LMS–EPSRC D URHAM S YMPOSIUM , 8–16 TH J ULY 2014 Building Bridges: Connections and Challenges in Modern Approaches to Numerical PDEs Approximation by plane and circular waves Andrea Moiola D EPARTMENT OF M ATHEMATICS AND S TATISTICS , U NIVERSITY OF R EADING R. Hiptmair (ETH Zürich) and I. Perugia (Vienna)
Time-harmonic PDEs, waves and Trefftz methods Consider time-harmonic PDEs, e.g., Helmholtz and Maxwell eq.s − ∆ u − ω 2 u = 0 , ∇ × ( ∇ × E ) − ω 2 E = 0 , ω > 0 . Their solutions are “waves”, oscillates with wavelength λ = 2 π/ω . At high frequencies, ω ≫ 1 , (piecewise) polynomial approximation is very expensive, standard FEMs are not good. Desired: more accuracy for less DOFs. Possible strategy: Trefftz methods are finite element schemes such that test and trial functions are solutions of Helmholtz (or Maxwell. . . ) 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 ) = . E.g.: TDG/PWDG, UWVF , VTCR, DEM, (m)DGM, FLAME, WBM, MFS, LS, PUM/PUFEM, GFEM. . . 1
Time-harmonic PDEs, waves and Trefftz methods Consider time-harmonic PDEs, e.g., Helmholtz and Maxwell eq.s − ∆ u − ω 2 u = 0 , ∇ × ( ∇ × E ) − ω 2 E = 0 , ω > 0 . Their solutions are “waves”, oscillates with wavelength λ = 2 π/ω . At high frequencies, ω ≫ 1 , (piecewise) polynomial approximation is very expensive, standard FEMs are not good. Desired: more accuracy for less DOFs. Possible strategy: Trefftz methods are finite element schemes such that test and trial functions are solutions of Helmholtz (or Maxwell. . . ) 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 ) = . E.g.: TDG/PWDG, UWVF , VTCR, DEM, (m)DGM, FLAME, WBM, MFS, LS, PUM/PUFEM, GFEM. . . 1
Typical Trefftz basis functions for Helmholtz 1 plane waves, (PWs) d ∈ S N − 1 x �→ e i ω x · d e il ψ J l ( ω | x | ) , Y m l ( x 2 circular / spherical waves, | x | ) j l ( ω | x | ) 3 corner waves, 4 fundamental solutions/multipoles, 5 wavebands, 6 evanescent waves, . . . 1 2 3 4 5 6 2
Best approximation estimates The analysis of any plane wave Trefftz method requires best approximation estimates: − ∆ u − ω 2 u = 0 in (bdd., Lip.) D ⊂ R N , u ∈ H k + 1 ( D ) , d 1 , . . . , d p ∈ S N − 1 , diam ( D ) = h , p ∈ N , � � p � � � α ℓ e i ω d ℓ · x � � inf � u − ≤ C ǫ ( h , p ) � u � H k + 1 ( D ) , � � α ∈ C p � � ℓ = 1 H j ( D ) h → 0 with explicit p →∞ 0 . ǫ ( h , p ) − − − → Goal: precise estimates on ǫ ( h , p ) ◮ for plane and circular/spherical waves; ◮ both in h and p (simultaneously); ◮ in 2 and 3 dimensions; ◮ with explicit bounds in the wavenumber ω ; ◮ (suitable for hp -schemes); ◮ for Helmholtz, Maxwell, elasticity, plates,. . . 3
Best approximation estimates The analysis of any plane wave Trefftz method requires best approximation estimates: − ∆ u − ω 2 u = 0 in (bdd., Lip.) D ⊂ R N , u ∈ H k + 1 ( D ) , d 1 , . . . , d p ∈ S N − 1 , diam ( D ) = h , p ∈ N , � � p � � � α ℓ e i ω d ℓ · x � � inf � u − ≤ C ǫ ( h , p ) � u � H k + 1 ( D ) , � � α ∈ C p � � ℓ = 1 H j ( D ) h → 0 with explicit p →∞ 0 . ǫ ( h , p ) − − − → Goal: precise estimates on ǫ ( h , p ) ◮ for plane and circular/spherical waves; ◮ both in h and p (simultaneously); ◮ in 2 and 3 dimensions; ◮ with explicit bounds in the wavenumber ω ; ◮ (suitable for hp -schemes); ◮ for Helmholtz, Maxwell, elasticity, plates,. . . 3
Previous results & outline Only few results available: ◮ [C ESSENAT AND D ESPRÉS 1998], using Taylor polynomials, h -convergence, 2D, L 2 -norm, order is not sharp; ◮ [M ELENK 1995], using Vekua theory, no ω -dependence, p -convergence for plane w., h and p for circular w., 2D. We follow the general strategy of Melenk. Outline: ◮ algebraic best approximation estimates: ◮ Vekua theory; ◮ approximation by circular and spherical waves; ◮ approximation by plane waves; ◮ exponential estimates for hp -schemes; ◮ (extension to Maxwell equations). 4
Previous results & outline Only few results available: ◮ [C ESSENAT AND D ESPRÉS 1998], using Taylor polynomials, h -convergence, 2D, L 2 -norm, order is not sharp; ◮ [M ELENK 1995], using Vekua theory, no ω -dependence, p -convergence for plane w., h and p for circular w., 2D. We follow the general strategy of Melenk. Outline: ◮ algebraic best approximation estimates: ◮ Vekua theory; ◮ approximation by circular and spherical waves; ◮ approximation by plane waves; ◮ exponential estimates for hp -schemes; ◮ (extension to Maxwell equations). 4
Part I Vekua theory
Vekua theory in N dimensions We need an old (1940s) tool from PDE analysis: Vekua theory. D ⊂ R N , open, star-shaped wrt. 0 , ω > 0 . Define two continuous functions: M 1 , M 2 : D × [ 0 , 1 ] → R √ N − 2 √ M 1 ( x , t ) = − ω | x | t � � √ 1 − t J 1 ω | x | 1 − t , 2 0 √ N − 3 M 2 ( x , t ) = − i ω | x | t � � � √ 1 − t J 1 i ω | x | t ( 1 − t ) . 2 J 1 ( t ) , J 1 ( it ) The Vekua operators V 1 , V 2 : C 0 ( D ) → C 0 ( D ) , � 1 V j [ φ ]( x ) := φ ( x ) + M j ( x , t ) φ ( t x ) d t ∀ x ∈ D , j = 1 , 2 . 0 5
4 properties of Vekua operators V 2 = ( V 1 ) − 1 1 ( − ∆ − ω 2 ) V 1 [ φ ] = 0 2 ∆ φ = 0 ⇐ ⇒ Main idea of Vekua theory: V 2 ← − − − − − − Harmonic functions Helmholtz solutions − − − − − − → V 1 Continuity in ( ω -weighted) Sobolev norms, explicit in ω 3 [ H j ( D ) , W j , ∞ ( D ) , j ∈ N ] P = Harmonic V 1 [ P ] = circular/spherical wave ⇐ ⇒ 4 polynomial � � e il ψ J l ( ω r ) , Y m l ( x | x | ) j l ( ω | x | ) � �� � � �� � 2 D 3 D 6
Part II Approximation by circular waves
Vekua operators & approximation by GHPs u ∈ H k + 1 ( D ) , − ∆ u − ω 2 u = 0 , ↓ V 2 can be approximated V 2 [ u ] is harmonic = ⇒ by harmonic polynomials (harmonic Bramble–Hilbert in h , C omplex analysis in p -2D [Melenk], new result in p -3D), ↓ V 1 u can be approximated by GHPs: � harmonic generalized � harmonic = circular/spherical waves. := V 1 polynomials polynomials 7
The approximation by GHPs: h -convergence � � � u − V 1 [ P ] inf j ,ω, D ≤ C inf P � V 2 [ u ] − P � j ,ω, D contin. of V 1 , � harmonic � � � �� � polynomials P ∈ = V 1 [ V 2 [ u ] − P ] of degree ≤ L ≤ C h k + 1 − j ǫ ( L ) � V 2 [ u ] � k + 1 ,ω, D harmonic approx. results , ≤ C h k + 1 − j ǫ ( L ) � u � k + 1 ,ω, D contin. of V 2 . For the h -convergence, Bramble–Hilbert theorem is enough: it provides a harmonic polynomial! The constant C depends on ω h , not on ω alone: 3 C = C · ( 1 + ω h ) j + 6 e 4 ω h . 8
Harmonic approximation: p -convergence Assume D is star-shaped wrt B ρ 0 . � log ( L + 2 ) � λ ( k + 1 − j ) In 2 dimensions, ǫ ( L ) = . sharp p -estimate! [M ELENK ] : L + 2 If D convex, λ = 1 . Otherwise λ = min ( re-entrant corner of D ) /π . In 2D, use complex analysis: R 2 ↔ C , harmonic ↔ holomorphic. —— We can prove an analogous result in N dimensions: ǫ ( L ) = L − λ ( k + 1 − j ) , where λ > 0 is a geometric unknown parameter. If u is the restriction of a solution in a larger domain (2 or 3D), the convergence in L is exponential. 9
Harmonic approximation: p -convergence Assume D is star-shaped wrt B ρ 0 . � log ( L + 2 ) � λ ( k + 1 − j ) In 2 dimensions, ǫ ( L ) = . sharp p -estimate! [M ELENK ] : L + 2 If D convex, λ = 1 . Otherwise λ = min ( re-entrant corner of D ) /π . In 2D, use complex analysis: R 2 ↔ C , harmonic ↔ holomorphic. —— We can prove an analogous result in N dimensions: ǫ ( L ) = L − λ ( k + 1 − j ) , where λ > 0 is a geometric unknown parameter. If u is the restriction of a solution in a larger domain (2 or 3D), the convergence in L is exponential. 9
Part III Approximation by plane waves
The approximation of GHPs by plane waves Link between plane waves and circular/spherical waves: Jacobi–Anger expansion � e iz cos θ = i l J l ( z ) e il θ z ∈ C , θ ∈ R , 2D l ∈ Z l � � i l j l ( r ) Y m e ir ξ · η ξ , η ∈ S 2 , r ≥ 0 . = 4 π l ( ξ ) Y m l ( η ) 3D � �� � � �� � l ≥ 0 m = − l plane wave GHP We need the other way round: GHP ≈ linear combination of plane waves ◮ truncation of J–A expansion, ◮ careful choice of directions (in 3D), → explicit error bound, 2 . ∼ h k q − q ◮ solution of a linear system, ◮ residual estimates, 10
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