Multiscale methods for time-harmonic acoustic and elastic wave propagation

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1 Multiscale methods for time-harmonic acoustic and elastic wave propagation Dietmar Gallistl (joint work with D. Brown and D. Peterseim) Institut für Angewandte und Numerische Mathematik Karlsruher Institut für Technologie (KIT) Workshop on Analysis and Numerics of Acoustic and Electromagnetic Problems RICAM, October 26 D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p.

2 Outline Introduction Helmholtz problem An ideal method Multiscale Petrov-Galerkin FEM Elastic wave propagation Numerical experiments D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 2

3 Introduction Helmholtz problem An ideal method Multiscale Petrov-Galerkin FEM Elastic wave propagation Numerical experiments

4 Model problem: high-frequency acoustic scattering I Ω Rd bounded polytope, diam Ω I Ω = ΓD ΓR I wave number κ > real parameter I incident wave uin I we seek u = uin + uscat I positive and bounded material parameters A(x), n(x), β (x) D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 3

5 Model problem: high-frequency acoustic scattering I Ω Rd bounded polytope, diam Ω I Ω = ΓD ΓR I wave number κ > real parameter I incident wave uin I we seek u = uin + uscat I positive and bounded material parameters A(x), n(x), β (x) div(a u) κ 2 n u = f in Ω u = on ΓD Ω (A u) ν iβ κu = g on ΓR := Ω \ ΓD D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 3

6 Pollution effect d Helmholtz: u xx κ 2 u = in [,], u in (x) = exp( iκx) Results for P-FEM (fixed resolution H := const/κ) 2 PFEM P-best ratio V-error κ D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 4

7 Pollution effect d Helmholtz: u xx κ 2 u = in [,], u in (x) = exp( iκx) Results for P-FEM (fixed resolution H := const/κ) 2 PFEM P-best ratio V-error κ D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 4

8 Pollution effect d Helmholtz: u xx κ 2 u = in [,], u in (x) = exp( iκx) Results for P-FEM (fixed resolution H := const/κ) 2 PFEM P-best ratio V-error κ D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 4

9 Pollution effect d Helmholtz: u xx κ 2 u = in [,], u in (x) = exp( iκx) Results for P-FEM (fixed resolution H := const/κ) 2 PFEM P-best ratio error(fem) κs error(bestapprox) [Babuška-Sauter 2] V-error κ D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 4

10 Pollution effect d Helmholtz: u xx κ 2 u = in [,], u in (x) = exp( iκx) Results for P-FEM (fixed resolution H := const/κ) 2 PFEM P-best ratio mspg Goal: error(mspg) error(bestapprox) C V-error κ D. Gallistl and D. Peterseim. Stable multiscale Petrov-Galerkin finite element method for high frequency acoustic scattering. Comput. Methods Appl. Mech. Eng., 25. D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 4

11 Other Approaches hp-fem with p logκ Melenk, Sauter Trefftz methods, Plane wave methods Hiptmair, Moiola, Perugia DG FEM Farhat, Tezaur; Feng, Wu Ultra-weak formulation Cessenat, Després; Buffa, Monk Discontinuous Petrov-Galerkin Demkowicz, Gopalakrishnan D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 5

12 Introduction Helmholtz problem An ideal method Multiscale Petrov-Galerkin FEM Elastic wave propagation Numerical experiments

13 Variational formulation Hilbert space V := H D (Ω;C) := {v H (Ω;C) : v ΓD = } Continuous sesquilinear form on V V a(v,w) := (A v, w) L 2 (Ω) κ 2 (nv,w) L 2 (Ω) iκ(βv,w) L 2 (Γ R ) D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 6

14 Variational formulation Hilbert space V := H D (Ω;C) := {v H (Ω;C) : v ΓD = } Continuous sesquilinear form on V V a(v,w) := (A v, w) L 2 (Ω) κ 2 (nv,w) L 2 (Ω) iκ(βv,w) L 2 (Γ R ) Weak formulation seeks u V such that a(u,v) = (f,v) L 2 (Ω) + (g,v) L 2 (Γ R ) for all v V. D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 6

15 Variational formulation Hilbert space V := H D (Ω;C) := {v H (Ω;C) : v ΓD = } Continuous sesquilinear form on V V a(v,w) := (A v, w) L 2 (Ω) κ 2 (nv,w) L 2 (Ω) iκ(βv,w) L 2 (Γ R ) Weak formulation seeks u V such that a(u,v) = (f,v) L 2 (Ω) + (g,v) L 2 (Γ R ) for all v V. Data f L 2 (Ω;C) and g L 2 (Γ R ;C) Norm v V := κ 2 v 2 L 2 (Ω) + v 2 L 2 (Ω) D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 6

16 Well-posedness Assumption (Polynomial well-posedness) There exist a constant c(ω) and a polynomial p such that c(ω) p(κ) inf sup v V\{} w V\{} Ra(v, w) v V w V ( ) D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 7

17 Well-posedness Assumption (Polynomial well-posedness) There exist a constant c(ω) and a polynomial p such that c(ω) p(κ) inf sup v V\{} w V\{} Ra(v, w) v V w V ( ) Homogeneous material: assumption not satisfied in general [Betcke et al 2] pure impedence problem + Ω convex γ(κ,ω) κ [Melenk 995] pure impedence problem + Ω Lipschitz γ(κ,ω) κ 7/2 [Esterhazy-Melenk 22] star-shaped scatterer γ(κ,ω) κ [Hetmaniuk 27] D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 7

18 Well-posedness Assumption (Polynomial well-posedness) There exist a constant c(ω) and a polynomial p such that c(ω) p(κ) inf sup v V\{} w V\{} Ra(v, w) v V w V ( ) Heterogeneous material: Theorem There is a class of smooth coefficients A, n such that ( ) holds. D. Brown, D. Gallistl, and D. Peterseim. Multiscale Petrov-Galerkin method for high-frequency heterogeneous Helmholtz equations. Arxiv preprint, 25. D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 7

19 Introduction Helmholtz problem An ideal method Multiscale Petrov-Galerkin FEM Elastic wave propagation Numerical experiments

20 Finite element spaces Coarse scale H /κ G H coarse mesh V H := Q (G H ) V standard Q FE space Fine scale h /κ 2 G h refinement of G H V h := Q (G h ) V standard Q FE space D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 8

21 Finite element spaces Coarse scale H /κ G H coarse mesh V H := Q (G H ) V standard Q FE space Fine scale h /κ 2 G h refinement of G H V h := Q (G h ) V standard Q FE space Quasi-interpolation I H : V h V H quasi-local projection Stability and L 2 -approximation H v I H v L 2 (T) + I H v L 2 (T) C IH v L 2 (N(T)) Example: I H := E H Π H Π H... piecewise L 2 projection onto Q (G H ) E H... nodal averaging operator D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 8

22 Subscale correction Fine scale remainder (null space of I H ): W h := {v h V h : I H (v h ) = } Subscale corrector C : V H W h : Given v H V H, C v H solves a(w,c v H ) = a(w,v H ) for all w W h D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 9

23 Subscale correction Fine scale remainder (null space of I H ): W h := {v h V h : I H (v h ) = } Subscale corrector C : V H W h : Given v H V H, C v H solves a(w,c v H ) = a(w,v H ) for all w W h C D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 9

24 Subscale correction Fine scale remainder (null space of I H ): W h := {v h V h : I H (v h ) = } Subscale corrector C : V H W h : Given v H V H, C v H solves a(w,c v H ) = a(w,v H ) for all w W h Lemma Under the resolution condition Hκ, the corrector problems are well-posed (even coercive). D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 9

25 Ideal Petrov-Galerkin method Standard trial space V H Corrected test space Ṽ H := ( C )V H D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p.

26 Ideal Petrov-Galerkin method Standard trial space V H Corrected test space Ṽ H := ( C )V H Re Re Re Im Im Im Λ z C Λ z Λ z = ( C )Λ z D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p.

27 Ideal Petrov-Galerkin method Standard trial space V H Corrected test space Ṽ H := ( C )V H Re Re Re Im Im Im Λ z C Λ z Λ z = ( C )Λ z D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p.

28 Ideal Petrov-Galerkin method Standard trial space V H Corrected test space Ṽ H := ( C )V H Re Re Re Im Im Im Λ z C Λ z Λ z = ( C )Λ z D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p.

29 Ideal Petrov-Galerkin method Standard trial space V H Corrected test space Ṽ H := ( C )V H Re Re Re Im Im Im Λ z C Λ z Λ z = ( C )Λ z D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p.

30 Ideal Petrov-Galerkin method Standard trial space V H Corrected test space Ṽ H := ( C )V H Re Re Re Im Im Im Λ z C Λ z Λ z = ( C )Λ z D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p.

31 Ideal Petrov-Galerkin method Standard trial space V H Corrected test space Ṽ H := ( C )V H Re Re Re Im Im Im Λ z C Λ z Λ z = ( C )Λ z D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p.

32 Ideal Petrov-Galerkin method Standard trial space V H Corrected test space Ṽ H := ( C )V H Re Re Re Im Im Im Λ z C Λ z Λ z = ( C )Λ z Ideal multiscale Petrov-Galerkin FEM seeks u ms H V H such that a(u ms H,ṽ H ) = (f,ṽ H ) L 2 (Ω) + (g,ṽ H ) L 2 (Γ R ) for all ṽ H Ṽ H. D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p.

33 Ideal Petrov-Galerkin method Standard trial space V H Corrected test space Ṽ H := ( C )V H Re Re Re Im Im Im Connections to existing methods Variational multiscale method [Hughes et al] Local orthogonal decomposition [Malqvist, Peterseim] D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p.

34 Stability and quasi-optimality Since u h I H u h W h, we have a(u h I H u }{{} h,( C )v H ) = for all v H V H. W h D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p.

35 Stability and quasi-optimality Since u h I H u h W h, we have a(u h I H u h, ( C )v H }{{} =ṽ H Ṽ H ) = for all v H V H. D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p.

36 Stability and quasi-optimality Since u h I H u h W h, we have a(u h I H u h, ( C )v H }{{} =ṽ H Ṽ H ) = for all v H V H. Hence: a(i H u h,ṽ H ) = a(u h,ṽ H ) = (f,ṽ H ) L 2 (Ω) + (g,ṽ H ) L 2 (Γ R ) D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p.

37 Stability and quasi-optimality Since u h I H u h W h, we have a(u h I H u h, ( C )v H }{{} =ṽ H Ṽ H ) = for all v H V H. Hence: a(i H u h,ṽ H ) = a(u h,ṽ H ) = (f,ṽ H ) L 2 (Ω) + (g,ṽ H ) L 2 (Γ R ) = I H u h solves Petrov-Galerkin method D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p.

38 Stability and quasi-optimality Since u h I H u h W h, we have a(u h I H u h, ( C )v H }{{} =ṽ H Ṽ H ) = for all v H V H. Hence: a(i H u h,ṽ H ) = a(u h,ṽ H ) = (f,ṽ H ) L 2 (Ω) + (g,ṽ H ) L 2 (Γ R ) = I H u h solves Petrov-Galerkin method Theorem The resolution condition Hκ implies and γ(κ,ω) inf sup v H V H \{} ṽ H Ṽ H \{} Ra(v H,ṽ H ) v H V ṽ H V u h u ms H V = ( I H )u h V min v H V H u h v H V. D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p.

39 Introduction Helmholtz problem An ideal method Multiscale Petrov-Galerkin FEM Elastic wave propagation Numerical experiments

40 Exponential decay of the correctors Theorem There is < β < such that for all vertices z and m N we have ( C )Λ z L 2 (Ω\B m H (z)) β m Λ z L 2 (Ω) Λ z = ( C )Λ z.5 Λ z D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 2

41 Localized fine-scale correctors Re Im D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 3

42 Localized fine-scale correctors Parameter m N m-th order element patches Ω T := N m (T) D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 4

43 Localized fine-scale correctors Parameter m N m-th order element patches Ω T := N m (T) D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 4

44 Localized fine-scale correctors Parameter m N m-th order element patches Ω T := N m (T) h H Localized test functions Ṽ H := ( C m )V H D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 4

45 Multiscale Petrov-Galerkin FEM Theorem (Well-posedness) Let h, Hκ and m logγ(κ,ω) logκ. Then a) Stability γ(κ,ω) inf sup v H V H \{} ṽ H Ṽ H \{} Ra(v H,ṽ H ) v H V ṽ H V b) Quasi-optimality u u ms H V min v H V H u v H V. D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 5

46 Introduction Helmholtz problem An ideal method Multiscale Petrov-Galerkin FEM Elastic wave propagation Numerical experiments

47 Time-harmonic elastic wave model We seek a displacement field u : Ω R d such that σ(u) = Cε(u) div(σ(u)) κ 2 u = f in Ω, for the strain ε := symd and the elasticity tensor CM := 2µM + λ trmi d d. σ(u) ν iκu = g on Γ R, u = on Γ D D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 6

48 Time-harmonic elastic wave model We seek a displacement field u : Ω R d such that σ(u) = Cε(u) div(σ(u)) κ 2 u = f in Ω, for the strain ε := symd and the elasticity tensor CM := 2µM + λ trmi d d. σ(u) ν iκu = g on Γ R, u = on Γ D Variational formulation employs V := {v (H (Ω)) d : v ΓD = } with norm V := κ 2 2 L 2 (Ω) + C/2 ε 2 L 2 (Ω). Sesquilinear form on V a(v,w) := (Cε(v),ε(w)) L 2 (Ω) (κ 2 v,w) L 2 (Ω) (iκv,w) L 2 (Γ R Ω). D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 6

49 Multiscale method and error analysis Analogous to the acoustic case. Provided h, Hκ and m logγ(κ,ω) logκ, then the method is stable and quasi-optimal under the assumption of polynomial well-posedness c(ω) p(κ) inf sup v V\{} w V\{} Ra(v, w). v V w V ( ) D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 7

50 Multiscale method and error analysis Analogous to the acoustic case. Provided h, Hκ and m logγ(κ,ω) logκ, then the method is stable and quasi-optimal under the assumption of polynomial well-posedness c(ω) p(κ) inf sup v V\{} w V\{} Ra(v, w). v V w V ( ) Only few (conditional) results available, see Cummings & Feng (25). D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 7

51 Multiscale method and error analysis Analogous to the acoustic case. Provided h, Hκ and m logγ(κ,ω) logκ, then the method is stable and quasi-optimal under the assumption of polynomial well-posedness c(ω) p(κ) inf sup v V\{} w V\{} Ra(v, w). v V w V ( ) Only few (conditional) results available, see Cummings & Feng (25). Brown&DG 26: Polynomial stability holds if Γ R = Ω. D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 7

52 Wavenumber-explicit stability bounds Theorem (Brown&DG 26) If Γ R = Ω, then c(ω) κ 7/2 inf sup v V\{} w V\{} Ra(v, w). v V w V D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 8

53 Wavenumber-explicit stability bounds Theorem (Brown&DG 26) If Γ R = Ω, then c(ω) κ 7/2 inf sup v V\{} w V\{} Ra(v, w). v V w V Proof: generalizes techniques of Melenk & Sauter (2) and Melenk & Esterhazy (22). Newton-potential estimates κ N κ (f ) H 2 (Ω) + N κ (f ) H (Ω) + κ N κ (f ) L 2 (Ω) C f L 2 (Ω), with N κ := G κ f and the Kupradze matrix G κ. D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 8

54 Introduction Helmholtz problem An ideal method Multiscale Petrov-Galerkin FEM Elastic wave propagation Numerical experiments

55 Numerical experiment for heterogeneous material Data: Ω = (,) 2, f = point sources ( + κ 2 n)u = f in Ω u ν iκu = on Ω Parameters: κ = { 2 7 n = /4 H variable m variable h = 2 9 } D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 9

56 Numerical experiment for heterogeneous material Data: Ω = (,) 2, f = point sources ( + κ 2 n)u = f in Ω u ν iκu = on Ω Parameters: κ = { 2 7 n = /4 H variable m variable h = 2 9 } D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 9

57 Numerical experiment for heterogeneous material Data: Ω = (,) 2, f = point sources ( + κ 2 n)u = f in Ω u ν iκu = on Ω V-norm error - -2 PFEM m = m = 2 m = 3 P-best O(H) -2 - H Parameters: κ = { 2 7 n = /4 H variable m variable h = 2 9 } D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 9

58 Numerical experiment: 2D elasticity (scattering) Data: Ω = (,) 2 \ [.375,.625] 2, f = point source, λ = = µ Parameters: κ = 2 7 H variable m variable h = 2 displacement D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 2

59 Numerical experiment: 2D elasticity (scattering) Data: Ω = (,) 2 \ [.375,.625] 2, f = point source, λ = = µ FEM mspg m = mspg m = 2 mspg m = 3 O(H) Parameters: κ = 2 7 H variable m variable h = V-norm errors H D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 2

60 Numerical experiment: 2D elasticity (scattering) Data: Ω = (,) 2 \ [.375,.625] 2, f = point source, λ = = µ L 2 -norm errors H FEM mspg m = mspg m = 2 mspg m = 3 O(H 2 ) Parameters: κ = 2 7 H variable m variable h = 2 D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 2

61 Numerical experiment: 3D elasticity (scattering) Data: Ω = (,) 3, f =, λ = = µ, polynomial solution, Γ R = Ω.5.5 FEM mspg m = mspg m = 2 O(H) Parameters: κ = 2 5 H variable m variable h = 2 6 V-norm errors.4.2 H D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 2

62 Numerical experiment: 3D elasticity (scattering) Data: Ω = (,) 3, f =, λ = = µ, polynomial solution, Γ R = Ω FEM mspg m = mspg m = 2 O(H 2 ) Parameters: κ = 2 5 H variable m variable h = 2 6 L 2 -norm errors.4.2 H D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 2

63 3D elasticity: pointwise errors Data: Ω = (,) 3, Γ R = Ω f =, λ = = µ, polynomial solution, -3 8 poinwise error: FEM (left) vs. multiscale (right) D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 22

64 Summary Multiscale Petrov-Galerkin FEM for the acoustic and elastic Helmholtz problems with large wave number Under reasonable assumptions on the parameters, the method is pollution-free In homogeneous (or more general periodic) media, the fine scale test functions depend only on local mesh-configurations and the seemingly high cost for the computation of the test functions can be drastically reduced on structured meshes Numerical experiments in two and three space dimensions support these observations. D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 23

65 References D. Brown, D. Gallistl, and D. Peterseim. Multiscale Petrov-Galerkin method for high-frequency heterogeneous Helmholtz equations. Lect. Notes Comput. Sci. Eng., 26. D. Brown and D. Gallistl, Multiscale sub-grid correction method for time-harmonic high-frequency elastodynamics with wavenumber explicit bounds. ArXiv e-prints, 26. D. Gallistl and D. Peterseim. Stable multiscale Petrov-Galerkin finite element method for high frequency acoustic scattering. CMAME, 295:-7, 25. D. Peterseim. Eliminating the pollution effect in Helmholtz problems by local subscale correction. Math. Comp., 26. A. Målqvist and D. Peterseim. Localization of elliptic multiscale problems. Math. Comp., 24. Thank you for your attention. D. Gallistl (KIT) Multiscale FEMs for waves RICAM 26 p. 24

Stable multiscale Petrov-Galerkin finite element method for high frequency acoustic scattering

Wegelerstraße 6 53115 Bonn Germany phone +49 228 73-3427 fax +49 228 73-7527 www.ins.uni-bonn.de D. Gallistl, D. Peterseim Stable multiscale Petrov-Galerkin finite element method for high frequency acoustic