B RITISH C OMPUTATIONAL PDE S C OLLOQUIUM : N EW T RENDS ICMS E DINBURGH , 23–24 J ANUARY 2014 Is the Helmholtz equation really sign-indefinite? Andrea Moiola D EPARTMENT OF M ATHEMATICS AND S TATISTICS , U NIVERSITY OF R EADING Joint work with Euan A. Spence (Bath)
What people say about sign-indefiniteness Is the Helmholtz equation really sign-indefinite? “...the Helmholtz operator for scattering problems is a highly indefinite complex-valued linear operator.” (2013) “The main difficulty of the analysis is caused by the strong indefiniteness of the Helmholtz equation.” (2009) “Problems in high-frequency scattering of acoustic or electromagnetic waves are highly indefinite.” (2013) Sure??? 2
What people say about sign-indefiniteness Is the Helmholtz equation really sign-indefinite? “...the Helmholtz operator for scattering problems is a highly indefinite complex-valued linear operator.” (2013) “The main difficulty of the analysis is caused by the strong indefiniteness of the Helmholtz equation.” (2009) “Problems in high-frequency scattering of acoustic or electromagnetic waves are highly indefinite.” (2013) Sure??? 2
The Helmholtz equation The main character: the Helmholtz equation ∆ u + k 2 u = − f in Ω ⊂ R d , d = 2 , 3 , k > 0 . Why is it interesting? very general: ( k = ω/ c ) 1 ∂ 2 U ∂ t 2 − c 2 ∆ U = c 2 F → Helmholtz wave equation equation; U ( x , t ) = ℜ{ u ( x ) e − i ω t } time-harmonic regime plenty of applications; 2 easy to write, difficult to solve numerically (for k ≫ 1 ): 3 ◮ oscillating solutions expensive to approximate; → ◮ numerical dispersion / pollution effect; ◮ sign-indefinite? 3
The Helmholtz equation The main character: the Helmholtz equation ∆ u + k 2 u = − f in Ω ⊂ R d , d = 2 , 3 , k > 0 . Why is it interesting? very general: ( k = ω/ c ) 1 ∂ 2 U ∂ t 2 − c 2 ∆ U = c 2 F → Helmholtz wave equation equation; U ( x , t ) = ℜ{ u ( x ) e − i ω t } time-harmonic regime plenty of applications; 2 easy to write, difficult to solve numerically (for k ≫ 1 ): 3 ◮ oscillating solutions expensive to approximate; → ◮ numerical dispersion / pollution effect; ◮ sign-indefinite? 3
The Helmholtz equation The main character: the Helmholtz equation ∆ u + k 2 u = − f in Ω ⊂ R d , d = 2 , 3 , k > 0 . Why is it interesting? very general: ( k = ω/ c ) 1 ∂ 2 U ∂ t 2 − c 2 ∆ U = c 2 F → Helmholtz wave equation equation; U ( x , t ) = ℜ{ u ( x ) e − i ω t } time-harmonic regime plenty of applications; 2 easy to write, difficult to solve numerically (for k ≫ 1 ): 3 ◮ oscillating solutions expensive to approximate; → ◮ numerical dispersion / pollution effect; ◮ sign-indefinite? 3
Variational formulations BVPs for (linear elliptic) PDEs are often posed in variational form: (VF) find u ∈ V such that a ( u , w ) = F ( w ) ∀ w ∈ V , V Hilbert space, a ( · , · ) : V × V → R bilinear form, F : V → R continuous linear functional. They can be approximated using a Galerkin discretisation: (GD) find u N ∈ V N s.t. ∀ w N ∈ V N , a ( u N , w N ) = F ( w N ) V N ⊂ V finite dimensional space, dim ( V N ) = N . 4
Variational formulations BVPs for (linear elliptic) PDEs are often posed in variational form: (VF) find u ∈ V such that a ( u , w ) = F ( w ) ∀ w ∈ V , V Hilbert space, a ( · , · ) : V × V → R bilinear form, F : V → R continuous linear functional. They can be approximated using a Galerkin discretisation: (GD) find u N ∈ V N s.t. ∀ w N ∈ V N , a ( u N , w N ) = F ( w N ) V N ⊂ V finite dimensional space, dim ( V N ) = N . 4
Continuity & coercivity Most desirable properties for (VF), ∃ C c , α > 0 : | a ( u , w ) | ≤ C c � u � V � w � V ∀ u , w ∈ V , continuity , | a ( w , w ) | ≥ α � w � 2 coercivity . ∀ w ∈ V , V (“Sign-definite” := coercive; “sign-indefinite” := not coercive.) Consequences of continuity & coercivity (Lax–Milgram, Céa): ◮ well-posedness of (VF): ∃ ! u ∈ V , � u � V ≤ � F � V ′ /α ; ◮ well-posedness of any (GD): ∃ ! u N ∈ V N , � u N � V ≤ � F � V ′ /α ; ◮ quasi-optimality of any (GD): � u − u N � V ≤ C c w N ∈V N � u − w N � V ; inf α ◮ good properties for (GD) linear system. Coercivity is a property of the bilinear form—no PDEs here. 5
Continuity & coercivity Most desirable properties for (VF), ∃ C c , α > 0 : | a ( u , w ) | ≤ C c � u � V � w � V ∀ u , w ∈ V , continuity , | a ( w , w ) | ≥ α � w � 2 coercivity . ∀ w ∈ V , V (“Sign-definite” := coercive; “sign-indefinite” := not coercive.) Consequences of continuity & coercivity (Lax–Milgram, Céa): ◮ well-posedness of (VF): ∃ ! u ∈ V , � u � V ≤ � F � V ′ /α ; ◮ well-posedness of any (GD): ∃ ! u N ∈ V N , � u N � V ≤ � F � V ′ /α ; ◮ quasi-optimality of any (GD): � u − u N � V ≤ C c w N ∈V N � u − w N � V ; inf α ◮ good properties for (GD) linear system. Coercivity is a property of the bilinear form—no PDEs here. 5
Back to PDEs Typical example: Standard (VF) of Dirichlet problem for the Laplace equation ( ∆ u = − f ) is continuous + coercive (+ symmetric): that’s why Laplace’s is an easy PDE! More interesting example: � ∆ u + k 2 u = − f in Ω , Impedance Helmholtz BVP ∂ u ∂ n − iku = g on ∂ Ω . � � ( −∇ u · ∇ w + k 2 uw ) d x + ik a ( u , w ) := uw d s , Ω ∂ Ω � � HVF F ( w ) := − f w d x − g w d s , Ω ∂ Ω L 2 (Ω) + k 2 � w � 2 V := H 1 (Ω) , � w � 2 1 , k , Ω := �∇ w � 2 L 2 (Ω) . (Note: now everything is complex-valued.) 6
Back to PDEs Typical example: Standard (VF) of Dirichlet problem for the Laplace equation ( ∆ u = − f ) is continuous + coercive (+ symmetric): that’s why Laplace’s is an easy PDE! More interesting example: � ∆ u + k 2 u = − f in Ω , Impedance Helmholtz BVP ∂ u ∂ n − iku = g on ∂ Ω . � � ( −∇ u · ∇ w + k 2 uw ) d x + ik a ( u , w ) := uw d s , Ω ∂ Ω � � HVF F ( w ) := − f w d x − g w d s , Ω ∂ Ω L 2 (Ω) + k 2 � w � 2 V := H 1 (Ω) , � w � 2 1 , k , Ω := �∇ w � 2 L 2 (Ω) . (Note: now everything is complex-valued.) 6
Is Helmholtz sign-indefinite? For k 2 ≥ λ 1 > 0 (1st Laplace–Dirichlet eigenvalue), a ( · , · ) is continuous but not coercive in H 1 (Ω) . Other techniques are applicable based on Fredholm alternative (Gårding inequality, Schatz’s argument. . . ) ⇒ well-posedness of (HVF), ⇒ well-posedness of (HGD) and quasi-optimality for “ N large enough” only. Does this imply that the Helmholtz equation is sign-indefinite? NO! The standard variational formulation (HVF) of the BVP is sign-indefinite, but not the equation itself. New question: is there any continuous & coercive variational formulation equivalent to the Helmholtz impedance BVP? 7
Is Helmholtz sign-indefinite? For k 2 ≥ λ 1 > 0 (1st Laplace–Dirichlet eigenvalue), a ( · , · ) is continuous but not coercive in H 1 (Ω) . Other techniques are applicable based on Fredholm alternative (Gårding inequality, Schatz’s argument. . . ) ⇒ well-posedness of (HVF), ⇒ well-posedness of (HGD) and quasi-optimality for “ N large enough” only. Does this imply that the Helmholtz equation is sign-indefinite? NO! The standard variational formulation (HVF) of the BVP is sign-indefinite, but not the equation itself. New question: is there any continuous & coercive variational formulation equivalent to the Helmholtz impedance BVP? 7
How to find a coercive Helmholtz formulation? ◮ Modus operandi: in general it holds � u � V ≤ α − 1 � F � V ′ ; coercivity explicit stability constant ⇒ Fredholm unknown stability constant ⇒ � u � V ≤ C � F � V ′ . ◮ A clue: M ELENK , C UMMINGS &F ENG , H ETMANIUK proved (almost) explicit stability bounds for (HVF). ? ⇒ ◮ A suspicion: maybe there’s a “hidden coercivity” behind. . . ◮ How to find an evidence? reverse engineer Melenk’s proof to define a variational formulation by applying the main tools used there: Rellich identities and multipliers. ◮ 1 st surprise: it works! ◮ 2 nd surprise: it is derived exactly as the standard (HVF). 8
How to find a coercive Helmholtz formulation? ◮ Modus operandi: in general it holds � u � V ≤ α − 1 � F � V ′ ; coercivity explicit stability constant ⇒ Fredholm unknown stability constant ⇒ � u � V ≤ C � F � V ′ . ◮ A clue: M ELENK , C UMMINGS &F ENG , H ETMANIUK proved (almost) explicit stability bounds for (HVF). ? ⇒ ◮ A suspicion: maybe there’s a “hidden coercivity” behind. . . ◮ How to find an evidence? reverse engineer Melenk’s proof to define a variational formulation by applying the main tools used there: Rellich identities and multipliers. ◮ 1 st surprise: it works! ◮ 2 nd surprise: it is derived exactly as the standard (HVF). 8
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