n u 2 L 2 (Γ) + k (1 δ)u 2 L 2 (Γ) u 2 DG,+ := u 2 DG + k β 1/2 {u } 2 L 2 (E I ) + k 1 α 1/2 {u } 2 L 2 (E I ) + k δ 1/2 u 2 L 2 (Γ) Theorem (Buffa/Monk, Hiptmair/Moiola/Perugia) Im A(u, u) = u 2 DG u V N A(u, v) C u DG v DG,+ u, v V N e L2 () = sup convergence theory: L 2 estimates = sup ϕ sup ϕ (e, ϕ) L2 () (adj. problem v k 2 v = ϕ) ϕ L 2 () K T (e, v k2 v) L 2 (K) ϕ L 2 ( ϕ L 2 () C e DG ϕ L 2 K T ( 1 + kh kh k β 1/2 v 2 L 2 ( K) +k 1 α 1/2 v 2 L 2 ( K) ) e DG u k 2 e = 0 elementwise trace estimates (elementwise) and quasi-uniformity of mesh α, β, δ = chosen constants 0 (independent of h, k, p) convex, in order to use a priori estimates ) 1/2 In particular, therefore, the DG method is quasi-optimal in DG. k v L 2 + v H 1 + k 1 v H 2 C ϕ L

17 Intro hp-fem nonstandard FEM BIEs Intro hp-fem nonstandard FEM BIEs Theorem (Monk/Buffa, Hiptmair/Moiola/Perugia) use p plane waves elementwise, p 2s + 1 Let kh be bounded for the L2 -estimate: TN quasi-uniform and Ω R2 convex Then, for α, β, δ constant (independent of h, k, p): ku un kdg Ck 1/2 hs 1/2 s 1 kku un kl2 (Ω) Ch where kuk2s,k,ω = Ps j=0 k log p p log p p s 1/2 s 1/2 kuks+1,k,ω kuks+1,k,ω 2(s j) u 2 H j (Ω) (1) remark: for Hiptmair/Moiola/Perugia choice of α, β, δ: optimal rates in k kdg but same convergence result in L2. Intro h-version performance (smooth sol.): left: k = 4, right: k = 64 hp-fem geometry: Ω = (0, 1)2, exact solution: H0 (k x x0 ), x0 = ( 1/4, 0)> 5 plane waves per element source: Gittelson/Hiptmair/Perugia nonstandard FEM BIEs 66 Intro hp-fem nonstandard FEM BIEs summary for volume-based methods 1 Introduction 2 classical hp-fem convergence of hp-fem regularity 3 some nonstandard FEM 4 boundary integral equations (BIEs) introduction to BIEs hp-bem regularity through decompositions numerical examples (classical hp-bem) example of a non-standard BEM standard hp-fem: quasi-optimality can be achieved with a fixed number of a DOF per wavelength, if high order methods are used proof relies on the fact that one has a Ga rding and k-explicit regularity estimates for the adjoint problem nonstandard approximation spaces: significant progress has been made to understand the approximation properties of these spaces stability: available (so far) only for discretizations for which coericivity can be shown 67 68

18 exterior Dirichlet problem (sound soft scattering) Sommerfeld radiation condition Γ d {2, 3} Γ analytic u k 2 u = 0 in R d \, u = g on Γ :=, at reformulations as 2nd kind BIEs 1 Brakhage-Werner : find ϕ s.t. Aϕ = g 2 Burton-Miller : n u solves A n u = f (f given in terms of g) fact: A and A : L 2 (Γ) L 2 (Γ) boundedly invertible 69 G k (z) := potential operators eik z 2ik d = 1, i 4 H(1) 0 (k z ) d = 2, e ik z 4π z d = 3. Newton potential N k (f) := single layer potential Ṽk(f) := double layer potential Kk (f) := facts: Ṽkf and K k f satisfy G k (x y)f(y) dy R d G k (x y)f(y) ds y Γ n(y) y G k (x y)f(y) ds y. the (homogeneous) Helmholtz equation piecewise the Sommerfeld radiation condition at Γ Γ 70 the 4 operators the BIOs V k, K k, K k, D k V k, K k, K k, D k : functions on Γ functions on Γ are defined by taking traces: n Γ single layer V k : double layer K k : adjoint double layer K k : hypersingular op. D k : γ ext 1 u := (n u) Γ V k ϕ := γ0 ext Ṽ k ϕ ( K k)ϕ := γ ext 0 K k ϕ ( K k)ϕ := γ1 ext Ṽ k ϕ jump relations: [Ṽkϕ] = 0, D k ϕ := γ ext 1 K k ϕ [ n Ṽ k ϕ] = ϕ [ K k ϕ] = ϕ, [ n Kk ϕ] = representation formula and Calderón identities representation formula/green s identity Let u solve the homogeneous Helmholtz eqn in R d \ (and Sommerfeld radiation condition). Then: u(x) = ( K k γ ext 0 u)(x) (Ṽkγ ext 1 u)(x) x R d \ taking the trace γ ext 0 and the conormal trace γ ext 1 on Γ leads to Calderón identities ( ) 1 γ0 ext u = 2 Id +K k γ0 ext u V k γ ext γ ext 1 u = D k γ ext 0 u + ( 1 2 Id K k 1 u ) γ ext 1 u 72

19 indirect methods Ansatz: the solution of the Dirichlet problem is sought as a potential (first attempt): u = Ṽkϕ for an unknown density ϕ. BIE V k ϕ = g on Γ However: V k not injective for some k (second attempt) u = K k ϕ. Again no good solvability theory for all k (combined field ansatz) u = (iηṽk + K k )ϕ for some parameter η R \ {0}. Brakhage-Werner g = γ ext 0 u = iηv k ϕ + ( ) K k ϕ =: Aϕ A : L 2 (Γ) L 2 (Γ) is boundedly invertible for every η R \ {0} 73 direct methods starting point: Calderón projector: ( 1 γ0 ext u = 2 Id +K k γ ext 1 u = D k γ ext 0 u + ) γ ext 0 u V k γ ext ( 1 2 Id K k 1 u ( iη) ) γ1 ext u linear combination yields [iη( 12 ] K k) D k γ0 ext u = [iηv k + ( 12 ] + K k ) γ1 ext u =: A γ1 ext u. Dirichlet problem: given γ ext 0 u = g, solve for γ ext 1 u. Representation formula gives u: u = Ṽkγ ext 1 u + K k γ ext 0 u. 74 A = K k + iηv k, the BIOs A and A coupling parameter η with η k. A = K k + iηv k Γ smooth = A and A are compact perturbations of 1 2 Id Fredholm theory available question how does k enter the mapping properties of A, A and their inverses A 1, (A ) 1? 75 given (V N ) N N L 2 (Γ) Galerkin discretizations find ϕ N V N s.t. Aϕ N, v L 2 (Γ) = f, v L 2 (Γ) v V N asymptotic quasioptimality: N 0 s.t. N N 0 ϕ ϕ N L 2 (Γ) 2 inf v V N ϕ v L 2 (Γ) question: how does N 0 depend on k? hp-bem spaces S p,0 (T h ) T h = mesh on Γ, mesh width h element maps analytic (+ suitable scaling properties) S p,0 (T h ) L 2 (Γ) S p,0 (T h ) = piecewise (mapped) polynomials of degree p 76

20 Theorem (Quasi-optimality of hp-bem) Assumption: (adjoint) well-posedness: (A ) 1 L 2 L 2 Ckα Then: c 1, c 2 = c 2 (α) independent of k s.t. the scale resolution condition kh p c 1 and p c 2 log k implies Corollary ϕ ϕ N L 2 (Γ) 2 inf ϕ v v S p,0 L (T h ) 2 (Γ) Selecting p = O(log k) and h p k leads to quasi-optimality for an hp-bem space of dimension N k d 1. remarks on assumption of well-posedness Assumption of well-posedness for some α R there holds (A ) 1 L 2 L 2 Ckα α = 0 for star shaped domains (Chandler-Wilde & Monk) often observed in practice (A ) 1 L 2 L 2 Ceγkm : for certain trapping domains and k m (Betcke/Chandler-Wilde/Graham/Langdon/Lindner) possible to show: ϕ ϕ N L 2 (Γ) (1 + ε h,p ) inf ϕ v v S p,0 L (T h ) 2 (Γ) 77 where ε h,p 0 if kh p 0 (and p log k) 78 regularity through decomposition idea: decompose operators into a part with k-independent bounds part with smoothing properties and k-explicit bounds example: A 1 = A 1 + A 1 A 1 order zero operator; k-independent bounds for A 1 A 1 maps into space of analytic functions example: A = K k + iηv k = K 0 + R + A A: maps into space of analytic functions; k-explicit bounds R: small, order -1, k-explicit bounds 79 Theorem decomposition of V k Let Γ be analytic and choose q (0, 1). Then: where (i) S V : L 2 (Γ) H 3 (Γ) and V k = V 0 + S V + A V S V L 2 L 2 qk 1, S V H 1 L 2 q, S V H 3 L2 k2 (ii) A V : L 2 (Γ) space of analytic functions and n A V ϕ L 2 (Γ) k 3/2 max{k, n} n γ n ϕ H 3/2 (Γ) n N 0 analogous result for K k 80

21 decomposition of Ṽk study: Ṽk χṽ0 (χ = smooth cut-off fct, χ 1 near Γ) given ϕ H 1/2 (Γ) set u := Ṽkϕ and u 0 := Ṽ0ϕ H 1 (B R ). Then δ := u χu 0 = Ṽkϕ χṽ0ϕ solves δ k 2 δ = k 2 u 0 χ + 2 χ u 0 + u 0 χ =: f [δ] = 0 [ n δ] = 0 on Γ, δ satisfies radiation condition δ = N k (f) = N k (H ηk f) + N k (L ηk f), where H ηk and L ηk are the high and low pass filters, η > 1. L ηk f analytic = N k (L ηk f) analytic N k (H ηk f) is an element of H 3 (B R ) (since f H 1 (B R )) small in H 1 (B R ) for large η > 1: N k (H ηk f) H 1 (R d ) Ck 1 H ηk f L 2 (R d ) C 1 ηk 2 f H 1 (R d ) C 1 η ϕ H 1/2 (Γ). Theorem (decomposition of Ṽk) Let Γ be analytic, s 1, and choose q (0, 1). Then: Ṽ k = χṽ0 + S V + ÃV where (i) S V : H 1/2+s (Γ) H 2 (B R ) H 3+s (B R \ Γ) and S V ϕ H s (B R \Γ) q 2 (q/k) 1+s s ϕ H 1/2+s (Γ), 0 s 3 + s (ii) à V : H 1/2+s (Γ) space of p.w. analytic functions and n à V ϕ L 2 (B R \Γ) γ n k max{k, n} n ϕ H 3/2 (Γ) n N Theorem Let Γ be analytic. where decomposition of A 1 Assume: A 1 L 2 L 2 Ckα for some α 0 Then: A 1 = A 1 + A 1 (i) A 1 : L 2 (Γ) L 2 (Γ) with A 1 L 2 L2 C independent of k (ii) A 1 : L 2 (Γ) space of analytic functions with n A 1 ϕ L 2 (Γ) Ck β max{k, n} n ϕ L 2 (Γ) n N 0 for suitable β 0. Remark: Analogous decomposition for (A ) 1 83 sketch of the proof the operators V k and K k can be decomposed as V k = V 0 + R V + A V and K k = K 0 + R K + A K, where R V L 2 L 2 qk 1, R K L 2 L 2 q. hence, decompose A = K k + iηv k as (use η = O(k)) A = 1 ( ) K k + iηv k = 2 + K 0 + iv 0 + R + A where =: (A 0 + R) + A =: Â0 + A 1 A 0 : L 2 (Γ) L 2 (Γ) is boundedly invertible 2 R L2 L2 q with arbitrary q (0, 1) 3 A maps into a space of analytic functions 4  0 : L 2 (Γ) L 2 (Γ) is boundedly invertible (with norm independent of k) 84

22 sketch of the proof, II A = Â0 + A Â 0 : L 2 (Γ) L 2 (Γ) is boundedly invertible A maps into a space of analytic functions Then A 1 = Â A AÂ 1 0 =: A 1 + A 1 is the desired decomposition of A 1 if we can show that A 1 maps analytic functions to analytic functions more specifically: structure of A: A is constructed by taking traces of potentials Aϕ = [z] for a piecewise analytic function (depending on ϕ) will need that A 1 maps traces of jumps of piecewise analytic functions to jumps of piecewise analytic functions 85 Theorem (analytic data) Let be analytic. Let f be the jump of a piecewise analytic function. Let ϕ L 2 (Γ) solve ( K k + iηv k )ϕ = Aϕ = f Then ϕ = [u] for a piecewise analytic function u. ideas of the proof: 1 define u = K k ϕ + iηṽkϕ. 2 jump conditions: ϕ = [u]. 3 γ ext 0 u = ( K k + iηv k )ϕ = f get bounds for u on B R \ 4 [u] = ϕ and [ n u] = iηϕ implies [ n u] + iη[u] = 0. 5 once u R d \ is known, we have an elliptic equation in with Robin boundary data estimates for u. 86 set-up of numerical examples circle (radius 1) A = K k + ikv k mesh T h is quasi-uniform and h 1/k for fixed mesh T h, degree p ranges from 1 to 14. Galerkin projector P h,p : L 2 (Γ) S p,0 (T h ) approximate quasi-optimality constant Id P h,p L 2 L 2 sup (Id P h,p )v L 2 v S 20,0 (T h ) v L 2 Galerkin error circle, r=1, η = k k=1024 k=256 k=128 k=64 k=32 k=16 k=8 k=4 singular value circle, r = 1, η = k σ max (L 1 A L T ) σ min (L 1 A L T ) O(k 1/2 ) indications for A L 2 L 2 and (A ) 1 L 2 L 2 usually η = k (some computations: η = 1) recall scale resolution condition: and p c log k. kh p small p+1 Number of elements N = k Galerkin Error = Id P h,p k quasioptimality constant: C opt = 1 + Galerkin Error 2 88

23 Galerkin error ellipse, a=1, b=1/4, η = k ellipse (semi-axes 1, 1/4) k=256 k=128 k=64 k=32 k=16 k=8 k= p+1 singular value Number of elements N = k Galerkin Error = Id P h,p 2 1 ellipse, a = 1, b = 1/4, η = k σ max (L 1 A L T ) σ min (L 1 A L T ) O(k 0.4 ) 10 2 k Galerkin error circle in circle (radii: 1/2, 1/4) circle in circle, r1=1/2, r2 =1/4, η = k k=512 k=256 k=128 k=64 k=32 k=16 k=8 k= p+1 singular value Number of elements N = 2k Galerkin Error = Id P h,p 2 1 circle in circle, r1 = 1/2, r2 = 1/4, η = k σ max (L 1 A L T ) σ min (L 1 A L T ) 10 2 k Galerkin error C shape, r=1/2, η = k C-shaped domain (η = k) m=3 m=6 m=12 m=24 singular value C shape, r=1/2, η = k σ max (L 1 A L T ) σ min (L 1 A L T ) O(k 0.9 ) Galerkin error C shape, r=1/2, η = 1 C-shaped domain (η = 1) m=48 m=24 m=12 m=6 m=3 C shape, r=1/2, η = 1 σ max (L 1 A L T ) σ min (L 1 A L T ) O(k 1.9 ) p k r/ p k r/3 Number of elements N = 20m k = 3πm/r r r/3 r/3 Number of elements N = 20m k = 3πm/r r r/3 r/3 r = 0.5 r =

24 circle in circle (radii 1/2, 1/4, η = 1) Conclusions for classical hp-bem Galerkin error 10 2 circle in circle, r1=1/2, r2 = 1/4, η = 1 k=512 k=256 k=128 k=64 k=32 k=16 k=8 k=4 singular value circle in circle, r1 = 1/2, r2 = 1/4, η = 1 decomposition of A and A 1 into parts with k-independent bounds parts with smoothing properties and k-explicit bounds hp-bem quasi-optimal with k-independent constant if kh p is sufficiently small and p c log k p+1 Number of elements N = 2k σ max (L 1 A L T ) σ min (L 1 A L T ) 10 2 k quasi-optimality for problem size N = O(k d 1 ) possible (select p = O(log k) and h = O(p/k)) often observe quasi-optimality already for kh p small caveat: the continuous problem needs to be well-posed, i.e., A 1 grows only polynomially in k non-standard BEM: a sound soft scattering problem u i = polygonal obstacle u i (x) := exp(ik d x), d = 1 u k 2 u = 0 in R d \ u = 0 on u s := u u i satisfies( r ik)u s = o(r (d 1)/2 ), goal: determine n u on question: find space V N from which n u can be approximated well multiscale approximation spaces Kirchhoff approximation of n u: n u Ψ := Ansatz { 2 n u i on lit side 0 on shadow side n u = Ψ + kφ for a function φ to be determined u i 95 96

25 n u for k = 10 and k = geometric mesh T L with L layers σ L σ 3 σ 2 σ 0 1 σ 3 σ σ 2 (0, 1) σ σ L 0 L1 N 1 k Theorem sharp gradient at corners highly oscillatory piecewise smooth V + N := exp(iks) p.w. polynomials of degree p on T L, V N := exp( iks) p.w. polynomials of degree p on T L, p L log k. Then: V N := V + N + V N satisfies inf φ v L2 ( ) k 1/2 e bp, v V N dimv N p relative error in L 2 for φ p=2 nonstandard BEM: k = 10 and k = p=3 non standard BEM p=4 p=5 k=10 k=10240 p= DOF composite Filon quadrature k-robust exponential convergence (absolute error) cost of quadrature formula independent of k cpu time 10 3 non standard BEM (Matlab) k=10 k= polynomial degree p conclusions for nonstandard BEM possible to design special approximation spaces that incorporate both the oscillatory nature of the solution and the corner singularities approximation properties are only weakly dependent on k possible to design (in 2D) exponentially convergent quadrature rule to set up BEM stiffness matrix with work depending only weakly on k