O XFORD — C OMPUTATIONAL M ATHEMATICS AND A PPLICATIONS S EMINAR — 15 TH M AY 2014 Plane Wave DG Methods: Exponential Convergence of the hp -version Andrea Moiola D EPARTMENT OF M ATHEMATICS AND S TATISTICS , U NIVERSITY OF R EADING R. Hiptmair, Ch. Schwab (ETH Zürich) and I. Perugia (Vienna)
The Helmholtz equation Simplest model of linear & time-harmonic waves: in bdd. Ω ⊂ R N , N = 2 , 3 , ω > 0 , − ∆ u − ω 2 u = 0 (+ impedance/Robin b.c.) Why is it interesting? Very general, related to any linear wave phenomena: 1 � ∂ 2 U wave equation: → Helmholtz ∂ t 2 − ∆ U = 0 � u ( x ) e − i ω t � equation; time-harmonic regime: U ( x , t ) = ℜ plenty of applications; 2 easy to write. . . but difficult to solve numerically ( ω ≫ 1 ): 3 ◮ oscillating solutions approximation issue, → ◮ numerical dispersion / pollution effect stability issue. → 2
The Helmholtz equation Simplest model of linear & time-harmonic waves: in bdd. Ω ⊂ R N , N = 2 , 3 , ω > 0 , − ∆ u − ω 2 u = 0 (+ impedance/Robin b.c.) Why is it interesting? Very general, related to any linear wave phenomena: 1 � ∂ 2 U wave equation: → Helmholtz ∂ t 2 − ∆ U = 0 � u ( x ) e − i ω t � equation; time-harmonic regime: U ( x , t ) = ℜ plenty of applications; 2 easy to write. . . but difficult to solve numerically ( ω ≫ 1 ): 3 ◮ oscillating solutions approximation issue, → ◮ numerical dispersion / pollution effect stability issue. → 2
The Helmholtz equation Simplest model of linear & time-harmonic waves: in bdd. Ω ⊂ R N , N = 2 , 3 , ω > 0 , − ∆ u − ω 2 u = 0 (+ impedance/Robin b.c.) Why is it interesting? Very general, related to any linear wave phenomena: 1 � ∂ 2 U wave equation: → Helmholtz ∂ t 2 − ∆ U = 0 � u ( x ) e − i ω t � equation; time-harmonic regime: U ( x , t ) = ℜ plenty of applications; 2 easy to write. . . but difficult to solve numerically ( ω ≫ 1 ): 3 ◮ oscillating solutions approximation issue, → ◮ numerical dispersion / pollution effect stability issue. → 2
Difficulty #1: oscillations Time-harmonic solutions are inherently oscillatory: a lot of DOFs needed for any polynomial discretisation! [Helmholtz BVP , picture by T. Betcke] Wavenumber ω = 2 π/λ is the crucial parameter ( λ =wavelength). 3
Difficulty #2: pollution effect Big issue in FEM solution for high wavenumbers: pollution effect � � � � � � � � � Galerkin error � � � ≥ C ω a � � � � a > 0 , ω → ∞ . � � � � � best approximation error � � � It affects every (low order) method in h : [B ABUŠKA , S AUTER 2000] . ⇓ Oscillating solutions pollution effect + = standard FEM are too expensive at high frequencies! Special schemes required, p - and hp -versions preferred. Z IENKIEWICZ , 2000: “Clearly, we can consider that this problem remains unsolved and a completely new method of approximation is needed to deal with the very short-wave solution.” 4
Difficulty #2: pollution effect Big issue in FEM solution for high wavenumbers: pollution effect � � � � � � � � � Galerkin error � � � ≥ C ω a � � � � a > 0 , ω → ∞ . � � � � � best approximation error � � � It affects every (low order) method in h : [B ABUŠKA , S AUTER 2000] . ⇓ Oscillating solutions pollution effect + = standard FEM are too expensive at high frequencies! Special schemes required, p - and hp -versions preferred. Z IENKIEWICZ , 2000: “Clearly, we can consider that this problem remains unsolved and a completely new method of approximation is needed to deal with the very short-wave solution.” 4
Trefftz methods Piecewise polynomials used in FEM are “general purpose” functions, can we use discrete spaces tailored for Helmholtz? Yes: Trefftz methods are finite element schemes such that test and trial functions are solutions of the Helmholtz equation in each element K of the mesh T h , e.g.: � � v ∈ L 2 (Ω) : − ∆ v − ω 2 v = 0 in each K ∈ T h V p ⊂ T ( T h ) = . Main idea: more accuracy for less DOFs. 5
Typical Trefftz basis functions for Helmholtz 1 plane waves (PWs), d ∈ S N − 1 x �→ e i ω x · d 2 circular / spherical waves (CWs), 3 corner waves, 4 fundamental solutions/multipoles, 5 wavebands, 6 evanescent waves, . . . 1 2 3 4 5 6 6
Wave-based methods Trefftz schemes require discontinuous functions. How to “match” traces across interelement boundaries? Plenty of Trefftz schemes for Helmholtz, Maxwell and elasticity: ◮ Least squares: method of fundamental solutions (MFS), wave-based method (WBM); ◮ Lagrange multipliers: discontinuous enrichment (DEM); ◮ Partition of unity method (PUM/PUFEM), non-Trefftz; ◮ Variational theory of complex rays (VTCR); ◮ Discontinuous Galerkin (DG): Ultraweak variational formulation (UWVF). We are interested in a family of Trefftz-discontinuous Galerkin (TDG) methods that includes the UWVF of Cessenat–Després. 7
Outline ◮ TDG method for Helmholtz: formulation and a priori ( p -version) convergence ◮ Approximation theory for plane and spherical waves ◮ Exponential convergence of the hp -TDG 8
Part I TDG method for the Helmholtz equation 9
TDG: derivation — I Consider Helmholtz equation with impedance (Robin) b.c.: 1 in Ω ⊂ R N bdd., Lip., N = 2 , 3 − ∆ u − ω 2 u = 0 ∈ L 2 ( ∂ Ω); ∇ u · n + i ω u = g introduce a mesh T h on Ω ; 2 multiply the Helmholtz equation with a test function v and 3 integrate by parts on a single element K ∈ T h : � � ( ∇ u · ∇ v − ω 2 uv ) d V − ( n · ∇ u ) v d S = 0 ; K ∂ K integrate by parts again: ultraweak step 4 � � ( − u ∆ v − ω 2 uv ) d V + ( − n · ∇ u v + u n · ∇ v ) d S = 0 ; K ∂ K 10
TDG: derivation — I Consider Helmholtz equation with impedance (Robin) b.c.: 1 in Ω ⊂ R N bdd., Lip., N = 2 , 3 − ∆ u − ω 2 u = 0 ∈ L 2 ( ∂ Ω); ∇ u · n + i ω u = g introduce a mesh T h on Ω ; 2 multiply the Helmholtz equation with a test function v and 3 integrate by parts on a single element K ∈ T h : � � ( ∇ u · ∇ v − ω 2 uv ) d V − ( n · ∇ u ) v d S = 0 ; K ∂ K integrate by parts again: ultraweak step 4 � � ( − u ∆ v − ω 2 uv ) d V + ( − n · ∇ u v + u n · ∇ v ) d S = 0 ; K ∂ K 10
TDG: derivation — I Consider Helmholtz equation with impedance (Robin) b.c.: 1 in Ω ⊂ R N bdd., Lip., N = 2 , 3 − ∆ u − ω 2 u = 0 ∈ L 2 ( ∂ Ω); ∇ u · n + i ω u = g introduce a mesh T h on Ω ; 2 multiply the Helmholtz equation with a test function v and 3 integrate by parts on a single element K ∈ T h : � � ( ∇ u · ∇ v − ω 2 uv ) d V − ( n · ∇ u ) v d S = 0 ; K ∂ K integrate by parts again: ultraweak step 4 � � ( − u ∆ v − ω 2 uv ) d V + ( − n · ∇ u v + u n · ∇ v ) d S = 0 ; K ∂ K 10
TDG: derivation — I Consider Helmholtz equation with impedance (Robin) b.c.: 1 in Ω ⊂ R N bdd., Lip., N = 2 , 3 − ∆ u − ω 2 u = 0 ∈ L 2 ( ∂ Ω); ∇ u · n + i ω u = g introduce a mesh T h on Ω ; 2 multiply the Helmholtz equation with a test function v and 3 integrate by parts on a single element K ∈ T h : � � ( ∇ u · ∇ v − ω 2 uv ) d V − ( n · ∇ u ) v d S = 0 ; K ∂ K integrate by parts again: ultraweak step 4 � � ( − u ∆ v − ω 2 uv ) d V + ( − n · ∇ u v + u n · ∇ v ) d S = 0 ; K ∂ K 10
TDG: derivation — II choose a discrete Trefftz space V p ( K ) and replace traces 5 on ∂ K with numerical fluxes � u p and � σ p : (discrete solution) in K , u → u p ∇ u on ∂ K ; u → � u p , i ω → � σ p use the Trefftz property: ∀ v p ∈ V p ( K ) 6 � � � u p ( − ∆ v p − ω 2 v p ) d V + u p ∇ v p · n d S − � i ω � σ p · n v p d S = 0 . � �� � K ∂ K ∂ K � �� � = 0 TDG eq. on 1 element Two things to set: discrete space V p and numerical fluxes � σ p . u p , � 11
TDG: derivation — II choose a discrete Trefftz space V p ( K ) and replace traces 5 on ∂ K with numerical fluxes � u p and � σ p : (discrete solution) in K , u → u p ∇ u on ∂ K ; u → � u p , i ω → � σ p use the Trefftz property: ∀ v p ∈ V p ( K ) 6 � � � u p ( − ∆ v p − ω 2 v p ) d V + u p ∇ v p · n d S − � i ω � σ p · n v p d S = 0 . � �� � K ∂ K ∂ K � �� � = 0 TDG eq. on 1 element Two things to set: discrete space V p and numerical fluxes � σ p . u p , � 11
TDG: the space V p The abstract error analysis works for every discrete Trefftz space! ( { d ℓ } p Possible choice: plane wave space ℓ = 1 ⊂ S N − 1 ) � p � � v ∈ L 2 (Ω) : v | K ( x ) = α ℓ e i ω x · d ℓ , α ℓ ∈ C , ∀ K ∈ T h V p ( T h ) = . ℓ = 1 p := number of basis plane waves (DOFs) in each element. 12
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