Multiscale methods for time-harmonic acoustic and elastic wave - PowerPoint PPT Presentation

Multiscale methods for time-harmonic acoustic and elastic wave propagation Dietmar Gallistl (joint work with D. Brown and D. Peterseim) Institut f¨ ur Angewandte und Numerische Mathematik Karlsruher Institut f¨ ur Technologie (KIT) Workshop on Analysis and Numerics of Acoustic and Electromagnetic Problems RICAM, 17 –22. October 2016 D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 1

Outline Introduction Helmholtz problem An ideal method Multiscale Petrov-Galerkin FEM Elastic wave propagation Numerical experiments D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 2

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

Model problem: high-frequency acoustic scattering ◮ Ω ⊂ R d bounded polytope, diam Ω ≈ 1 ◮ ∂ Ω = Γ D ∪ Γ R ◮ wave number κ > 0 real parameter ◮ incident wave u in ◮ we seek u = u in + u scat ◮ positive and bounded material parameters A ( x ) , n ( x ) , β ( x ) D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 3

Model problem: high-frequency acoustic scattering ◮ Ω ⊂ R d bounded polytope, diam Ω ≈ 1 ◮ ∂ Ω = Γ D ∪ Γ R ◮ wave number κ > 0 real parameter ◮ incident wave u in ◮ we seek u = u in + u scat ◮ positive and bounded material parameters A ( x ) , n ( x ) , β ( x ) − div ( A ∇ u ) − κ 2 nu = in Ω f u = 0 on Γ D ⊆ ∂ Ω ( A ∇ u ) · ν − i βκ u = on Γ R : = ∂ Ω \ Γ D g D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 3

Pollution effect 1 d Helmholtz: − u xx − κ 2 u = 0 in [ − 1 , 1 ] , u in ( x ) = exp ( − i κ x ) Results for P 1 -FEM (fixed resolution H : = const / κ ) 10 2 P1FEM P1-best ratio 10 1 V-error 10 0 10 -1 10 0 10 1 10 2 10 3 κ D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 4

Pollution effect 1 d Helmholtz: − u xx − κ 2 u = 0 in [ − 1 , 1 ] , u in ( x ) = exp ( − i κ x ) Results for P 1 -FEM (fixed resolution H : = const / κ ) 10 2 P1FEM P1-best ratio 10 1 V-error 10 0 10 -1 10 0 10 1 10 2 10 3 κ D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 4

Pollution effect 1 d Helmholtz: − u xx − κ 2 u = 0 in [ − 1 , 1 ] , u in ( x ) = exp ( − i κ x ) Results for P 1 -FEM (fixed resolution H : = const / κ ) 10 2 P1FEM P1-best ratio 10 1 V-error 10 0 10 -1 10 0 10 1 10 2 10 3 κ D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 4

Pollution effect 1 d Helmholtz: − u xx − κ 2 u = 0 in [ − 1 , 1 ] , u in ( x ) = exp ( − i κ x ) Results for P 1 -FEM (fixed resolution H : = const / κ ) 10 2 P1FEM P1-best ratio 10 1 error(FEM) error(bestapprox) � κ s V-error 10 0 [Babuˇ ska-Sauter 2000] 10 -1 10 0 10 1 10 2 10 3 κ D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 4

Pollution effect 1 d Helmholtz: − u xx − κ 2 u = 0 in [ − 1 , 1 ] , u in ( x ) = exp ( − i κ x ) Results for P 1 -FEM (fixed resolution H : = const / κ ) 10 2 P1FEM P1-best ratio msPG Goal: 10 1 V-error error(msPG) error(bestapprox) ≤ C 10 0 10 -1 10 0 10 1 10 2 10 3 κ D. Gallistl and D. Peterseim. Stable multiscale Petrov-Galerkin finite element method for high frequency acoustic scattering. Comput. Methods Appl. Mech. Eng. , 2015. D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 4

Other Approaches ◮ hp -FEM with p ≈ log κ Melenk, Sauter ◮ Trefftz methods, Plane wave methods Hiptmair, Moiola, Perugia ◮ DG FEM Farhat, Tezaur; Feng, Wu ◮ Ultra-weak formulation Cessenat, Despr´ es; Buffa, Monk ◮ Discontinuous Petrov-Galerkin Demkowicz, Gopalakrishnan D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 5

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

Variational formulation ◮ Hilbert space V : = H 1 D ( Ω ; C ) : = { v ∈ H 1 ( Ω ; C ) : v | Γ D = 0 } ◮ Continuous sesquilinear form on V × V a ( v , w ) : = ( A ∇ v , ∇ w ) L 2 ( Ω ) − κ 2 ( nv , w ) L 2 ( Ω ) − i κ ( β v , w ) L 2 ( Γ R ) D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 6

Variational formulation ◮ Hilbert space V : = H 1 D ( Ω ; C ) : = { v ∈ H 1 ( Ω ; C ) : v | Γ D = 0 } ◮ Continuous sesquilinear form on V × V a ( v , w ) : = ( A ∇ v , ∇ w ) L 2 ( Ω ) − κ 2 ( nv , w ) L 2 ( Ω ) − i κ ( β v , 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 2016 p. 6

Variational formulation ◮ Hilbert space V : = H 1 D ( Ω ; C ) : = { v ∈ H 1 ( Ω ; C ) : v | Γ D = 0 } ◮ Continuous sesquilinear form on V × V a ( v , w ) : = ( A ∇ v , ∇ w ) L 2 ( Ω ) − κ 2 ( nv , w ) L 2 ( Ω ) − i κ ( β v , 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 2016 p. 6

Well-posedness Assumption (Polynomial well-posedness) There exist a constant c ( Ω ) and a polynomial p such that c ( Ω ) ℜ a ( v , w ) p ( κ ) ≤ ( ⋆ ) inf sup � v � V � w � V v ∈ V \{ 0 } w ∈ V \{ 0 } D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 7

Well-posedness Assumption (Polynomial well-posedness) There exist a constant c ( Ω ) and a polynomial p such that c ( Ω ) ℜ a ( v , w ) p ( κ ) ≤ ( ⋆ ) inf sup � v � V � w � V v ∈ V \{ 0 } w ∈ V \{ 0 } Homogeneous material: ◮ assumption not satisfied in general [Betcke et al 2011] ◮ pure impedence problem + Ω convex ⇒ γ ( κ , Ω ) � κ [Melenk 1995] ◮ pure impedence problem + Ω Lipschitz ⇒ γ ( κ , Ω ) � κ 7 / 2 [Esterhazy-Melenk 2012] ◮ star-shaped scatterer ⇒ γ ( κ , Ω ) � κ [Hetmaniuk 2007] D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 7

Well-posedness Assumption (Polynomial well-posedness) There exist a constant c ( Ω ) and a polynomial p such that c ( Ω ) ℜ a ( v , w ) p ( κ ) ≤ ( ⋆ ) inf sup � v � V � w � V v ∈ V \{ 0 } w ∈ V \{ 0 } 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 , 2015. D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 7

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

Finite element spaces Coarse scale H � 1 / κ ◮ G H coarse mesh ◮ V H : = Q 1 ( G H ) ∩ V standard Q 1 FE space Fine scale h � 1 / κ 2 ◮ G h refinement of G H ◮ V h : = Q 1 ( G h ) ∩ V standard Q 1 FE space D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 8

Finite element spaces Coarse scale H � 1 / κ ◮ G H coarse mesh ◮ V H : = Q 1 ( G H ) ∩ V standard Q 1 FE space Fine scale h � 1 / κ 2 ◮ G h refinement of G H ◮ V h : = Q 1 ( G h ) ∩ V standard Q 1 FE space Quasi-interpolation ◮ I H : V h → V H quasi-local projection ◮ Stability and L 2 -approximation H − 1 � v − I H v � L 2 ( T ) + � ∇ I H v � L 2 ( T ) ≤ C I H � ∇ v � L 2 ( N ( T )) ◮ Example: I H : = E H ◦ Π H Π H ... piecewise L 2 projection onto Q 1 ( G H ) E H ... nodal averaging operator D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 8

Subscale correction Fine scale remainder (null space of I H ): W h : = { v h ∈ V h : I H ( v h ) = 0 } 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 2016 p. 9

Subscale correction Fine scale remainder (null space of I H ): W h : = { v h ∈ V h : I H ( v h ) = 0 } 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 1 1 0.8 0.8 0.6 0.6 0.4 0.4 �→ C ∞ 0.2 0.2 0 0 0 0.5 1 0 0.2 0.4 0.6 0.8 1 D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 9

Subscale correction Fine scale remainder (null space of I H ): W h : = { v h ∈ V h : I H ( v h ) = 0 } 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 H κ � 1 , the corrector problems are well-posed (even coercive). D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 9

Ideal Petrov-Galerkin method ◮ Standard trial space V H ◮ Corrected test space � V H : = ( 1 − C ∞ ) V H D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 10

Ideal Petrov-Galerkin method ◮ Standard trial space V H ◮ Corrected test space � V H : = ( 1 − C ∞ ) V H 1 1 1 Re Re Re Im Im Im 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ˜ Λ z = ( 1 − C ∞ ) Λ z Λ z C ∞ Λ z D. Gallistl (KIT) Multiscale FEMs for waves RICAM 2016 p. 10